Entropy of a quantum channel

Gilad Gour1, ∗ and Mark M. Wilde2, †

1Department of Mathematics and Statistics, Institute for Quantum Science and Technology,University of Calgary, Alberta, Canada T2N 1N4

2Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA

The von Neumann entropy of a quantum state is a central concept in physics and informationtheory, having a number of compelling physical interpretations. There is a certain perspective thatthe most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states,unitary evolutions, measurements, and discarding of quantum systems can each be regarded ascertain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningfulnotion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state ρcan be formulated as the difference of the number of physical qubits and the “relative entropydistance” between ρ and the maximally mixed state, here we define the entropy of a channel Nas the difference of the number of physical qubits of the channel output with the “relative entropydistance” between N and the completely depolarizing channel. We prove that this definition satisfiesall of the axioms, recently put forward in [Gour, IEEE Trans. Inf. Theory 65, 5880 (2019)], requiredfor a channel entropy function. The task of quantum channel merging, in which the goal is forthe receiver to merge his share of the channel with the environment’s share, gives a compellingoperational interpretation of the entropy of a channel. We define Renyi and min-entropies of achannel and prove that they satisfy the axioms required for a channel entropy function. Amongother results, we also prove that a smoothed version of the min-entropy of a channel satisfies theasymptotic equipartition property.

CONTENTS

I. Introduction 2

II. Entropy of a quantum channel 3A. Properties of the entropy of a quantum

channel 31. Non-decrease under the action of a

uniformity preserving superchannel 32. Additivity 43. Reduction to states and normalization 5

B. Alternate representations for the entropy ofa channel 5

III. Renyi entropy of a quantum channel 6A. Properties of the Renyi entropy of a

quantum channel 7B. Alternate representations for the Renyi

entropy of a quantum channel 7

IV. Min-entropy of a quantum channel 8A. Alternate representation for the

min-entropy of a channel in terms ofconditional min-entropies 9

B. Relation of min-entropy of a channel to itsextended min-entropy 10

V. Asymptotic Equipartition Property 10

∗ gour@ucalgary.ca† mwilde@lsu.edu

VI. Quantum channel merging 12A. Converse bound 13B. Achievability bound 14C. Quantum channel merging capacity is equal

to the entropy of a channel 16

VII. Examples 16A. Finite-dimensional channels 17B. Energy-constrained entropy of a channel 18C. Bosonic Gaussian channels 18

VIII. Generalized channel entropies from generalizeddivergences 19A. Collapse of entropy functions derived from

quantum relative entropy 21B. Collapse of entropy functions derived from

max-relative entropy 21C. Entropy functions derived from Renyi

relative entropies 21D. Entropy functions derived from Choi and

adversarial Choi divergences 22

IX. Conclusion and outlook 23

Acknowledgments 24

References 24

A. Max-mutual information of a channel and theasymptotic equipartition property 27

B. Data processing of the Choi divergence underparticular superchannels 28

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C. Optimizing the adversarial channel divergence 29

I. INTRODUCTION

In his foundational work on quantum statistical me-chanics, von Neumann extended the classical Gibbs en-tropy concept to the quantum realm [vN32]. This exten-sion, known as the von Neumann or quantum entropy,plays a key role in physics and information theory. It isdefined by the following formula [vN32]:

H(A)ρ ≡ −TrρA log2 ρA, (1)

where ρA is the state of a system A. The entropy has op-erational interpretations in terms of quantum data com-pression [Sch95] and optimal entanglement manipulationrates of pure bipartite quantum states [BBPS96], wherethe choice of base two for the logarithm becomes clear. Inrecent developments of quantum thermodynamics, it wasshown that the free energy, namely, the difference of theenergy and the product of the temperature and the vonNeumann entropy, can be interpreted as the rate at whichwork can be extracted from a large number of copies of aquantum system in a thermal bath at fixed temperature,by using only thermal operations [BaHO+13].

By defining the quantum relative entropy of a state ρAand a positive semi-definite operator σA as [Ume62]

D(ρA‖σA) ≡ TrρA [log2 ρA − log2 σA], (2)

if supp(ρA) ⊆ supp(σA) and D(ρA‖σA) = +∞ other-wise, we can rewrite the formula for quantum entropy asfollows:

H(A)ρ = log2 |A| −D(ρA‖πA), (3)

where |A| denotes the dimension of the system A andπA ≡ IA/ |A| denotes the maximally mixed state. Inthis way, we can think of entropy as quantifying the dif-ference of the number of physical qubits contained in thesystem A and the “relative entropy distance” of the stateρA to the maximally mixed state πA. This way of think-ing about quantum entropy is relevant in the resourcetheory of purity [OHHH02, HHH+03, HHO03, OHH+03,Dev05, KD07], in which the goal is to distill local purestates from a given state (or vice versa) by allowing localunitary operations for free. Furthermore, the quantumrelative entropy D(ρA‖πA) has an operational meaningas the optimal rate at which the state ρA can be distin-guished from the maximally mixed state πA in the Steinsetting of quantum hypothesis testing [HP91, ON00]. Inwhat follows, we use the formula in (3) as the basis fordefining the entropy of a quantum channel.

For some time now, there has been a growing realiza-tion that the fundamental constituents of quantum me-chanics are quantum channels. Recall that a quantumchannel NA→B is a completely positive, trace preservingmap that takes a quantum state for system A to one for

system B [Hol12]. Indeed, all the relevant componentsof the theory, including quantum states, measurements,unitary evolutions, etc., can be written as quantum chan-nels. A quantum state can be understood as a prepa-ration channel, sending a trivial quantum system to anon-trivial one prepared in a given state. A quantummeasurement can be understood as a quantum channelthat sends a quantum system to a classical one; and ofcourse a unitary evolution is a kind of quantum chan-nel, as well as the discarding of a quantum system. Onemight even boldly go as far as to say that there is reallyonly a single postulate of quantum mechanics, and it isthat “everything is a quantum channel.” With this per-spective, one could start from this unified postulate andthen understand from there particular kinds of channels,i.e., states, measurements, and unitary evolutions.

Due to the fundamental roles of quantum channels andthe entropy of a quantum state, as highlighted above, it isthus natural to ask whether there is a meaningful notionof the entropy of a quantum channel, i.e., a quantifier ofthe uncertainty of a quantum channel. As far as we areaware, this question has not been fully addressed in priorliterature, and it is the aim of the present paper to pro-vide a convincing notion of a quantum channel’s entropy.To define such a notion, we look to (3) for inspiration.As such, we need generalizations of the quantum relativeentropy and the maximally mixed state to the setting ofquantum channels:

1. The quantum relative entropy of channels NA→Band MA→B is defined as [CMW16, LKDW18]

D(N‖M) ≡ supρRA

D(NA→B(ρRA)‖MA→B(ρRA)), (4)

where the optimization is with respect to bipar-tite states ρRA of a reference system R of arbitrarysize and the channel input system A. Due to statepurification, the data-processing inequality [Lin75],and the Schmidt decomposition theorem, it sufficesto optimize over states ρRA that are pure and suchthat system R is isomorphic to system A. Thisobservation significantly reduces the complexity ofcomputing the channel relative entropy.

2. The channel that serves as a generalization of themaximally mixed state is the channel RA→B thatcompletely randomizes or depolarizes the inputstate as follows:

RA→B(XA) = TrXAπB , (5)

where XA is an arbitrary operator for system A.That is, its action is to discard the input and re-place with a maximally mixed state πB .

With these notions in place, we can now define theentropy of a quantum channel:

3

Definition 1 (Entropy of a quantum channel) LetNA→B be a quantum channel. Its entropy is defined as

H(N ) ≡ log2 |B| −D(N‖R), (6)

where D(N‖R) is the channel relative entropy in (4) andRA→B is the completely randomizing channel in (5).

We remark here that, in analogy to the operational in-terpretation for D(ρA‖πA) mentioned above, it is knownthat D(N‖R) is equal to the optimal rate at which thechannel NA→B can be distinguished from the completelyrandomizing channel RA→B , by allowing for any possiblequantum strategy to distinguish the channels [CMW16].Again, this statement holds in the Stein setting of quan-tum hypothesis testing (see [CMW16] for details).

The remainder of our paper contains arguments ad-vocating for this definition of a channel’s entropy. Inthe next section, we show that it satisfies the three ba-sic axioms, put forward in [Gou19], for any functionto be called an entropy function for a quantum chan-nel, including non-decrease under the action of a ran-dom unitary superchannel, additivity, and normaliza-tion. After that, we provide several alternate representa-tions for the entropy of a channel, the most significant ofwhich is the completely bounded entropy of [DKJR06].From there, we define the α-Renyi entropy of a chan-nel, prove that it satisfies the basic axioms for certainvalues of the Renyi parameter α, and provide alternaterepresentations for it. In Section IV, we define the min-entropy of a channel, establish that it satisfies the ba-sic axioms, and provide alternate representations for it.In Section V, we define the smoothed min-entropy ofa channel, and then we prove an asymptotic equipar-tition property, which relates the smoothed min-entropyof a channel to its entropy. Section VI delivers an op-erational interpretation of a channel’s entropy in termsof an information-theoretic task that we call quantumchannel merging, which generalizes the well known taskof quantum state merging [HOW05, HOW07]. We cal-culate channel entropies for several example channels inSection VII, which include erasure, dephasing, depolar-izing, and Werner–Holevo channels. In the same section,we introduce the energy-constrained and unconstrainedentropies of a quantum channel and calculate them forthermal, amplifier, and additive-noise bosonic Gaussianchannels. In Section VIII, we discuss other entropies of achannel, noting that several of them collapse to the (vonNeumann) entropy of a channel. We finally conclude inSection IX with a summary and some open questions.

Note on related work—After completing the results inour related preprint [GW18], we noticed [Yua18, Eq. (6)],in which Yuan proposed to define the entropy of a quan-tum channel in the same way as we have proposed inDefinition 1. Yuan’s work is now published as [Yua19].

II. ENTROPY OF A QUANTUM CHANNEL

Proceeding with Definition 1 for the entropy of a quan-tum channel, we now establish several of its properties,and then we provide alternate representations for it.

A. Properties of the entropy of a quantum channel

In [Gou19], it was advocated that a function of a quan-tum channel is an entropy function if it satisfies non-decrease under random unitary superchannels, additivity,and normalization. As shown in the next three subsec-tions, the entropy of a channel, as given in Definition 1,satisfies all three axioms, and in fact, it satisfies strongerproperties that imply these.

1. Non-decrease under the action of a uniformitypreserving superchannel

Before addressing the first axiom, let us first brieflyreview the notion of superchannels [CDP08b], which arelinear maps that take as input a quantum channel andoutput a quantum channel. To define them, let L(A →B) denote the set of all linear maps from L(A) to L(B).Similarly, let L(C → D) denote the set of all linear mapsfrom L(C) to L(D). Let Θ : L(A→ B)→ L(C → D) de-note a linear supermap, taking L(A→ B) to L(C → D).A quantum channel is a particular kind of linear map,and any linear supermap Θ that takes as input an ar-bitrary quantum channel ΨA→B ∈ L(A → B) and is re-quired to output a quantum channel ΦC→D ∈ L(C → D)should preserve the properties of complete positivity(CP) and trace preservation (TP). That is, the supermapshould be CPTP preserving. Furthermore, for the su-permap to be physical, the same should be true whenit acts on subsystems of bipartite quantum channels,so that the supermap id⊗Θ should be CPTP preserv-ing, where id represents an arbitrary identity supermap.A supermap satisfying this property is said to be com-pletely CPTP preserving and is then called a superchan-nel. It was proven in [CDP08b] that any superchannelΘ : L(A → B) → L(C → D) can be physically realizedas follows. If

ΦC→D = Θ[ΨA→B ] (7)

for an arbitrary input channel ΨA→B ∈ L(A → B) andsome output channel ΦC→D ∈ L(C → D), then the phys-ical realization of the superchannel Θ is as follows:

ΦC→D = ΩBE→D (ΨA→B ⊗ idE) ΛC→AE , (8)

where ΛC→AE : L(C)→ L(AE) is a pre-processing chan-nel, system E corresponds to some memory or environ-ment system, and ΩBE→D : L(BE) → L(D) is a post-processing channel.

4

A uniformity preserving superchannel Θ is a super-channel that takes the completely randomizing channelRA→B in (5) to another completely randomizing channelRC→D, such that |A| = |C| and |B| = |D|, i.e.,

Θ(RA→B) = RC→D. (9)

For such superchannels, we have the following:

Proposition 2 Let NA→B be a quantum channel, andlet Θ be a uniformity preserving superchannel as definedabove. Then the entropy of a channel does not decreaseunder the action of such a superchannel:

H(Θ(N )) ≥ H(N ). (10)

Proof. This follows from the fact that the channel rel-ative entropy is non-increasing under the action of anarbitrary superchannel [Gou19, Yua18]. That is, for twochannels NA→B and MA→B , and a superchannel Ξ, thefollowing inequality holds

D(N‖M) ≥ D(Ξ(N )‖Ξ(M)). (11)

Applying this, we find that

H(N ) = log2 |B| −D(N‖R) (12)

≤ log2 |B| −D(Θ(N )‖Θ(R)) (13)

= log2 |B| −D(Θ(N )‖R) (14)

= log2 |D| −D(Θ(N )‖R) (15)

= H(Θ(N )). (16)

The second equality follows by definition from (9).

In [Gou19], a superchannel Υ was called a random uni-tary superchannel if its action on a channel NA→B canbe written as

Υ(NA→B) =∑x

pX(x)VxB→D NA→B UxC→A, (17)

where UxC→A and VxB→D are unitary channels and pX(x)is a probability distribution. In [Gou19], it was provedthat a random unitary superchannel is a special kind ofuniformity preserving superchannel. Thus, due to Propo-sition 2, it follows that the entropy of a channel, as givenin Definition 1, satisfies the first axiom from [Gou19] re-quired for an entropy function.

2. Additivity

In this subsection, we prove that the entropy of achannel is additive, which is the second axiom pro-posed in [Gou19] for a channel entropy function. Theproof is related to many prior additivity results from[AC97, Ali04, DKJR06, CMW16, BHKW18].Proposition 3 (Additivity) Let N and M be quan-tum channels. Then the channel entropy is additive inthe following sense:

H(N ⊗M) = H(N ) +H(M). (18)

Proof. This can be understood as a consequence ofthe additivity results from [CMW16, BHKW18], whichin turn are related to the earlier additivity resultsfrom [AC97, Ali04, DKJR06]. For channels NA1→B1

and MA2→B2, and corresponding randomizing channels

R(1)A1→B1

and R(2)A2→B2

, we have by definition that

H(N ⊗M)

= log2(|B1| |B2|)−D(N ⊗M‖R(1) ⊗R(2)) (19)

= log2 |B1|+ log2 |B2| −D(N ⊗M‖R(1) ⊗R(2)),(20)

and so the result follows if

D(N ⊗M‖R(1) ⊗R(2)) = D(N‖R(1)) +D(M‖R(2)).(21)

Note that the inequality “≥” for (21) trivially follows,and so it remains to prove the inequality “≤” for (21).To this end, let ψRA1A2

be an arbitrary pure state, anddefine

ρR′A1≡MA2→B2

(ψRA1A2), (22)

σR′A1≡ RA2→B2

(ψRA1A2), (23)

where system R′ ≡ RB2. Then we find that

D((NA1→B1⊗MA2→B2

)(ψRA1A2)‖(RA1→B1

⊗RA2→B2)(ψRA1A2

))

= D(NA1→B1(ρR′A1

)‖RA1→B1(σR′A1

)) (24)

≤ D(NA1→B1(ρR′A1

)‖RA1→B1(ρR′A1

)) +D(ρR′A1‖σR′A1

) (25)

= D(NA1→B1(ρR′A1

)‖RA1→B1(ρR′A1

)) +D(MA2→B2(ψRA1A2

)‖RA2→B2(ψRA1A2

)) (26)

≤ supρR′A1

D(NA1→B1(ρR′A1

)‖RA1→B1(ρR′A1

))

5

+ supψRA1A2

D(MA2→B2(ψRA1A2

)‖RA2→B2(ψRA1A2

)) (27)

= D(NA1→B1‖RA1→B1

) +D(MA2→B2‖RA2→B2

). (28)

The first inequality follows from the same steps given inthe proof of [BHKW18, Lemma 38]. This concludes theproof.

Another approach to establishing additivity is to em-ploy the first identity of Proposition 5 (in Section II B)and [AC97, Eq. (3.28)], the latter of which was indepen-dently formulated in [DKJR06, Section 2.3].

3. Reduction to states and normalization

We now prove that the entropy of a channel reduces tothe entropy of a state if the channel is one that replacesthe input with a given state.

Proposition 4 (Reduction to states) Let the chan-nel NA→B be a replacer channel, defined such thatNA→B(ρA) = σB for all states ρA and some state σB.Then the following equality holds

H(N ) = H(B)σ. (29)

Proof. For any input ψRA, the output is NA→B(ψRA) =ψR ⊗ σB , and we find that

D(NA→B(ψRA)‖RA→B(ψRA)) = D(ψR ⊗ σB‖ψR ⊗ πB)

= D(σB‖πB). (30)

This implies that

H(N ) = log2 |B| −D(N‖R) (31)

= log |B| −D(σB‖πB) (32)

= H(B)σ, (33)

concluding the proof.

A final axiom (normalization) for a channel entropyfunction [Gou19] is that it should be equal to zero forany channel that replaces the input with a pure stateand it should be equal to the logarithm of the outputdimension for any channel that replaces the input withthe maximally mixed state. Clearly, Proposition 4 im-plies the normalization property if the replaced state ismaximally mixed or pure.

We note here that “the entropy of a channel” was alsodefined in [RZF11, Rog11], but the definition given theredoes not satisfy “reduction to states” or the basic axiomof normalization. For this reason, it cannot be consid-ered an entropy function according to the approach of[Gou19].

B. Alternate representations for the entropy of achannel

The entropy of a quantum channel has at least threealternate representations, in terms of the completelybounded entropy of [DKJR06], the entropy gain of itscomplementary channel [Ali04], and the maximum out-put entropy of the channel conditioned on its environ-ment. We recall these various channel functions now.

Recall that the completely bounded entropy of a quan-tum channel NA→B is defined as [DKJR06]

HCB,min(N ) ≡ infρRA

H(B|R)ω, (34)

where H(B|R)ω ≡ H(BR)ω −H(R)ω is the conditionalentropy of the state ωRB = NA→B(ρRA) and the sys-tem R is unbounded. However, due to data processing,purification, and the Schmidt decomposition theorem, itfollows that

HCB,min(N ) = infψRA

H(B|R)ω, (35)

where ψRA is a pure bipartite state with system R iso-morphic to the channel input system A.

Due to the Stinespring representation theorem [Sti55],every channel NA→B can be realized by the action of anisometric channel UNA→BE and a partial trace as follows:

NA→B = TrE UNA→BE . (36)

If we instead trace over the channel output B, this real-izes a complementary channel of NA→B :

N cA→E ≡ TrB UNA→BE . (37)

Using these notions, we can define the entropy gain of acomplementary channel of NA→B as follows [Ali04]:

G(N cA→E) ≡ inf

ρA[H(E)τ −H(A)ρ] , (38)

where τBE ≡ UNA→BE(ρA). The entropy gain has beeninvestigated for infinite-dimensional quantum systems in[Hol10, Hol11a, Hol11b]. We can also define the maxi-mum output entropy of the channel conditioned on itsenvironment as

supρA

H(B|E)τ , (39)

where again τBE ≡ UNA→BE(ρA).We now prove that the entropy of a channel, as given in

Definition 1, is equal to the completely bounded entropy,the entropy gain of a complementary channel, and thenegation of the maximum output entropy of the channelconditioned on its environment.

6

Proposition 5 Let NA→B be a quantum channel, andlet UNA→BE be an isometric channel extending it, as in(36). Then

H(N ) = HCB,min(N ) = G(N cA→E) = − sup

ρA

H(B|E)τ ,

(40)where τBE ≡ UNA→BE(ρA). It then follows that

|H(N )| ≤ log2 |B| . (41)

Proof. Using the identity D(ρ‖cσ) = D(ρ‖σ) − log2 c,for a constant c > 0, and the fact that the conditionalentropy H(B|R)N (ψ) = −D(NA→B(ψRA)‖ψR ⊗ IB), wefind that

H(N ) = log2 |B| −D(N‖R) (42)

= log2 |B| − supψRA

D(NA→B(ψRA)‖RA→B(ψRA))

(43)

= log2 |B| − supψRA

D(NA→B(ψRA)‖ψR ⊗ πB) (44)

= − supψRA

D(NA→B(ψRA)‖ψR ⊗ IB) (45)

= infψRA

H(B|R)N (ψ) (46)

= HCB,min(N ). (47)

We can then conclude the dimension bound in (41)from the fact that it holds uniformly for the condi-tional entropy |H(B|R)| ≤ log2 |B|. Defining τRBE =UNA→BE(ψRA), from the identity

H(B|R)τ = H(BR)τ −H(R)τ = H(E)τ −H(A)ρ, (48)

for ρA = TrRψRA, and where we used τR = ψR, wehave that

H(N ) = G(N cA→E) ≡ inf

ρA[H(E)τ −H(A)ρ] . (49)

We finally conclude that

H(N ) = − supρA

H(B|E)τ , (50)

which follows from the identity (duality of conditionalentropy)

H(B|R)ω = −H(B|E)U(ψA). (51)

This concludes the proof.

We note here, as observed in [DKJR06], that the di-mension lower bound H(N ) ≥ − log2 |B| is saturated forthe identity channel, while the dimension upper boundH(N ) ≤ log2 |B| is saturated for the completely random-izing (depolarizing) channel, which sends every state tothe maximally mixed state.

III. RENYI ENTROPY OF A QUANTUMCHANNEL

In this section, we define the Renyi entropy of a chan-nel, following the same approach discussed in the intro-duction. That is, we first write the Renyi entropy of astate as the difference of the number of physical qubitsand the Renyi relative entropy of the state to the maxi-mally mixed state. Then we define the Renyi entropy of achannel in the same way as in Definition 1, but replacingthe channel relative entropy with the sandwiched Renyichannel relative entropy from [CMW16].

The Renyi entropy of a quantum state ρA of system Ais defined for α ∈ (0, 1) ∪ (1,∞) as

Hα(A)ρ ≡1

1− αlog2 TrραA (52)

=1

1− αlog2 ‖ρA‖

αα , (53)

where ‖X‖α ≡ [Tr|X|α]1/α and |X| ≡√X†X for an

operator X. The Renyi relative entropy of quantumstates can be defined in two different ways, known asthe Petz–Renyi relative entropy [Pet85, Pet86] and thesandwiched Renyi relative entropy [MLDS+13, WWY14].The sandwiched Renyi relative entropy is defined forα ∈ (0, 1)∪ (1,∞), a state ρ, and a positive semi-definiteoperator σ as

Dα(ρ‖σ) ≡ 1

α− 1log2 Tr

(σ(1−α)/2αρσ(1−α)/2α

)α,

(54)whenever either α ∈ (0, 1) or supp(ρ) ⊆ supp(σ) andα > 1. Otherwise, it is set to +∞. The sand-wiched Renyi relative entropy obeys the data process-ing inequality for ρ and σ as above, a quantum chan-nel N , and α ∈ [1/2, 1) ∪ (1,∞) [FL13] (see also[Bei13, MO15, MLDS+13, WWY14, Wil18]):

Dα(ρ‖σ) ≥ Dα(N (ρ)‖N (σ)). (55)

It converges to the quantum relative entropy in the limitα→ 1 [MLDS+13, WWY14]:

limα→1

Dα(ρ‖σ) = D(ρ‖σ). (56)

By inspection, the Renyi entropy of a state can be writtenas

Hα(A)ρ = log2 |A| −Dα(ρA‖πA). (57)

The sandwiched Renyi channel divergence of channelsNA→B and MA→B is defined for α ∈ [1/2, 1) ∪ (1,∞)as [CMW16]

Dα(N‖M) ≡ supρRA

Dα(NA→B(ρRA)‖MA→B(ρRA)),

(58)where the optimization is with respect to bipartite statesρRA of a reference system R of arbitrary size and the

7

channel input system A. Due to state purification, thedata-processing inequality in (55), and the Schmidt de-composition theorem, it suffices to optimize over statesρRA that are pure and such that system R is isomorphicto system A.

We now define the Renyi entropy of a quantum channelas follows:

Definition 6 (Renyi entropy of a q. channel) LetNA→B be a quantum channel. For α ∈ [1/2, 1) ∪ (1,∞),the Renyi entropy of the channel N is defined as

Hα(N ) ≡ log2 |B| −Dα(N‖R), (59)

where RA→B is the completely randomizing channelfrom (5).

We remark here that Dα(N‖R), for α > 1, has an op-erational interpretation as the strong converse exponentfor discrimination of the channel NA→B from the com-pletely randomizing channel RA→B , when consideringany possible channel discrimination strategy [CMW16].

One could alternatively define a different Renyi en-tropy of a channel according to the above recipe, but interms of the Petz–Renyi relative entropy. However, it isunclear whether the additivity property is generally sat-isfied for the resulting Renyi entropy of a channel, andso we do not consider it further here, instead leaving thisquestion open.

A. Properties of the Renyi entropy of a quantumchannel

The Renyi entropy of a channel obeys the three desiredaxioms from [Gou19], and in fact, the proofs are essen-tially the same as the previous ones, but instead usingproperties of the sandwiched Renyi relative entropy.

Proposition 7 Let NA→B be a quantum channel, andlet Θ be a uniformity preserving superchannel as definedabove. Then for all [1/2, 1) ∪ (1,∞):

Hα(Θ(N )) ≥ Hα(N ). (60)

Proof. We follow the same steps as in (12)–(16), butmaking the substitutions H → Hα and D → Dα. Also,we use the fact that, for [1/2, 1)∪ (1,∞), the sandwichedRenyi channel divergence does not increase under the ac-tion of a superchannel, as shown in [Gou19].

Proposition 8 (Additivity) Let N and M be quan-tum channels. Then the channel Renyi entropy is addi-tive in the following sense for α ∈ (1,∞):

Hα(N ⊗M) = Hα(N ) +Hα(M). (61)

Proof. The proof here follows the same approach givenin the proof of Proposition 3, making the substitutionsH → Hα and D → Dα. The steps in (24)–(28) follow

from the same steps given in the proof of Proposition 41of [BHKW18], which in turn rely upon the additivityresult from [DKJR06]. See also [CMW16] in this context.

Proposition 9 (Reduction to states) Let the chan-nel NA→B be a replacer channel, defined such thatNA→B(ρA) = σB for all states ρA and some state σB.Then the following equality holds for all α ∈ (0, 1) ∪(1,∞):

Hα(N ) = Hα(B)σ. (62)

Proof. The proof is essentially the same as the proofof Proposition 4, making the substitutions H → Hα andD → Dα.

We can then conclude that the Renyi entropy of achannel satisfies the normalization axiom from the factthat Hα(B)σ = log |B| if σB is maximally mixed andH(B)σ = 0 if σB is pure.

B. Alternate representations for the Renyi entropyof a quantum channel

Just as we showed in Section II B that there are alter-nate representations for the entropy of a quantum chan-nel, here we do the same for the Renyi entropy of a chan-nel. We define the conditional Renyi entropy of a bipar-tite state ρAB as

Hα(A|B)ρ|ρ ≡ −Dα(ρAB‖IA ⊗ ρB), (63)

where Dα(ρ‖σ) is the sandwiched Renyi relative entropyfrom (54). The conditional Petz–Renyi entropy of a bi-partite state ρAB is defined as

Hα(A|B)ρ ≡ − infσBDα(ρAB‖IA ⊗ σB), (64)

where the Petz–Renyi relative entropyDα(ρ‖σ) is definedfor α ∈ (0, 1) ∪ (1,∞) as [Pet85, Pet86]

Dα(ρ‖σ) ≡ 1

α− 1log2 Tr

ρασ1−α , (65)

whenever either α ∈ (0, 1) or supp(ρ) ⊆ supp(σ) and α >1. Otherwise, it is set to +∞. The Petz–Renyi relativeentropy obeys the data processing inequality for ρ and σas above, a quantum channel N , and α ∈ (0, 1) ∪ (1, 2][Pet85, Pet86]:

Dα(ρ‖σ) ≥ Dα(N (ρ)‖N (σ)). (66)

The completely bounded 1 → α norm of a quantumchannel is defined for α ≥ 1 as [DKJR06]

‖NA→B‖CB,1→α ≡ supρR

∥∥∥ρ1/2αR NA→B(ΓRA)ρ

1/2αR

∥∥∥α,

(67)

8

where the optimization is with respect to a density oper-ator ρR and ΓRA ≡ |Γ〉〈Γ|RA denotes the projection ontothe following maximally entangled vector:

|Γ〉RA ≡∑i

|i〉R|i〉A, (68)

where |i〉Ri and |i〉Ai are orthonormal bases andsystem R is isomorphic to the channel input system A.

We can now state the alternate representations for theRenyi entropy of a channel:

Proposition 10 Let NA→B be a quantum channel, andlet UNA→BE be an isometric channel extending it, as in(36). Then for α ∈ (0, 1) ∪ (1,∞),

Hα(N ) = infψRA

Hα(B|R)ω|ω = − supρA

Hβ(B|E)τ . (69)

where the first optimization is with respect to bipar-tite pure states with system R isomorphic to system A,ωRB ≡ NA→B(ψRA), τBE ≡ UNA→BE(ρA), and β = 1/α.For α ∈ [1/2, 1) ∪ (1,∞),

|Hα(N )| ≤ log2 |B| . (70)

For α ∈ (1,∞), we have that

Hα(N ) =α

1− αlog2 ‖NA→B‖CB,1→α . (71)

Proof. To establish the equality

Hα(N ) = infψRA

Hα(B|R)ω|ω, (72)

we follow the same reasoning as in (42)–(46), but makingthe substitutions H → Hα and D → Dα. To establishthe equality

infψRA

Hα(B|R)ω|ω = − supρA

Hβ(B|E)τ , (73)

we employ the identity [TBH14, Theorem 2]

Hα(B|R)ω|ω = −Hβ(B|E)τ . (74)

To establish the dimension bounds, consider from dataprocessing that

Hα(B|R)ω|ω ≤ Hα(B)ω ≤ log2 |B| , (75)

where the second inequality follows from a dimensionbound for the Renyi entropy. To establish the other di-mension bound, let us employ the identity [TBH14, The-orem 2] again

Hα(B|R)ω|ω = −Hβ(B|E)τ (76)

≥ − infσEDβ(τBE‖IB ⊗ σE) (77)

≥ −Hβ(B)τ (78)

≥ − log2 |B| . (79)

The first inequality is stated in [TBH14, Corollary 4],and the second follows from data processing of the Petz–Renyi relative entropy under measurements, which holdsfor β ∈ (0, 1) ∪ (1,∞), as shown in [Hay06, Section 2.2](note that a measurement in the eigenbasis of τB com-bined with the partial trace over system E is a particularkind of measurement).

To establish the connection to the completely boundednorm for α > 1, we invoke [CMW16, Lemma 8] to findthat

Hα(N )

= log2 |B| −Dα(N‖R) (80)

= log2 |B| −α

α− 1log∥∥∥Ω

π(1−α)/αB

NA→B∥∥∥

CB,1→α(81)

=α

1− αlog ‖NA→B‖CB,1→α , (82)

where

Ωπ(1−α)/αB

(XB) ≡ π(1−α)/2αB XBπ

(1−α)/2αB (83)

= |B|(α−1)/αXB , (84)

concluding the proof.

Again, the dimension lower bound is saturated for theidentity channel, while the dimension upper bound is sat-urated for the completely depolarizing channel.

IV. MIN-ENTROPY OF A QUANTUMCHANNEL

The min-entropy of a quantum state ρA of a system Ais defined as [Ren05]

Hmin(A)ρ ≡ − log2 ‖ρ‖∞ (85)

= limα→∞

Hα(A)ρ. (86)

The max-relative entropy of a state ρ with a positivesemi-definite operator σ is defined as [Dat09]

Dmax(ρ‖σ) ≡ infλ : ρ ≤ 2λσ

(87)

= log2

∥∥∥σ−1/2ρσ−1/2∥∥∥∞, (88)

whenever supp(ρ) ⊆ supp(σ), and otherwise, it is set to+∞. It is known that [MLDS+13]

Dmax(ρ‖σ) = limα→∞

Dα(ρ‖σ). (89)

Observe that the min-entropy of a quantum state ρcan be written as the difference of the number of physicalqubits for the system A and the max-relative entropy ofρ to the maximally mixed state πA:

Hmin(A)ρ = log2 |A| −Dmax(ρA‖πA). (90)

Thus following the spirit of previous developments, wedefine the min-entropy of a channel as follows:

9

Definition 11 (Min-entropy of a quantum channel)We define the min-entropy of a quantum channel NA→Baccording to the recipe given in the introduction of ourpaper:

Hmin(N ) ≡ log2 |B| −Dmax(N‖R), (91)

where Dmax(N‖R) is the max-channel divergence[CMW16, LKDW18] and RA→B is the completely ran-domizing channel from (5).

The max-channel divergence is defined for two ar-bitrary channels NA→B and MA→B as [CMW16,LKDW18]

Dmax(N‖M)

≡ supρRA

Dmax(NA→B(ρRA)‖MA→B(ρRA)) (92)

= Dmax(NA→B(ΦRA)‖MA→B(ΦRA)). (93)

The latter equality, that an optimal state is the max-imally entangled state ΦRA, was proved in [BHKW18,Lemma 12] (see also [GFW+18, Eq. (45)] and [BHKW18,Remark 13] in this context). In fact, an optimal state isany pure bipartite state with full Schmidt rank (reducedstate has full support).

Due to the limit in (89) and the equality in (93), itfollows that

Dmax(N‖M) = limα→∞

Dα(N‖M). (94)

As such, we can immediately conclude that the min-entropy of a channel Hmin(N ) is equal to the followinglimit

Hmin(N ) = limα→∞

Hα(N ), (95)

and that it satisfies non-decrease under a uniformity pre-serving superchannel, additivity, and reduction to states(i.e., for a replacer channel, it reduces to the min-entropyof the replacing state), which, as stated previously, implythe three axioms from [Gou19].

A. Alternate representation for the min-entropy ofa channel in terms of conditional min-entropies

The conditional min-entropy of a bipartite quantumstate ρAB is defined as [Ren05]

Hmin(A|B)ρ ≡ − infσBDmax(ρAB‖IA ⊗ σB). (96)

We can also define the following related quantity:

Hmin(A|B)ρ|ρ ≡ −Dmax(ρAB‖IA ⊗ ρB), (97)

and clearly we have that

Hmin(A|B)ρ ≥ Hmin(A|B)ρ|ρ. (98)

The identities in (35) and (40), as well as the defi-nition of conditional min-entropy, inspire the followingquantity:

H↑min(N ) = infψRA

Hmin(B|R)ω. (99)

In the above, ωRB ≡ NA→B(ψRA) and ψRA is a purestate with system R isomorphic to the channel input sys-tem A.

This quantity might seem different from the min-entropy of a channel, but the following proposition states

that H↑min(N ) is actually equal to the min-entropy of thechannel Hmin(N ), thus simplifying the notion of min-entropy of a quantum channel:

Proposition 12 Let NA→B be a quantum channel.Then

Hmin(N ) = infψRA

Hmin(B|R)ω|ω (100)

= Hmin(B|R)ΦN |ΦN (101)

= H↑min(N ), (102)

where ωRB ≡ NA→B(ψRA) and ψRA is a pure statewith system R isomorphic to the channel input systemA. Also, the state ΦNRB = NA→B(ΦRA) is the Choi stateof the channel.

Proof. The first equality follows from the same stepsin the proof of Proposition 10 (see the reasoning around(72)). The second equality, i.e.,

Hmin(N ) = Hmin(B|R)ΦN |ΦN , (103)

follows by the observation in (93).

The proof of the equality Hmin(N ) = H↑min(N ) fol-lows from semi-definite programming duality, similar towhat was done previously for conditional min-entropy in[KRS09]. Consider that

H↑min(N )

= infψRA

Hmin(B|R)NA→B(ψRA) (104)

= infψRA

[− infσRDmax(NA→B(ψRA)‖σR ⊗ IB)

](105)

= infψRA

[− infσRlog2 TrσR : NA→B(ψRA) ≤ σR ⊗ IB

](106)

= − log2 supψRA

infσRTrσR : NA→B(ψRA) ≤ σR ⊗ IB .

(107)

Considering the innermost part of the last line above asthe following semi-definite program

infσRTrσR : NA→B(ψRA) ≤ σR ⊗ IB , (108)

10

its dual is given by

supXRB

TrXRBNA→B(ψRA) : XR ≤ IR, XRB ≥ 0 .

(109)

Now writing the pure state ψRA as ψRA = ρ1/2R ΓRAρ

1/2R

for some density operator ρR, this means we can rewritethe last line of the first block (without the negative log-arithm) as

supρR,XRB

Trρ1/2

R XRBρ1/2R NA→B(ΓRA) : XR ≤ IR, XRB ≥ 0, TrρR = 1, ρR ≥ 0

. (110)

We now define X ′RB ≡ ρ1/2R XRBρ

1/2R , which implies that XRB = ρ

−1/2R X ′RBρ

−1/2R , and the above is equal to

supρR,X′RB

TrX ′RBNA→B(ΓRA) : X ′R ≤ ρR, TrρR = 1, X ′RB ≥ 0, ρR ≥ 0

= supX′RB

TrX ′RBNA→B(ΓRA) : TrX ′R = 1, X ′RB ≥ 0 (111)

= supX′RB

TrX ′RBNA→B(ΓRA) : TrX ′RB = 1, X ′RB ≥ 0 (112)

= ‖NA→B(ΓRA)‖∞ (113)

So we conclude that

infψRA

Hmin(B|R)NA→B(ψRA)

= − log2 ‖NA→B(ΓRA)‖∞ (114)

= − log2 inf λ : NA→B(ΓRA) ≤ λIRB (115)

= − log2 inf λ : NA→B(ΦRA) ≤ λπR ⊗ IB (116)

= Hmin(B|R)ΦN |ΦN (117)

= Hmin(N ), (118)

where ΦN = NA→B(ΦRA) and the last equality followsfrom (103).

B. Relation of min-entropy of a channel to itsextended min-entropy

The extended min-entropy of a channel is definedas [Gou19]

Hextmin(N ) ≡ Hmin(B|R)ω, (119)

where ωRA = NA→B(ΦRA), with ΦRA the maximallyentangled state. It is not clear to us whether Hext

min(N ) isgenerally equal to the min-entropy of a channel Hmin(N ).However, due to (103), we conclude that

Hextmin(N ) ≥ Hmin(N ). (120)

V. ASYMPTOTIC EQUIPARTITIONPROPERTY

The smoothed conditional min-entropy of a bipartitestate ρAB is defined for ε ∈ (0, 1) as (see, e.g., [Tom15])

Hεmin(A|B)ρ ≡ sup

P (ρAB ,ρAB)≤εHmin(A|B)ρ, (121)

where the optimization is with respect to all subnormal-ized states ρAB (satisfying ρAB ≥ 0, TrρAB ≤ 1, andρAB 6= 0) and the sine distance (also called purified dis-tance) of quantum states ρ and σ [Ras02, Ras03, GLN05,Ras06] is defined in terms of the fidelity [Uhl76] as

P (ρ, σ) ≡√

1− F (ρ, σ), (122)

F (ρ, σ) ≡∥∥√ρ√σ∥∥2

1. (123)

The definition of fidelity is generalized to subnormalizedstates ω and τ as follows [TCR10]:

F (ω, τ) ≡ F (ω ⊕ [1− Trω], τ ⊕ [1− Trτ]), (124)

where the right-hand side is the usual fidelity of states(that is, we just add an extra dimension to ω and τand complete them to states). The smoothed conditionalmin-entropy satisfies the following asymptotic equiparti-tion property [TCR09] (see also [Tom15]), which is oneway that it connects with the conditional entropy of ρAB :

limn→∞

1

nHε

min(An|Bn)ρ⊗n = H(A|B)ρ. (125)

The purified channel divergence of two channelsNA→Band MA→B is defined as [LKDW18]

P (N ,M) ≡ supρRA

P (NA→B(ρRA),MA→B(ρRA)), (126)

Again, due to state purification, the data-processing in-equality for P (ρ, σ), and the Schmidt decomposition the-orem, it suffices to optimize over states ρRA that are pureand such that system R is isomorphic to system A. Wethen use this notion for smoothing the min-entropy of achannel:

11

Definition 13 (Smoothed min-entropy of a channel)The smoothed min-entropy of a channel is defined forε ∈ (0, 1) as

Hεmin(N ) ≡ sup

P (N ,N )≤εHmin(N ), (127)

where P (N , N ) is the purified channel divergence[LKDW18].

In the following theorem, we prove that the smoothedmin-entropy of a channel satisfies an asymptotic equipar-tition theorem that generalizes (125).

Theorem 14 (Asymptotic equipartition property)For all ε ∈ (0, 1), the following inequality holds

limn→∞

1

nHε

min(N⊗n) ≥ H(N ). (128)

We also have that

limε→0

limn→∞

1

nHε

min(N⊗n) ≤ H(N ). (129)

Proof. We first prove the inequality in (128). Let ωRnAndenote the de Finetti state [CKR09], defined as

ωRnAn ≡∫d(σRA) σ⊗nRA, (130)

where σRA is a pure state with system R isomorphic tothe channel input system A, and d(σRA) denotes theHaar measure on pure states. Let ωR′RnAn denote thepurification of the de Finetti state, with the purifying

system R′ satisfying the inequality |R′| ≤ (n+ 1)|A|2−1

[CKR09]. The reduced state ωAn is permutation in-

variant and has full rank. Let ωNn

R′RnBn denote thestate resulting from the action of the quantum channel

NnAn→Bn on the input state ωR′RnAn , and let ωN

⊗n

R′RnBn

denote the state resulting from the action of the quan-tum channel N⊗nA→B on the input state ωR′RnAn . LetCPTP(An → Bn) denote the set of all quantum chan-nels from input system An to output system Bn. LetPerm(An → Bn) denote the set of all permutation co-variant quantum channels from input system An to out-

put system Bn. Define ψNn

RBn to be the state resulting

from the action of the channel NnAn→Bn on the input

state ψRAn . Then consider that

Hεmin(N⊗n)

= supNn∈CPTP(An→Bn):

P (N⊗n,Nn)≤ε

infψRAn

Hmin(Bn|R)ψNn |ψNn (131)

≥ supNn∈Perm(An→Bn):

P (N⊗n,Nn)≤ε

infψRAn

Hmin(Bn|R)ψNn |ψNn (132)

= supNn∈Perm(An→Bn):

P (N⊗n,Nn)≤ε

Hmin(Bn|RnR′)ωNn |ωNn (133)

≥ supNn∈Perm(An→Bn):

P (ωN⊗n

,ωNn

)≤ε′

Hmin(Bn|RnR′)ωNn |ωNn (134)

The first equality follows from Definition 13. The firstinequality follows by restricting the maximization topermutation-covariant channels. The second equality fol-lows because the reduced state ωAn has full rank and byapplying the remark after (93). The second inequalityfollows by applying the postselection technique with

ε′ ≡ ε (n+ 1)−2(|A|2−1) . (135)

Note that the factor of two in the exponent of (135) isnecessary because we are employing the sine distance asthe channel distance measure. Continuing, we have that

Eq. (134)

= supNn∈CPTP(An→Bn):

P (ωN⊗n

,ωNn

)≤ε′

Hmin(Bn|RnR′)ωNn |ωNn (136)

= supσR′RnBn :

P (N⊗n(ωR′RnAn ),σR′RnBn )≤ε′,σR′Rn=ωR′Rn

Hmin(Bn|RnR′)σ|σ

(137)

≥ supσR′RnBn :

P (N⊗n(ωR′RnAn ),σR′RnBn )≤2ε′/3

Hmin(Bn|RnR′)σ|σ

− log2

(8 + [ε′/3]

2

[ε′/3]2

). (138)

The first equality follows from reasoning similar to thatgiven for Lemma 11 in Appendix B of [FWTB19], i.e.,that a permutation-covariant channel is optimal amongall channels, due to the fact that the original channelN⊗n is permutation covariant. In our case, it followsby employing the fact that the channel min-entropy doesnot decrease under the action of a uniformity preservingsuperchannel (see the discussion after (95)), and the su-perchannel that randomly performs a permutation at thechannel input and the inverse permutation at the channeloutput is one such superchannel. The second equality isa consequence of the fact that the following two sets areequal:NnAn→Bn(φRAn) :

P (NnAn→Bn(φRAn),N⊗nA→B(φRAn)) ≤ ε,

NnAn→Bn ∈ CPTP

=ωRBn ∈ D(HRBn) : P (ωRBn ,N⊗nA→B(φRAn)) ≤ ε,

ωR = φR

, (139)

which follows from applying Lemma 10 in Appendix Bof [FWTB19]. The inequality follows from Theorem 3of [ABJT19] (while noting that the state ρAB defined

12

therein satisfies ρB = ρB , so that the proof Theorem 3of [ABJT19] applies to our situation).

Continuing, and by applying [Tom15, Proposition 6.5],we find that

supσR′RnBn :

P (N⊗n(ωR′RnAn ),σR′RnBn )≤2ε′/3

Hmin(Bn|RnR′)σ|σ

(140)

≥ Hα(Bn|RnR′)ωN⊗n |ωN⊗n +

log2

(1−

√1− (2ε′/3)

2

)α− 1

(141)

≥ Hα(N⊗n) +

log2

(1−

√1− (2ε′/3)

2

)α− 1

(142)

= nHα(N ) +

log2

(1−

√1− (2ε′/3)

2

)α− 1

(143)

The second inequality follows from the definition of theRenyi entropy of a channel (Definition 6), and the equal-ity follows from the additivity of the Renyi entropy ofa channel (Proposition 8). Putting everything above to-gether, we conclude the following bound:

1

nHε

min(N ) ≥ Hα(N )− 1

nlog2

(8 + [ε′/3]

2

[ε′/3]2

)

+

log2

(1−

√1− (2ε′/3)

2

)n(α− 1)

. (144)

Taking the limit as n→∞, we conclude that the follow-ing inequality holds for all α > 1:

limn→∞

1

nHε

min(N ) ≥ Hα(N ). (145)

Since this inequality holds for all α > 1, we can take thelimit as α→ 1 to conclude that

limn→∞

1

nHε

min(N ) ≥ H(N ). (146)

This concludes the proof of the inequality in (128).

To arrive at the second inequality in (129), let Nn bea channel such that

P (N⊗n, Nn) ≤ ε. (147)

Now let φRAn be an arbitrary state. We then have fromthe definition in (126) that

P (N⊗nA→B(φRAn), NnAn→Bn(φRAn)) ≤ ε. (148)

Defining the states

ωRBn ≡ NnAn→Bn(φRAn), (149)

ωRBn ≡ N⊗nA→B(φRAn), (150)

we find that

Hmin(Nn) ≤ Hmin(Bn|R)ω|ω (151)

≤ H(Bn|R)ω (152)

≤ H(Bn|R)ω + ε2n log2 |B|+ g2(ε), (153)

where

g2(ε) ≡ (ε+ 1) log2(ε+ 1)− ε log2 ε. (154)

The second inequality follows from monotonicity of theconditional Renyi entropy with respect to α, and the lastfrom the uniform continuity bound in [Win16, Lemma 2].The above bound holds for any choice of φRAn , and sowe conclude that

Hmin(Nn) ≤ H(N⊗n) + ε2n log2 |B|+ g2(ε) (155)

= nH(N ) + ε2n log2 |B|+ g2(ε), (156)

where the equality follows from the additivity of the en-tropy of a channel (Proposition 3). Now, the inequality

has been shown for all Nn satisfying P (N⊗n, Nn) ≤ ε,and so we conclude, after dividing by n, that

1

nHε

min(N⊗n) ≤ H(N ) + 2ε log2 |B|+1

ng2(ε). (157)

Taking the limit as n→∞, we get that

limn→∞

1

nHε

min(N⊗n) ≤ H(N ) + 2ε log2 |B| . (158)

Now taking the limit as ε → 0, we arrive at the secondinequality in (129).

In Appendix A, we point out how an approach similarto that in the above proof leads to an alternate proofof the upper bound in [FWTB19, Theorem 8], regardingan asymptotic equipartition property for the smoothedmax-mutual information of a quantum channel.

VI. QUANTUM CHANNEL MERGING

Given a bipartite state ρBE , the goal of quantum statemerging is for Bob to use forward classical communica-tion to Eve, as well as entanglement, to merge his shareof the state with Eve’s share [HOW05, HOW07]. Theoptimal rate of entanglement consumed is equal to theconditional entropy H(B|E)ρ. Alternatively, the optimalrate of entanglement gained is equal to the conditionalentropy H(B|R)ψ, where ψRBE is a purification of ρBE .

In this section, we define a task, called quantumchannel merging, that generalizes state merging. Givena quantum channel NA→B with isometric extensionUNA→BE , the goal is for Bob to merge his share of thechannel with Eve’s share. We find here that the entan-glement cost of the protocol is equal to supρA H(B|E)ω,

13

where ωBE = UNA→BE(ρA). Equivalently, by employing(50), the entanglement gain of the protocol is equal toH(N ), the entropy of the channel NA→B . Thus, themain result of this section is a direct operational inter-pretation of the entropy of a channel as the entanglementgain in quantum channel merging. We note here that thecompletely bounded entropy of [DKJR06] (i.e., entropyof a channel) was recently interpreted in terms of a cryp-tographic task in [YHW19].

We now specify the quantum channel merginginformation-processing task in detail. Let NA→B be aquantum channel, and suppose that UNA→BE is an isomet-ric channel extending it. Here, we think of the isometric

channel UNA→BE as a broadcast channel (three-terminaldevice), which connects a source to the receivers Bob andEve. Suppose that a source generates an arbitrary stateψRAn and then sends the A systems through the isomet-ric channel (UNA→BE)⊗n, which transmits the B systemsto Bob and the E systems to Eve. The goal is for Bob touse free one-way local operations and classical communi-cation (one-way LOCC) in order to generate ebits at themaximum rate possible, while also merging his systemswith Eve’s.

Let n ∈ N, M ∈ Q, and ε ∈ [0, 1]. An (n,M, ε) pro-tocol for this task consists of a one-way LOCC channelPBnEnB0E0→BnEEnB1E1

such that

supψRAn

1

2

∥∥∥∥[id⊗nBE→BEE

(UNA→BE)⊗n](ψRAn)⊗ ΦLB1E1

− PBnEnB0E0→BnEEnB1E1

([(UNA→BE)⊗n(ψRAn)]⊗ ΦK

B0E0

)∥∥∥∥1

≤ ε, (159)

where ΦKB0E0

and ΦLB1E1

are maximally entangled states

of Schmidt rank K and L, respectively and M = L/K, sothat the number of ebits gained in the protocol is equalto log2M = log2 L− log2K. Figure 1 depicts the task ofquantum channel merging.

Definition 15 (Q. channel merging capacity) Arate R is achievable for quantum channel merging iffor all ε ∈ (0, 1], δ > 0, and sufficiently large n, thereexists an (n, 2n[R−δ], ε) protocol of the above form. Thequantum channel merging capacity CM (N ) is defined tobe the supremum of all achievable rates:

CM (N ) ≡sup R | R is achievable for channel merging on N .

(160)

A. Converse bound

Let us begin by considering the converse part, follow-ing the approach given in [HOW07] for quantum statemerging.

Proposition 16 Fix n,L,K ∈ N and ε ∈ [0, 1]. LetNA→B be a quantum channel. Then an (n,L/K, ε) quan-tum channel merging protocol for NA→B satisfies the fol-lowing bound:

1

n

[(1−

√ε) log2 L− log2K

]≤ H(N ) +

√ε log2 |A|+ g2(

√ε). (161)

Proof. We closely follow the approach given in [HOW07,Section IV-B], which established the converse part of

the quantum state merging theorem. Consider an ar-bitrary (n,L/K, ε) quantum channel merging protocol ofthe form described above. To prove the converse, we canreally employ any entanglement measure that reduces tothe entropy of entanglement for pure states and is asymp-totically continuous. So let us choose the entanglementof formation [BDSW96], which is defined for a bipartitestate ρAB as

EF (A;B)ρ ≡

inf

∑x

pX(x)H(A)ψx : ρAB =∑x

pX(x)ψxAB

,

(162)

where the infimum is with respect to all convex decompo-sitions of ρAB into pure states ψxAB . The entanglement offormation does not increase under the action of an LOCCchannel [BDSW96]. For the purposes of the converse, asin [HOW07, Section IV-B], we imagine that the referenceparty R is working together with Bob B, and they arespatially separated from Eve E. Let ωRBnEEnB1E1

and

ωRBnEEnB1E1denote the following respective states:

ωRBnEEnB1E1≡

[id⊗nBE→BEE

(UNA→BE)⊗n](ψRAn)⊗ ΦLB1E1

, (163)

ωRBnEEnB1E1≡

PBnEnB0E0→BnEEnB1E1([(UNA→BE)⊗n(ψRAn)]⊗ΦK

B0E0).

(164)

14

En

AnNV

Bn

En

AnNV

Bn

P ΦK ΦL

Bn

EnBn

Vs.

FIG. 1. The goal of quantum channel merging is for Bob to merge his share of the channel with Eve’s. Given a channel NA→B ,let VN ≡ (UNA→BE)⊗n, where UNA→BE is an isometric channel extending NA→B . By consuming a maximally entangled stateΦK of Schmidt rank K and applying a one-way LOCC protocol P , Bob and Eve can distill a maximally entangled state ΦL

of Schmidt rank L and transfer Bob’s systems Bn to Eve, in such a way that any third party having access to the inputs An

and the outputs Bn and En would not be able to distinguish the difference between the ideal situation on the left and thesimulation on the right. Theorem 18 states that the optimal asymptotic rate of entanglement gain is equal to the entropy ofthe channel N .

Define

f(n, ε, |A| , L) ≡√εn log2 |A|+

√ε log2 L+ g2(

√ε).(165)

We then have that

log2 L+H(R)ω

= H(B1)ω +H(R)ω (166)

= H(RB1)ω (167)

= EF (RB1; BnEEnE1)ω (168)

≤ EF (RB1; BnEEnE1)ω + f(n, ε, |A| , L). (169)

The first equality follows because log2 L = H(B1)ω fora maximally entangled state of Schmidt rank L. Thesecond equality follows because quantum entropy is ad-ditive with respect to product states. The third equalityfollows because the entanglement of formation reduces toentropy of entanglement for pure states. The inequalityis a consequence of the uniform continuity bound from[Win16, Corollary 4]. Continuing, we have that

EF (RB1; BnEEnE1)ω

≤ EF (RBnB0;EnE0)(UN )⊗n(ψ)⊗ΦK (170)

= H(RBnB0)(UN )⊗n(ψ)⊗ΦK (171)

= H(RBn)(UN )⊗n(ψ) +H(B0)ΦK (172)

= H(RBn)(UN )⊗n(ψ) + log2K. (173)

The first inequality follows from LOCC monotonicity ofthe entanglement of formation under the action of theone-way LOCC channel PBnEnB0E0→BnEEnB1E1

. The

last three equalities follow for reasons similar to whathave been given above. Putting everything together, wefind that

log2M = log2 L− log2K (174)

≤ H(Bn|R)(UN )⊗n(ψ) + f(n, ε, |A| , L). (175)

Since the protocol is required to work for every possible

input state ψRAn , we conclude the following bound

log2M ≤ infψRAn

H(Bn|R)(UN )⊗n(ψ) + f(n, ε, |A| , L)

(176)

= nH(N ) + f(n, ε, |A| , L), (177)

with the equality following from the additivity of the en-tropy of a channel [DKJR06] (recalled here as Proposi-tion 3). The inequality in the statement of the proposi-tion follows by dividing by n and rearranging.

B. Achievability bound

Now let us consider the achievability part.

Proposition 17 Fix n,L,K ∈ N and ε ∈ (0, 1). LetNA→B be a quantum channel. Then there exists an(n,L/K, ε) channel merging protocol for NA→B such thatits entanglement gain satisfies the following inequality forall α > 1:

1

n[log2 L− log2K] ≥ Hα(N )

− α

n (α− 1)

[4 log2(1/ε) + 4(|A|2 − 1) log2(n+ 1)

]− α

n (α− 1)[1/α+ 2 log2 13] . (178)

Proof. For the achievability part, we employ ideas usedin the theory of quantum channel simulation [BCR11,BBCW13, BRW14, Ber13]. In particular, the main chal-lenge of quantum channel merging over quantum statemerging is that it is necessary for the protocol to workfor every possible state ψRAn that could be input, notmerely for a fixed state input. In prior work on quantumchannel simulation [BCR11, BBCW13, BRW14, Ber13],this challenge has been met by appealing to the post-selection technique [CKR09, Theorem 1]. Here, we usethe same approach. In the context of the post-selection

15

technique, it is helpful to consult the unpublished note[Mat10] for further details.

Let ζAnAn denote the maximally mixed state of the

symmetric subspace of the AnAn systems [Har13], where

A is isomorphic to the channel input system A. Notethat this state can be written as [Har13, Proposition 6]

ζAnAn =

∫dψAA ψ⊗n

AA, (179)

where ψAA denotes a pure state and dψAA is the Haarmeasure over the pure states. This state is permutationinvariant; i.e., for a unitary channel Wπ

An ⊗ WπAn

cor-

responding to a permutation π, we have that ζAnAn =(Wπ

An ⊗WπAn

)(ζAnAn) for all π ∈ Sn, with Sn denoting

the symmetric group. Let ζR′AnAn be a purification ofζAnAn , and note that it can be chosen such that [Mat10]

ζR′AnAn = (WπR′ ⊗Wπ

An⊗Wπ

An)(ζAnAn), (180)

where WπR′ is some unitary, which implies that

(Wπ−1

R′ ⊗Wπ−1

An)(ζR′AnAn) =Wπ

An(ζR′AnAn). (181)

The first goal is to show the existence of a state merg-ing protocol for the state (UNA→BE)⊗n(ζR′AnAn). Asshown in [DBWR14, Theorem 5.2] (see also the earlier[Ber09, Proposition 4.7] in this context), there exists a

state merging protocol with error√

13ε′, with the entan-glement gain satisfying

log2 L− log2K ≥ Hε′

min(Bn|AnR′)(UN )⊗n(ζ)

− 2 log2

(1

ε′

). (182)

(To arrive at the inequality in (182), one needs touse the fact that P (ρ, σ) ≥ 1

2 ‖ρ− σ‖1 for any twostates.) That is, there exists a one-way LOCC channelPBnEnB0E0→BnEEnB1E1

such that the following inequal-

ity holds

1

2

∥∥∥∥[id⊗nBE→BEE

(UNA→BE)⊗n](ζR′AnAn)⊗ ΦLB1E1

− PBnEnB0E0→BnEEnB1E1([(UNA→BE)⊗n(ζR′AnAn)]⊗ ΦK

B0E0)

∥∥∥∥1

≤√

13ε′. (183)

Now our goal is for (159) to be satisfied for all possible states ψRAn . As a first step toward this goal, note that wecan symmetrize the protocol PBnEnB0E0→BnEEnB1E1

as follows

PBnEnB0E0→BnEEnB1E1≡ 1

n!

∑π∈Sn

(Wπ−1

BnE⊗Wπ−1

En

) PBnEnB0E0→BnEEnB1E1

(WπBn ⊗Wπ

En) , (184)

and the inequality in (183) is still satisfied, i.e.,

1

2

∥∥∥∥[id⊗nBE→BEE

(UNA→BE)⊗n](ζR′AnAn)⊗ ΦLB1E1

− PBnEnB0E0→BnEEnB1E1

([(UNA→BE)⊗n(ζR′AnAn)]⊗ ΦK

B0E0

)∥∥∥∥1

≤√

13ε′. (185)

This follows from the unitary invariance and convexity of the trace norm, the permutation covariance of the maps[id⊗n

BE→BEE(UNA→BE)⊗n] and (UNA→BE)⊗n:

∀π ∈ Sn :(Wπ−1

BnE⊗Wπ−1

En

) id⊗n

BE→BEE(UNA→BE)⊗n Wπ

An = id⊗nBE→BEE

(UNA→BE)⊗n, (186)

∀π ∈ Sn :(Wπ−1

Bn ⊗Wπ−1

En

) (UNA→BE)⊗n Wπ

An = (UNA→BE)⊗n, (187)

and the equality in (181). Furthermore, the symmetrization can be accomplished by one-way LOCC (Bob randomly

picks π, appliesWπBn , communicates the value to Eve, who appliesWπ

En at the input andWπ−1

BnE⊗Wπ−1

Enat the output),

and is thus free in our model. Since the symmetrized protocol, the target channel [id⊗nBE→BEE

(UNA→BE)⊗n], and the

channel (UNA→BE)⊗n are permutation covariant, we can now invoke the post-selection technique [CKR09, Theorem 1]

to conclude that as long as we choose ε′ = ε (n+ 1)−2(|A|2−1), then it is guaranteed that

16

supψRAn

1

2

∥∥∥∥[id⊗nBE→BEE

(UNA→BE)⊗n](ψRAn)⊗ ΦLB1E1

− PBnEnB0E0→BnEEnB1E1([(UNA→BE)⊗n(ψRAn)]⊗ ΦK

B0E0)

∥∥∥∥1

≤√

13ε. (188)

Propagating this choice of ε′ to the quantity in (182), this means that we require

log2 L− log2K ≥ Hε(n+1)

−2(|A|2−1)min (Bn|AnR′)(UN )⊗n(ζ) − 2 log2

(1

ε

)− 4

(|A|2 − 1

)log2 (n+ 1) . (189)

At this point, we invoke [Tom15, Eq. (6.92)], as well as the inequality 1−√

1− δ2 ≥ δ2/2 holding for all δ ∈ [0, 1], toconclude the following bound for α > 1:

Hε(n+1)

−2(|A|2−1)min (Bn|AnR′)(UN )⊗n(ζ)

≥ Hα(Bn|AnR′)(UN )⊗n(ζ)|(UN )⊗n(ζ) +2 log2(ε (n+ 1)

−2(|A|2−1))− 1

α− 1(190)

≥ infφRAn

Hα(Bn|R)ω|ω −2 log2(1/ε) + 4

(|A|2 − 1

)log2 (n+ 1)− 1

α− 1(191)

= Hα(N⊗n)−2 log2(1/ε) + 4

(|A|2 − 1

)log2 (n+ 1)− 1

α− 1(192)

= nHα(N )−2 log2(1/ε) + 4

(|A|2 − 1

)log2 (n+ 1)− 1

α− 1. (193)

where the first equality follows from Proposition 10,with ωRBn ≡ N⊗nA→B(φRAn), and the last equality crit-ically relies upon the additivity Hα(N⊗n) = nHα(N )from Proposition 8, which in turn directly follows fromthe main result of [DKJR06]. Putting everything to-gether, we conclude that for ε ∈ (0, 1/13), there exists

an (n,L/K,√

13ε) channel merging protocol for NA→Bsuch that its entanglement gain satisfies the following in-equality for all α > 1:

1

n[log2 L− log2K] ≥ Hα(N )

− α

n (α− 1)

[2 log2(

1

ε) + 4

(|A|2 − 1

)log2(n+ 1) +

1

α

].

(194)

We arrive at the statement of the proposition by a finalsubstitution ε′′ =

√13ε ∈ (0, 1), which implies that ε =

(ε′′)2/13 and 2 log2(1/ε) = 4 log2(1/ε′′) + 2 log2 13.

C. Quantum channel merging capacity is equal tothe entropy of a channel

We can now put together the previous two propositionsto conclude the following theorem:

Theorem 18 The quantum channel merging capacity of

a channel N is equal to its entropy:

CM (N ) = H(N ). (195)

Proof. By applying the limits n → ∞ and ε → 0, thefollowing bound is a consequence of Proposition 16:

CM (N ) ≤ H(N ). (196)

For an arbitrary α > 1, ε ∈ (0, 1), and δ > 0, wecan conclude from Proposition 17 that there exists an(n, 2n[Hα(N )−δ], ε) channel merging protocol by taking nsufficiently large. This implies that Hα(N ) is an achiev-able rate for all α > 1. However, since this statementis true for all α > 1, we can conclude that the ratesupα>1Hα(N ) = H(N ) is achievable also. This estab-lishes that CM (N ) ≥ H(N ).

VII. EXAMPLES

In this section, we provide formulas for the entropyof several fundamental channel models, including era-sure channels, dephasing channels, depolarizing channels,and Werner–Holevo channels. We also define the energy-constrained and unconstrained entropies of a channel anddetermine formulas for them for common bosonic channelmodels, including thermal, amplifier, and additive-noisechannels.

17

A. Finite-dimensional channels

A first observation to make is that, for any finite-dimensional channel, it is an “easy” optimization task tocalculate its entropy. This is a consequence of the iden-tity H(N ) = − supρA H(B|E)U(ρ) from Proposition 5and the concavity of conditional entropy [LR73b, LR73a](in this context, see also [AC97, Eq. (3.19)]). Thus,one can exploit numerical optimizations to calculate it[FSP18, FF18].

For channels with symmetry, it can be much easierto evaluate a channel’s entropy, following from some ob-servations from, e.g., [KW18, Section 6]. Let us beginby recalling the notion of a covariant channel NA→B[Hol02]. For a group G with unitary channel represen-tations UgAg and VgBg acting on the input system Aand output system B of the channel NA→B , the channelNA→B is covariant with respect to the group G if thefollowing equality holds for all g ∈ G:

NA→B UgA = VgB NA→B . (197)

If the averaging channel is such that 1|G|∑g U

gA(X) =

Tr[X]I/ |A| (implementing a unitary one-design), thenwe simply say that the channel NA→B is covariant. Itturns out that the entropy of a channel is simple to calcu-late for covariant channels, with the optimal ψRA in (35)being the maximally entangled state, or equivalently, theoptimal ρA in − supρA H(B|E)U(ρ) being the maximallymixed state.

Proposition 19 Let NA→B be a quantum channel thatis covariant with respect to a group G, in the sense of(197), and let UNA→BE be an isometric channel extendingit. Then it suffices to perform the optimization for theentropy of a channel over states that respect the symme-try of the channel:

H(N ) = − supρA=SA(ρA)

H(B|E)U(ρ), (198)

where the symmetrizing channel SA = 1|G|∑g∈G U

gA.

Thus, if a channel is covariant, then H(N ) =−H(B|E)U(π); i.e., the optimal state ρA is the maximallymixed state πA.

Proof. First recall from Proposition 5 that H(N ) =− supρA H(B|E)U(ρ). Let ρA be an arbitrary state. If achannel NA→B is covariant as in (197), then it is knownthat there exists a unitary channelWg

E such that [Hol06,Hol12]

UNA→BE UgA = (VgB ⊗W

gE) UNA→BE . (199)

See also [DBW17, Appendix A] for a simple proof. Thenwe find that

H(B|E)U(ρ) = H(B|E)(Vg⊗Wg)U(ρ) (200)

=1

|G|∑g∈G

H(B|E)(Vg⊗Wg)U(ρ) (201)

=1

|G|∑g∈G

H(B|E)(UUg)(ρ) (202)

≤ H(B|E)(US)(ρ). (203)

The first equality follows from invariance of conditionalentropy under the action of a local unitary (the equal-ity holds for all g ∈ G). The third equality follows fromchannel covariance. The inequality follows from concav-ity of conditional entropy [LR73b, LR73a].

A simple example of a channel that is covariant is thequantum erasure channel, defined as [GBP97]

Ep(ρ) ≡ (1− p)ρ+ p|e〉〈e|, (204)

where ρ is a d-dimensional input state, p ∈ [0, 1] is theerasure probability, and |e〉〈e| is a pure erasure state or-thogonal to any input state, so that the output state hasd+1 dimensions. A d-dimensional dephasing channel hasthe following action:

Dp(ρ) =

d−1∑`=0

p`Z`ρZ`†, (205)

where p is a vector containing the probabilities p` andZ has the following action on the computational basisZ|x〉 = e2πix/d|x〉. This channel is covariant with respectto the Heisenberg–Weyl group of unitaries, which is wellknown to form a unitary one-design. A particular kindof Werner–Holevo channel performs the following trans-formation on a d-dimensional input state ρ [WH02]:

W(d)(ρ) ≡ 1

d− 1(TrρI − T (ρ)) , (206)

where d ≥ 2 and T denotes the transpose map T (·) =∑i,j |i〉〈j|(·)|i〉〈j|. As observed in [WH02, Section II],

this channel is covariant. The d-dimensional depolariz-ing channel is a common model of noise in quantum in-formation, transmitting the input state with probability1− p ∈ [0, 1] and replacing it with the maximally mixedstate π ≡ I

d with probability p:

∆p(ρ) = (1− p) ρ+ pπ. (207)

By applying Proposition 19 and evaluating the result-ing entropy −H(B|E) for each of the above channelswhen the maximally mixed state π is input, we arriveat the following formulas:

H(Ep) = h2(p) + (p− 1) log2 d, (208)

H(Dp) = H(p)− log2 d, (209)

H(W(d)) = log2 [(d− 1)/2] , (210)

H(∆p) = −(

1− p+p

d2

)log2

(1− p+

p

d2

)−(d2 − 1

) pd2

log2

p

d2− log2 d, (211)

where H(p) is the Shannon entropy of the probabilityvector p.

18

B. Energy-constrained entropy of a channel

We can define the energy-constrained entropy of achannel for infinite-dimensional systems, by employingthe identity in Proposition 5 and the definition of condi-tional entropy from [Kuz11].

To review the definition from [Kuz11], recall that thequantum entropy of a state ρ acting on a separableHilbert space is defined as

H(ρ) ≡ Trη(ρ), (212)

where η(x) = −x log2 x if x > 0 and η(0) = 0. The tracein the above equation can be taken with respect to anycountable orthonormal basis of H [AL70, Definition 2].The quantum entropy is a non-negative, concave, lowersemicontinuous function on D(H) [Weh76]. It is also notnecessarily finite (see, e.g., [BV13]). When ρA is assignedto a system A, we write H(A)ρ ≡ H(ρA). Recall thatthe relative entropy of two states ρ and σ acting on aseparable Hilbert space is given by [Fal70, Lin73]

D(ρ‖σ) ≡

[ln 2]−1∑i,j

|〈φi|ψj〉|2[p(i) ln

(p(i)

q(j)

)+ q(j)− p(i)],

(213)

where ρ =∑i p(i)|φi〉〈φi| and σ =

∑j q(j)|ψj〉〈ψj | are

spectral decompositions of ρ and σ with |φi〉i and|ψj〉j orthonormal bases. The prefactor [ln 2]−1 is thereto ensure that the units of the quantum relative entropyare bits. For a bipartite state ρAB , the mutual informa-tion is defined as

I(A;B)ρ ≡ D(ρAB‖ρA ⊗ ρB). (214)

Finally, for a bipartite state ρAB such that H(A)ρ <∞,the conditional entropy is defined as [Kuz11]

H(A|B)ρ ≡ H(A)ρ − I(A;B)ρ, (215)

and it is known that H(A|B)ρ ∈ [−H(A)ρ, H(A)ρ][Kuz11].

A Gibbs observable is a positive semi-definite oper-ator G acting on a separable Hilbert space such thatTre−βG <∞ for all β > 0 [Hol03, Hol04, Hol12]. Thiscondition for a Gibbs observable means that there is al-ways a well defined thermal state.

Finally, we say that a quantum channel NA→B obeysthe finite-output entropy condition [Hol03, Hol04, Hol12]with respect to a Gibbs observable G if for all P ≥ 0, thefollowing inequality holds

supρA:TrGρA≤P

H(B)N (ρ) <∞. (216)

We now define the energy-constrained and uncon-strained channel entropy as follows:

Definition 20 Let NA→B be a quantum channel thatsatisfies the finite-output entropy condition with respectto a Gibbs observable G. For P ≥ 0, the energy-constrained entropy of NA→B is defined as

H(N , G, P ) ≡ infψRA:TrGψA≤P

H(B|R)ω, (217)

where ωRB ≡ NA→B(ψRA) and the optimization is withrespect to all pure bipartite states with system R isomor-phic to system A. The unconstrained entropy of NA→Bwith respect to G is then defined as

H(N , G) ≡ infP≥0

H(N , G, P ). (218)

C. Bosonic Gaussian channels

In this section, we evaluate the energy-constrainedand unconstrained entropy of several important bosonicGaussian channels [Hol12, Ser17], including the thermal,amplifier, and additive-noise channels. Here we take theGibbs observable to be the photon number operator n[Hol12, Ser17], and we note that each of these channelssatisfies the finite-output entropy condition mentionedabove. From a practical perspective, we should be mostinterested in these particular single-mode bosonic Gaus-sian channels, as these are of the greatest interest inapplications, as stressed in [Hol12, Section 12.6.3] and[HG12, Section 3.5]. Each of these are defined respec-tively by the following Heisenberg input-output relations:

b =√ηa+

√1− ηe, (219)

b =√Ga+

√G− 1e†, (220)

b = a+ (x+ ip) /√

2, (221)

where a, b, and e are the field-mode annihilation opera-tors for the sender’s input, the receiver’s output, and theenvironment’s input of these channels, respectively.

The channel in (219) is a thermalizing channel, inwhich the environmental mode is prepared in a thermalstate θ(NB) of mean photon number NB ≥ 0, defined as

θ(NB) ≡ 1

NB + 1

∞∑n=0

(NB

NB + 1

)n|n〉〈n|, (222)

where |n〉∞n=0 is the orthonormal, photonic number-state basis. When NB = 0, θ(NB) reduces to the vac-uum state, in which case the resulting channel in (219) iscalled the pure-loss channel. The parameter η ∈ (0, 1) isthe transmissivity of the channel, representing the aver-age fraction of photons making it from the input to theoutput of the channel. Let Lη,NB denote this channel.

The channel in (220) is an amplifier channel, and theparameter G > 1 is its gain. For this channel, the en-vironment is prepared in the thermal state θ(NB). IfNB = 0, the amplifier channel is called the pure-amplifierchannel. Let AG,NB denote this channel.

19

Finally, the channel in (221) is an additive-noise chan-nel, representing a quantum generalization of the clas-sical additive white Gaussian noise channel. In (221),x and p are zero-mean, independent Gaussian randomvariables each having variance ξ ≥ 0. Let Tξ denote thischannel. Note that the additive-noise channel arises fromthe thermal channel in the limit η → 1, NB → ∞, butwith (1− η)NB → ξ [GGL+04].

Kraus representations for the channels in (219)–(221)are available in [ISS11], which can be helpful for furtherunderstanding their action on input quantum states.

All of the above channels are phase-insensitive orphase-covariant Gaussian channels [Hol12, Ser17]. LetNS ≥ 0. Since the function supρ:Trnρ≤NS H(B|E)U(ρ)

we are evaluating is concave in the input and invariantunder local unitaries, [SWAT18, Remark 22] applies, im-plying that the optimal input state for the entropies ofthese channels is the bosonic thermal state θ(NS). Wethen find by employing well known entropy formulas from[HW01, GLMS03] (see also [WHG12] in this context)that

H(Lη,NB , n, NS) =

g2([D1 + (1− η) (NS −NB)− 1] /2)

+ g2([D1 − (1− η) (NS −NB)− 1] /2)− g2(NS),(223)

H(AG,NB , n, NS) =

g2([D2 + (G− 1) (NS +NB + 1)− 1] /2)

+ g2([D2 − (G− 1) (NS +NB + 1)− 1] /2)− g2(NS),(224)

H(Tξ, n, NS) = g2([D3 − (ξ + 1)] /2)

+ g2([D3 + ξ − 1] /2)− g2(NS), (225)

where g2 is the bosonic entropy function defined in (154)and

D1 ≡√

[(η + 1)NS + (1− η)NB + 1]2 − 4ηNS (NS + 1), (226)

D2 ≡√

[(G+ 1)NS + (G− 1) (NB + 1) + 1]2 − 4GNS (NS + 1), (227)

D3 ≡√

(ξ + 1)2

+ 4ξNS . (228)

Note that we arrived at the formula for H(Tξ, n, NS) byconsidering the limit discussed above. Furthermore, bythe same reasoning as given in [SWAT18, Section 6], thesefunctions are decreasing with increasing NS , and so wefind that

H(Lη,NB , n) = infNS≥0

H(Lη,NB , n, NS) (229)

= limNS→∞

H(Lη,NB , n, NS), (230)

H(AG,NB , n) = infNS≥0

H(AG,NB , n, NS) (231)

= limNS→∞

H(AG,NB , n, NS), (232)

H(Tξ, n) = infNS≥0

H(Tξ, n, NS) (233)

= limNS→∞

H(Tξ, n, NS), (234)

which leads to the following formulas for the uncon-strained entropies of the channels:

H(Lη,NB , n) = log2(1− η) + g2(NB), (235)

H(AG,NB , n) = log2(G− 1) + g2(NB), (236)

H(Tξ, n) = log2(ξ) +1

ln 2. (237)

A Mathematica file is available with the arXiv postingof this paper to automate these calculations, but we notehere that the expansion g2(x) = log2(x)+1/ ln 2+O(1/x)is helpful for this purpose. We also note that the formulasin (235)–(236) were presented in [PGPBL09, Eq. (2)] andthe formula in (237) was presented in [HW01, Section V].

VIII. GENERALIZED CHANNEL ENTROPIESFROM GENERALIZED DIVERGENCES

In this section, we discuss other possibilities for defin-ing generalized entropies of a quantum channel. Onemain concern might be how unique or distinguished ournotion of entropy of a channel from Definition 1 is, beingbased on the channel relative entropy of the channel ofinterest and the completely randomizing channel. As aconsequence of the fact that there are alternate ways ofdefining channel relative entropies, there could be alter-nate notions of channel entropies. However, we shouldrecall that one of the main reasons we have chosen thedefinition in Definition 1 is that the channel relative en-tropy appearing there has a particularly appealing opera-tional interpretation in the context of channel discrimina-

20

tion [CMW16]. That is, for what one might consider themost natural and general setting of quantum channel dis-crimination, the optimal rate for distinguishing a channelfrom the completely randomizing channel is given by thechannel relative entropy in (4) [CMW16]. As we show inwhat follows, there are further reasons to focus on ourdefinition of the entropy of a channel from Definition 1,as well as our definition of the min-entropy of a channelfrom Definition 11.

To begin the discussion, let S(C) denote the set ofquantum states for an arbitrary quantum system C. Letus recall that a function D : S(C)× S(C)→ R ∪ +∞is a generalized divergence [PV10, SW12] if for arbitraryHilbert spaces HA and HB , arbitrary states ρA, σA ∈S(A), and an arbitrary channel NA→B , the followingdata processing inequality holds

D(ρA‖σA) ≥ D(NA→B(ρA)‖NA→B(σA)). (238)

Examples of interest are in particular the quantum rela-tive entropy, the Petz-Renyi divergences, the sandwichedRenyi divergences, as considered in this paper.

Based on generalized divergences, one can define atleast two different channel divergences as a measure forthe distinguishability of two quantum channels NA→BandMA→B . Here we consider a function of two quantumchannels to be a channel divergence if it is monotoneunder the action of a superchannel.

1. Generalized channel divergence [LKDW18]:

D(N‖M) ≡ supρRA

D(NA→B(ρRA)‖MA→B(ρRA)). (239)

In the above, the optimization can be restricted topure states of systems R and A with R isomorphicto system A. The monotonicity of the generalizedchannel divergence under the action of a superchan-nel was proven in [Gou19].

2. Amortized channel divergence [BHKW18]:

DA(N‖M) ≡sup

ρRA,σRA

D(NA→B(ρRA)‖MA→B(σRA))−D(ρRA‖σRA).

(240)

The monotonicity of the amortized channel di-vergence under the action of a superchannel wasproven in [BHKW18].

We can consider other divergences as follows, but theyare not known to be monotone under the action of ageneral superchannel, and so we do not label them aschannel divergences:

1. Choi divergence:

DΦ(N‖M) ≡ D(NA→B(ΦRA)‖MA→B(ΦRA)). (241)

As we show in Appendix B, the Choi divergence ismonotone under the action of a superchannel con-sisting of mixtures of a unital pre-processing chan-nel and an arbitrary post-processing channel.

2. Adversarial divergence:

Dadv(N‖M) ≡ supρRA

infσRA

D(NA→B(ρRA)‖MA→B(σRA)).

(242)In the above, due to state purification, data pro-cessing, and the Schmidt decomposition, the maxi-mization can be restricted to pure states ρRA of sys-tems R and A with R isomorphic to system A. Theminimization should be taken over mixed statesσRA. For a proof of this fact, see Appendix C.

3. Adversarial Choi divergence:

Dadv,Φ(N‖M) ≡ infσRA

D(NA→B(ΦRA)‖MA→B(σRA)).

(243)

4. “No quantum memory” divergence:

supρA

D(NA→B(ρA)‖MA→B(ρA)). (244)

There could certainly even be other divergences toconsider. In our context, two effective ways of sin-gling out particular divergences as primary and othersas secondary are 1) whether the channel divergence hasa compelling operational interpretation for a channel dis-crimination task and 2) whether the channel divergenceleads to an entropy function that satisfies the axiomsfrom [Gou19].

Based on the recipe given in the introduction, from agiven divergence D′(N‖M) (any of the choices above),one could then define a generalized entropy function of achannel NA→B as

H(N ) ≡ log2 |B| −D′(N‖R), (245)

where RA→B is the completely randomizing channelfrom (5).

Taking the above approach to pruning entropy func-tions, we can already rule out the last one (“no quantummemory”), as done in [Gou19], because, after taking D tobe the most prominent case of quantum relative entropy,the resulting entropy function is the minimum outputentropy of a channel, which is known to be non-additive[Has09]. While an entropy arising from the Choi diver-gence leads to an entropy function satisfying the axiomsdesired for an entropy function, the Choi divergence it-self does not appear to have a compelling operationalinterpretation in the sense of being a “channel measure”because it simply reduces a channel discrimination prob-lem to a state discrimination problem (i.e., it does notmake use of the most general approach one could takefor discriminating arbitrary channels). This point couldbe debated, and we do return to entropy functions de-rived from Choi and adversarial Choi divergences in Sec-tion VIII D below.

21

A. Collapse of entropy functions derived fromquantum relative entropy

From the list above, by focusing on the operational andaxiomatic criteria listed above, this leaves us with thegeneralized channel divergence and the amortized chan-nel divergence. Here we also consider the adversarial di-vergence. Interestingly, after taking D to be the promi-nent case of quantum relative entropy and the channelM to be the completely randomizing channel, we findthe following collapse of the divergences:

D(N‖R) = DA(N‖R) = Dadv(N‖R). (246)

The first equality was shown in [CMW16, BHKW18], andwe show the second one now. From the definitions, wehave that Dadv(N‖M) ≤ D(N‖M) for any generalizeddivergence D and any channelM. So we show the oppo-site inequality for the special case of D = D andM = R.Let ρRA and σRA be arbitrary states. Then

D(NA→B(ρRA)‖RA→B(σRA))

= D(NA→B(ρRA)‖σR ⊗ πB) (247)

= −H(NA→B(ρRA))

− TrNA→B(ρRA) log2(σR ⊗ πB) (248)

= −H(NA→B(ρRA))− TrρR log2 σR− TrNA→B(ρA) log2 πB (249)

= −H(NA→B(ρRA))− TrρR log2 ρR+D(ρR‖σR)− TrNA→B(ρA) log2 πB (250)

= −H(NA→B(ρRA))

− TrNA→B(ρRA) log2(ρR ⊗ πB)+D(ρR‖σR)(251)

= D(NA→B(ρRA)‖RA→B(ρRA)) +D(ρR‖σR). (252)

Now taking an infimum over all σRA and invoking thenon-negativity of quantum relative entropy, we concludethat

infσRA

D(NA→B(ρRA)‖RA→B(σRA))

= D(NA→B(ρRA)‖RA→B(ρRA)). (253)

By taking a supremum over ρRA, we then conclude thatDadv(N‖R) = D(N‖R).

Thus, the collapse in (246), as well as the operationalinterpretation of D(N‖R) from [CMW16] and the factthat the resulting entropy function satisfies the axiomsfrom [Gou19], indicate that our choice of the entropy ofa quantum channel in Definition 1 is cogent.

B. Collapse of entropy functions derived frommax-relative entropy

Interestingly, a similar and further collapse occurswhen taking D to be the max-relative entropy:

Dmax(N‖R) = DΦmax(N‖R) (254)

= DAmax(N‖R) (255)

= Dadvmax(N‖R). (256)

The first two equalities were shown in [BHKW18, Propo-sition 10] for arbitrary channelsN andM. By employinga semi-definite programming approach as in the proof ofProposition 12, we can conclude the last equality. Thus,this collapse, as well as the facts that the max-relativeentropy Dmax(N‖M) is an upper bound on the rate atwhich any two channels can be distinguished in an ar-bitrary context [BHKW18, Corollary 18] and the result-ing entropy function Hmin(N ) satisfies the axioms from[Gou19], indicate that our choice of the min-entropy of aquantum channel in Definition 11 is also cogent.

C. Entropy functions derived from Renyi relativeentropies

In Section III, we defined the Renyi entropy of a chan-nel as in Definition 6, in terms of the sandwiched Renyirelative entropy. The following collapse is known forthe sandwiched Renyi relative entropy for α ∈ (1,∞)[CMW16, BHKW18]:

Dα(N‖R) = DAα (N‖R). (257)

However, it is not known whether these quantities areequal for α ∈ (0, 1) or whether they are equal to theadversarial divergence Dadv

α (N‖R) for any α ∈ (0, 1) ∪(1,∞). At the same time, one of the most compelling rea-sons to fix the definition of channel Renyi entropy as wehave done is that the channel divergence Dα(N‖R) hasboth a convincing operational interpretation in channeldiscrimination as the optimal strong converse exponentand the entropy function satisfies all of the desired axiomsfor an entropy function. Furthermore, the entropy func-tion Hα(N ) represents a useful bridge between the en-tropy and min-entropy of a quantum channel, due to thefacts that limα→1Hα(N ) = H(N ), limα→∞Hα(N ) =Hmin(N ), and Hα(N ) ≤ Hβ(N ) for α ≥ β ≥ 1.

One should notice that we did not define the Renyientropy of a channel in terms of the Petz–Renyi relativeentropy and the resulting channel divergence, amortizedchannel divergence, or adversarial divergence. One ofthe main reasons for this is that it is not known whetherthe resulting entropy functions are additive. Further-more, operational interpetations for these divergenceshave not been established, having been open since thepaper [CMW16] appeared. As such, it very well could bethe case that one could derive cogent notions of channelentropy from the Petz–Renyi relative entropy, but thisremains the topic of future work.

22

D. Entropy functions derived from Choi andadversarial Choi divergences

In this subsection, we discuss various entropy functionsderived from Choi and adversarial Choi divergences. Asemphasized previously, we note again here that the op-erational interpretations for these divergences are reallyabout state discrimination tasks rather than channel dis-crimination tasks. Nevertheless, the resulting entropyfunctions satisfy the axioms put forward in [Gou19].

By picking the divergence D to be the quantum relativeentropy D, we find that the Choi and adversarial Choidivergences are equal when discriminating an arbitrarychannel NA→B from the completely randomizing channelRA→B :

DΦ(N‖R) = Dadv,Φ(N‖R). (258)

The proof of this statement follows along the lines of(247)–(252). There is a simple operational interpre-tation for DΦ(N‖R) in terms of state discrimination[HP91, ON00], while an operational interpretation forDadv,Φ(N‖R) in terms of state discrimination was givenrecently in [HT16].

We could also pick the divergence D to be Petz–Renyi relative entropy Dα or the sandwiched Renyi rel-ative entropy Dα. The resulting Choi and adversarialChoi divergences are then generally not equal when dis-criminating an arbitrary channel NA→B from the com-pletely randomizing channel RA→B . There is an oper-

ational interpretation for DΦ

α(N‖R) for α ∈ (0, 1) interms of state discrimination [Hay07, Nag06] (error ex-ponent problem), and there is an operational interpre-tation for DΦ

α (N‖R) for α ∈ (1,∞) in terms of statediscrimination [MO15] (strong converse exponent prob-lem). Interestingly, [HT16] has given a meaningful op-erational interpretation for the adversarial Choi diver-

gences Dadv,Φ

α (N‖R) for α ∈ (0, 1) and Dadv,Φα (N‖R) for

α ∈ (1,∞) in terms of error exponent and strong converseexponent state discrimination problems, respectively.

For NA→B a quantum channel and ΦNRB ≡NA→B(ΦRA) the Choi state, the resulting channel en-tropy functions are then as follows:

HΦ(N ) ≡ H(B|R)ΦN = Hadv,Φ(N ), (259)

HΦα (N ) ≡ Hα(B|R)ΦN |ΦN , (260)

HΦ

α(N ) ≡ Hα(B|R)ΦN |ΦN , (261)

Hadv,Φα (N ) ≡ Hα(B|R)ΦN , (262)

Hadv,Φ

α (N ) ≡ Hα(B|R)ΦN . (263)

It then follows that all of the above entropy functions areadditive for α ∈ (0, 1)∪ (1,∞) (with the exception of ad-ditivity holding for Hadv,Φ

α (N ) for α ∈ [1/2, 1)∪ (1,∞)),due to the facts that the Choi state of a tensor-productchannel is equal to the tensor product of the Choi statesof the individual channels, as well as the additivity of theunderlying conditional entropies, for Hadv,Φ

α (N ) shown

in [Tom15] and for Hadv,Φ

α (N ) following from the quan-tum Sibson identity [SW12, Lemma 7] (see also [TBH14,Lemma 1]). Normalization and reduction to states (asin Proposition 4) follows for all of the above quanti-ties. What remains is monotonicity under random uni-tary superchannels, and what we can show is somethingstronger: monotonicity under doubly stochastic super-channels, the latter defined in [Gou19] as superchannelsΘ such that their adjoint Θ† is also a superchannel, wherethe adjoint is defined with respect to the inner productfor supermaps considered in [Gou19].

Theorem 21 Let Θ be a doubly stochastic superchannelgiven by

Θ [NA→B ] ≡ ΩBE→D NA→B ΛC→AE . (264)

with ΩBE→D and ΛC→AE quantum channels, E a quan-tum memory system, |A| = |C|, and |B| = |D|. Then,for H any of the entropy functions in (259)–(263), thefollowing inequality holds

H (Θ [NA→B ]) ≥ H(NA→B) . (265)

The inequality above holds for α ∈ [1/2, 1) ∪ (1,∞) forthe functions in (260) and (262) and for α ∈ (0, 1)∪(1, 2]for the functions in (261) and (263).

Proof. Recall from [Gou19] that, since Θ is doublystochastic, we have that

TrEΛC→AE(IC) = IA, (266)

ΩBE→D (IB ⊗ ρER) = ID ⊗ ρR . (267)

Let Θ be as above, and let us begin by considering theadversarial quantities for the ranges of α for which dataprocessing holds. Let ωR be an arbitrary state. LetξAER ≡ ΛC→AE (ΦCR), and note that the marginal ξAis the maximally mixed state due to (266) and the di-mension constraint |A| = |C|. Therefore, there exists aquantum channel ER→ER such that

ξAER = ER→ER(ΦAR) . (268)

Let σER ≡ ER→ER(ωR). With these notations set, andworking with the specific entropy function in (262), wefind that

Hadv,Φα (Θ [NA→B ]) ≥ −Dα(Θ [NA→B ] (ΦCR) ‖ID ⊗ σR) (269)

23

= −Dα(ΩBE→D NA→B ΛC→AE (ΦCR) ‖ID ⊗ σR) (270)

= −Dα(ΩBE→D NA→B ΛC→AE (ΦCR) ‖ΩBE→D (IB ⊗ σER)) (271)

≥ −Dα(NA→B ΛC→AE (ΦCR) ‖IB ⊗ σER) (272)

= −Dα(NA→B (ξAER) ‖IB ⊗ σER) (273)

= −Dα(ER→ER NA→B (ΦAR) ‖ER→ER(IB ⊗ ωR)) (274)

≥ −Dα(NA→B (ΦAR) ‖IB ⊗ ωR) . (275)

Since the inequality holds for an arbitrary state ωR, weconclude that

Hadv,Φα (Θ [NA→B ]) ≥ Hadv,Φ

α (NA→B), (276)

which is the inequality in (265) for the adversarial ChoiRenyi entropy Hadv,Φ

α (N ). The proof for the entropyfunctions in (259) and (263) goes the same way, since theabove proof only relied upon the data processing inequal-ity.

To arrive at the inequality in (265) for the entropyfunctions in (260)–(261), we exploit the same proof, butwe choose ωR to be the maximally mixed state. By trac-ing over systems AE in (268), we find that

πR = TrAEξAER = TrAEER→ER(ΦAR) (277)

= (TrE ER→ER)(πA), (278)

and so we conclude that the reduced channelTrE ER→ER is unital. This means that, by choos-ing σER ≡ ER→ER(ωR) again, we can conclude thatσR = πR. By applying the same steps as above, we thenfind that

HΦα (Θ [NA→B ]) = −Dα(Θ [NA→B ] (ΦCR) ‖ID ⊗ πR)

(279)

≥ −Dα(NA→B (ΦAR) ‖IB ⊗ πR) (280)

= HΦα (NA→B), (281)

which is the inequality in (265) for the entropy functionin (260). The proof for the entropy function in (261) thengoes the same way.

As a final remark to conclude this section, we note thatthe following limit holds

limα→∞

Hadv,Φα (NA→B) = Hext

min(NA→B), (282)

as a consequence of (89), and so the proof given aboverepresents a different way, from that given in [Gou19], forarriving at the conclusion that the extended min-entropyof a channel is non-decreasing under the action of a dou-bly stochastic superchannel.

IX. CONCLUSION AND OUTLOOK

In this paper, we have introduced a definition for theentropy of a quantum channel, based on the channel rela-tive entropy between the channel of interest and the com-pletely randomizing channel. Building on this approach,

we defined the Renyi and min-entropy of a channel. Weproved that these channel entropies satisfy the axioms forentropy functions, recently put forward in [Gou19]. Wealso proved that the entropy of a channel is equal to thecompletely bounded entropy of [DKJR06], and the Renyientropy of a channel is related to the completely bounded1 → p norm considered in [DKJR06]. The smoothedmin-entropy of a channel satisfies an asymptotic equipar-tition property that generalizes the same property forsmoothed min-entropy of quantum states [TCR09]. Weshowed that the entropy of a channel has an operationalinterpretation in terms of a task called quantum channelmerging, in which the goal is for the receiver to mergehis share of the channel with the environment’s share,and this task extends the known task of quantum statemerging [HOW05, HOW07] from states to channels. Weevaluated the entropy of a channel for several commonchannel models. Finally, we considered other generalizedentropies of a quantum channel and gave further evidencethat Definition 1 is a cogent approach for defining entropyof a quantum channel.

Going forward from here, one of the most interestingopen questions is to determine if there is a set of axiomsthat uniquely identifies the entropy of a quantum chan-nel, similar to how there is a set of axioms that uniquelycharacterizes Shannon entropy [Csi08]. We wonder thesame for the Renyi entropy of a channel, given that theRenyi entropies were originally identified [Ren61] by re-moving one of the axioms that uniquely characterizesShannon entropy. On a different front, one could alterna-tively define the entropy of n uses of a quantum channelin terms of an optimization over quantum co-strategies[GW07, Gut09] or quantum combs [CDP08a], and foranalyzing the asymptotic equipartition property in thisscenario, one could alternatively smooth with respectto the strategy norm of [CDP08a, Gut12]. The resultsof [CMW16] suggest that the asymptotic equipartitionproperty might still hold in this more complex scenario,but further analysis is certainly required. Note that arelated scenario has been considered recently in [CE16].Finally, if the Petz–Renyi channel divergence between anarbitrary channel and the completely depolarizing chan-nel is additive, then a Renyi channel entropy defined fromit would be convincing. This question about Petz–Renyichannel divergence has been open since [CMW16].

24

ACKNOWLEDGMENTS

We are grateful to the local organizers (Graeme Smithand Felix Leditzky) of the Rocky Mountain Summit onQuantum Information, held at JILA, Boulder, Coloradoduring June 2018. We are especially grateful to Xiao

Yuan for notifying us of a gap in a previous proof of (128).GG acknowledges support from the Natural Sciencesand Engineering Research Council of Canada (NSERC).MMW acknowledges support from the National ScienceFoundation under grant no. 1714215.

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Appendix A: Max-mutual information of a channeland the asymptotic equipartition property

In this appendix, we point out how the max-mutualinformation of a quantum channel is a limit of the sand-wiched Renyi mutual information of a channel, the latterhaving been defined in [GW15]. We then show how toarrive at an alternate proof of the asymptotic equiparti-tion property in [FWTB19, Theorem 8] by making useof this connection.

First recall that the sandwiched Renyi mutual infor-mation of a channel is defined for α ∈ (0, 1) ∪ (1,∞) as[GW15, Eq. (3.5)]

Iα(N ) ≡ maxψRA

Iα(R;B)ω, (A1)

where

ωRB ≡ NA→B(ψRA), (A2)

Iα(R;B)ω ≡ minσB

Dα(ωRB‖ωR ⊗ σB), (A3)

where Dα is the sandwiched Renyi relative entropy from(54). It was subsequently used in [CMW16]. The max-mutual information of a channel is equal to [FWTB19,Definition 4]

Imax(N ) ≡ maxψRA

Imax(R;B)ω, (A4)

Imax(R;B)ω ≡ minσB

Dmax(ωRB‖ωR ⊗ σB). (A5)

Proposition 22 For a quantum channel NA→B, the fol-lowing limit holds

Imax(N ) = limα→∞

Iα(N ). (A6)

Proof. To see this, consider that

limα→∞

Iα(N )

= limα→∞

maxψRA

minσB

Dα(NA→B(ψRA)‖ψR ⊗ σB) (A7)

= supα>1

maxψRA

minσB

Dα(NA→B(ψRA)‖ψR ⊗ σB) (A8)

≤ maxψRA

minσB

supα>1

Dα(NA→B(ψRA)‖ψR ⊗ σB) (A9)

= maxψRA

minσB

Dmax(NA→B(ψRA)‖ψR ⊗ σB) (A10)

= Imax(N ). (A11)

Now consider that

limα→∞

Iα(N )

= supα>1

maxψRA

minσB

Dα(NA→B(ψRA)‖ψR ⊗ σB) (A12)

28

≥ supα>1

minσB

Dα(NA→B(ΦRA)‖ΦR ⊗ σB) (A13)

= minσB

supα>1

Dα(NA→B(ΦRA)‖ΦR ⊗ σB) (A14)

= minσB

Dmax(NA→B(ΦRA)‖ΦR ⊗ σB) (A15)

= Imax(N ). (A16)

with the exchange of min and sup in the last line followingfrom [MH11, Corollary A.2]. The last equality followsfrom the remark after [FWTB19, Definition 4].

The smoothed max-mutual information of a quantumchannel NA→B is then defined for ε ∈ (0, 1) as [FWTB19,Definition 5]

Iεmax(N ) ≡ infN : P (N ,N )≤ε

Imax(N ). (A17)

(Here we smooth with respect to purified distance forconvenience.) We then have that [FWTB19, Theorem 8]

limn→∞

1

nIεmax(N⊗n) ≤ I(N ), (A18)

where I(N ) is the mutual information of a channel[AC97], defined as

I(N ) = limα→1

Iα(N ) = supψRA

I(R;B)ω, (A19)

where ωRB ≡ NA→B(ψRA).To arrive at an alternate proof of the upper bound in

[FWTB19, Theorem 8], consider that an application of[Tom15, Eq. (6.92)], definitions, and arguments similar tothose in the first part of Theorem 14 imply the followinginequality for all α > 1 and ε ∈ (0, 1):

Iεmax(N⊗n) ≤ Iα(N⊗n) + f(ε, α) (A20)

= nIα(N ) + f(ε, α), (A21)

where the equality follows from [GW15, Lemma 6] andf(ε, α) is a function of ε and α that vanishes when divid-ing by n and taking the large n limit. Dividing by n and

taking the limit n→∞, we find that

limn→∞

1

nIεmax(N⊗n) ≤ Iα(N ). (A22)

Since the inequality holds for all α > 1, we can take thelimit α → 1, apply (A19), and conclude the bound in(A18).

Appendix B: Data processing of the Choi divergenceunder particular superchannels

Proposition 23 Let Θ be a superchannel of the follow-ing form:

Θ(NA→B) =∑x

p(x) ΩxBE→D NA→B ΛxC→AE (B1)

where p(x) is a probability distribution, and for eachx the map ΩxBE→D is an arbitrary quantum channel,and ΛxC→AE is a unital quantum channel (hence |C| =|A||E|). Then the Choi divergence is monotone undersuch superchannels:

DΦ(N‖M) ≥ DΦ(Θ(N )‖Θ(M)). (B2)

Proof. To prove it, we first prove the monotonicity un-der any superchannel of the form

Υ(NA→B) = ΩBE→D NA→B ΛC→AE (B3)

with ΩBE→D an arbitrary quantum channel, and ΛC→AEa unital quantum channel. Indeed, denoting by ΛtAE→Cthe quantum channel obtained from ΛC→AE by takingthe transpose on each of its Kraus operators, and denot-ing by A, C, and E, replicas of systems A, C, and E, wefind that

DΦ(Υ(N )‖Υ(M))

= D(Υ(NA→B)(ΦCC)‖Υ(MA→B)(ΦCC)) (B4)

= D((ΩBE→D NA→B ΛC→AE)(ΦCC)‖(ΩBE→D MA→B ΛC→AE)(ΦCC)) (B5)

≤ D((NA→B ΛC→AE)(ΦCC)‖(MA→B ΛC→AE)(ΦCC)) (B6)

= D((ΛtAE→C NA→B)(ΦAA ⊗ ΦEE)‖(Λt

AE→C MA→B)(ΦAA ⊗ ΦEE)) (B7)

≤ D(NA→B(ΦAA ⊗ ΦEE)‖MA→B(ΦAA ⊗ ΦEE)) (B8)

= D(NA→B(ΦAA)‖MA→B(ΦAA)) (B9)

= DΦ(N‖M). (B10)

where, in both inequalities, we used data processing of the divergence D, for the second equality we used the

29

relation ΛC→AE(ΦCC) = ΛtAE→C(ΦAA ⊗ ΦEE), and for

the third equality we used the property D(ρ ⊗ ω‖σ ⊗ω) = D(ρ‖σ) [WWY14]. Now, to prove the monotonicity

under Θ as in (B1), we write Θ =∑x p(x)Υx, where each

Υx has the form (B3). With this notation, we find that

DΦ(Θ(N )‖Θ(M)) = D

(∑x

p(x)Υx(NA→B)(ΦCC)∥∥∥∑

x

p(x)Υx(MA→B)(ΦCC)

)= (B11)

D

(TrX

[∑x

p(x)Υx(NA→B)(ΦCC)⊗ |x〉〈x|X]∥∥∥TrX

[∑x

p(x)Υx(MA→B)(ΦCC)⊗ |x〉〈x|X])

(B12)

≤ D

(∑x

p(x)Υx(NA→B)(ΦCC)⊗ |x〉〈x|X∥∥∥∑

x

p(x)Υx(MA→B)(ΦCC)⊗ |x〉〈x|X

)(B13)

≤ D

(∑x

p(x)NA→B(ΦAA)⊗ |x〉〈x|X∥∥∥∑

x

p(x)MA→B(ΦAA)⊗ |x〉〈x|X

)(B14)

= D(NA→B(ΦAA)

∥∥∥MA→B(ΦAA))

(B15)

= DΦ(N‖M). (B16)

where, in the first inequality, we used the monotonicity ofthe divergence under data processing, and for the secondinequality, we used the monotonicity under maps of theform in (B3).

Appendix C: Optimizing the adversarial channeldivergence

By definition, we always have that

Dadv(N‖M) ≥ supψRA

infσRA

D(NA→B(ψRA)‖MA→B(σRA)),

(C1)where ψRA is pure with system R isomorphic tosystem A.

To see the claim after (242), let ρRA be an arbitrarystate with purification φR′RA. It thus holds that φR′RAis a purification of ρA, with R′R acting as the purifyingsystems. By taking a “canonical” purification of ρA thatis in direct correspondence with its eigendecomposition,there exists a purification ϕSA of ρA with system S iso-morphic to system A. Since the purification φR′RA is re-lated by an isometric channel US→R′R to the purificationϕSA as φR′RA = US→R′R(ϕSA) and applying the isomet-ric invariance of generalized divergences [TWW17], weconclude for an arbitrary state ωSA that

D(NA→B(ϕSA)‖MA→B(ωSA))

= D((US→R′R NA→B)(ϕSA)‖(US→R′R MA→B)(ωSA))(C2)

= D(NA→B(φR′RA)‖MA→B(US→R′R(ωSA))) (C3)

≥ D(NA→B(ρRA)‖MA→B((TrR′ US→R′R)(ωSA)))(C4)

≥ infσRA

D(NA→B(ρRA)‖MA→B(σRA)). (C5)

The first inequality is from data processing under thepartial trace over R′. Since the inequality holds for arbi-trary ωSA, we conclude that

infσRA

D(NA→B(ρRA)‖MA→B(σRA)) ≤

infωSA

D(NA→B(ϕSA)‖MA→B(ωSA)). (C6)

We can then take a supremum to conclude that

infσRA

D(NA→B(ρRA)‖MA→B(σRA)) ≤

supϕSA

infωSA

D(NA→B(ϕSA)‖MA→B(ωSA)). (C7)

Since the inequality holds for an arbitrary choice of ρRA,we conclude that

Dadv(N‖M) ≤ supϕSA

infωSA

D(NA→B(ϕSA)‖MA→B(ωSA)).

(C8)This concludes the proof.