Structure of Reversible Computation Determines the Self-Duality of Quantum Theory

Markus P. Muller1 and Cozmin Ududec1

1Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada(Received 6 January 2012; revised manuscript received 22 February 2012; published 27 March 2012)

Predictions for measurement outcomes in physical theories are usually computed by combining two

distinct notions: a state, describing the physical system, and an observable, describing the measurement

which is performed. In quantum theory, however, both notions are in some sense identical: outcome

probabilities are given by the overlap between two state vectors—quantum theory is self-dual. In this

Letter, we show that this notion of self-duality can be understood from a dynamical point of view. We

prove that self-duality follows from a computational primitive called bit symmetry: every logical bit can

be mapped to any other logical bit by a reversible transformation. Specifically, we consider probabilistic

theories more general than quantum theory, and prove that every bit-symmetric theory must necessarily be

self-dual. We also show that bit symmetry yields stronger restrictions on the set of allowed bipartite states

than the no-signalling principle alone, suggesting reversible time evolution as a possible reason for

limitations of nonlocality.

DOI: 10.1103/PhysRevLett.108.130401 PACS numbers: 03.65.Ta, 03.65.Ca, 03.65.Ud, 03.67.Lx

A central idea of every statistical physical theory is thedistinction between states and observables. If we perform ameasurement on a physical system, the state describesthe preparation of the system, while the observable corre-sponds to our choice of measurement. Combining the two,we obtain expectation values of measurement outcomes.

In principle, states and observables are fundamentallydistinct objects. However, in quantum theory, they turn outto be identical: transition probabilities between two statesj’i and jc i are given by the overlap

Prob ðc ! ’Þ ¼ jh’jc ij2 ¼ Trðj’ih’jjc ihc jÞ: (1)

More generally, the probability of obtaining an outcomedescribed by the projector or effect operator P, measuredon a (mixed) quantum state �, is given by Trð�PÞ. It isremarkable that state � and observable P are described bythe same mathematical objects: up to normalization, theyare both arbitrary positive semidefinite operators [1].This property of self-duality, which is most obvious inthe special case (1), lies at the very heart of quantumtheory, and can be understood as the main ingredient inthe Born rule.

In this Letter we show that this remarkable property canbe understood in information-theoretic terms: self-dualityis a consequence of a certain computational primitive thatwe call bit symmetry. Every theory that satisfies bit sym-metry—which we argue is necessary to allow for powerfulcomputation—must be self-dual. We also prove that bitsymmetry restricts the set of possible bipartite states in alltheories with nonlocality, including quantum theory.

General probabilistic theories.—Almost any conceiv-able statistical physical theory, including quantum theoryand classical probability theory as special cases, can bedescribed within the framework of general probabilistictheories [2–6]. The main physical notions are preparations,

transformations, and measurements. Any physical systemis described by a finite-dimensional real vector space A.The possible preparation procedures are represented by aset of normalized states �A � A (in quantum theory, A isthe set of self-adjoint operators on some Hilbert space,while �A is the set of density matrices). If we have twostates ’,! 2 �A, we can think of a device which prepareseither state ’ with probability p, or ! with probability1� p, yielding the state p’þ ð1� pÞ! [5]. Therefore,state spaces are convex. Similarly as in quantum theory,states will be called mixed if they can be written as aconvex combination of this form for some 0< p< 1 and’ � !, and otherwise pure. We also assume that statespaces are compact, which implies that every state can bewritten as a finite convex combination of pure states [3].It is important for calculations to include unnormalized

states in the framework, that is, elements of the form � �!for � � 0 and ! 2 �A. The set of all these elements iscalled Aþ. It is closed with respect to sums and convexcombinations—in convex geometry, sets of this kind arecalled cones. We assume that Aþ spans the whole space A.In quantum theory, Aþ is the set of positive semidefinitematrices.In order to describe observables, consider any measure-

ment with several possible outcomes that we perform on astate !. Denote by Eð!Þ the probability of obtaining oneparticular outcome. This must be a number between 0and 1, and it must respect probabilistic mixtures: E½p’þð1� pÞ!� ¼ pEð’Þ þ ð1� pÞEð!Þ; that is, E must belinear [2]. Linear maps E:A ! R (i.e., functionals) whichare non-negative on all of Aþ will be called effects, and theset of all effects is denoted A�þ. It is easy to see that A�þ isagain a cone—in convex geometry terms, it is called thedual cone of Aþ [7]. The normalization of states is deter-mined by the unit u, a particular element of A�þ which

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assigns the value one to all normalized states: uð!Þ ¼ 1 forall ! 2 �A [in quantum theory, we have uð�Þ � Trð�Þ].An effect E 2 A�þ is called a proper effect if 0�Eð!Þ�1for all states ! 2 �A.

In quantum theory, all effects can be written as maps� � Trð�PÞ, where P � 0 is a positive semidefinite ma-trix; it is proper iff P � 1. Identifying this effect with thematrix P, we see that A�þ can be identified with the set ofpositive semidefinite matrices, such that Aþ ’ A�þ. This isthe notion of self-duality which will be studied in moredetail in the next section. At this point, however, it isimportant to note that Aþ and A�þ can be very different ingeneral. As an example, consider a state space

�A :¼ fðx1; x2; 1ÞT 2 R3j � 1 � x1; x2 � 1g: (2)

This state space looks like a square. It contains four purestates, for example ! ¼ ð1; 1; 1ÞT and ’ ¼ ð�1;�1; 1ÞT ,and has unit uðxÞ :¼ x3. Using the standard inner producton A ¼ R3 and the pure state!, we can define a linear mapE! by E!ðxÞ :¼ h!; xi ¼ x1 þ x2 þ x3. Even though ! isa valid state, E! is not a valid effect: for example E!ð’Þ ¼�1 6�0. For the square state space, Aþ and A�þ cannot beidentified in this way—they will be different no matterwhich inner product we use [8].

Self-duality.—Building on the previous examples, wedefine a system A to be self-dual [9] iff there is some innerproduct h�; �i on A such that the set of effects (representedas vectors in A) agrees with the set of states, A�þ ¼ Aþ;that is,

A�þ ¼ f! � h!;’ij’ 2 Aþg:Quantum theory is self-dual. To see this, recall that for ann-level quantum system, the real vector space A is the setof self-adjoint n n matrices. Consider the Hilbert-Schmidt inner product on A, given by hX; Yi :¼ TrðXYÞ.As we have seen above, under this inner product, we canidentify Aþ and A�þ: both are the set of positive semi-definite matrices.

As another example, it can be shown that the square statespace (2) is not self-dual [8], as already indicated. Moregenerally, regular polygons with n vertices are self-dual ifand only if n is odd. This will become important below.

Bit symmetry.—In addition to preparations (states) andmeasurements (effects), physical theories also contain anotion of transformations. Transformations describe on theone hand possible physical time evolution, and on the otherhand possible computations that can be accomplished inthe respective theory. In this Letter, we will only considerreversible transformations. This is motivated by the factthat time evolution in our Universe seems to be fundamen-tally reversible, and also by the conceptual analogy to thereversible circuit model in quantum computation.

Transformations must be linear (since they must respectprobabilistic mixtures [2]), preserve the normalization, andmap states to states. For reversible transformations T, this

must also be true for their inverses. Consequently, theymust be symmetries of the state space: Tð�AÞ ¼ �A.Therefore, the set of reversible transformations on a systemA is a groupGA, which is a subgroup of all symmetries. Weassume that GA is compact, which may be motivated onphysical grounds [10]. In quantum theory, GA is the groupof unitaries.We are interested in a particular type of symmetry which

connects all logical bits. To this end, we call two states ’,! 2 �A perfectly distinguishable if there is a proper effectE such that Eð’Þ ¼ 0 and Eð!Þ ¼ 1—that is, if there is aconceivable measurement device that distinguishes ’ and! perfectly in a single run. Since all states c have 0 �Eðc Þ � 1, the states ’ and !must lie on opposite sides ofstate space: the set of vectors x 2 A with EðxÞ ¼ 1 respec-tively EðxÞ ¼ 0 are two parallel supporting hyperplanes,touching the state space in ’ and !, with the full statespace lying in between, as sketched in Fig. 1.Every pair of pure and perfectly distinguishable states ’

and! generates a logical bit: in terms of convex geometry,this is the face generated by ’ and !, that is, the smallestface [11] of �A containing both ’ and !. In quantumtheory, two pure states j’ih’j and j!ih!j are perfectlydistinguishable if and only if h’j!i ¼ 0. The logical bitthat they generate is not simply the line segment making uptheir convex hull, but contains all pure states of the form�j’i þ �j!i and their convex mixtures—that is, a fullBloch ball [12].

FIG. 1. Two state spaces: one is a square, the other a pentagon.Shown are pairs of perfectly distinguishable states !, ’ and !0,’0. For the square, there is no symmetry which maps the pair !,’ to the pair !0, ’0: the square state space is not bit symmetric.For the pentagon, the pair !, ’ is mapped to !0, ’0 by areflection across a symmetry axis. All pairs of perfectly distin-guishable pure states can be mapped to each other—the penta-gon is bit symmetric. The dotted lines denote the level sets of ameasurement effect E which distinguishes ! and ’ (and, acci-dentally, also !0 and ’0). That is, the line containing ! isfx:EðxÞ ¼ 1g, and the line containing ’ is fx:EðxÞ ¼ 0g. Forthe square state space, there are two types of inequivalent logicalbits: lines generated by adjacent pure states like !, ’, and thesquare itself which is generated by diametral states like !0, ’0.For the pentagon—and any other bit-symmetric theory—alllogical bits generated by pairs of perfectly distinguishable purestates are isometric (in this case, all pairs generate the fullpentagon).

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Now we are ready to define our main notion: a system Ais called bit symmetric, if one of the two following equiva-lent conditions holds: (1) If ’, ! are perfectly distinguish-able pure states, and so are ’0,!0, then there is a reversibletransformation T 2 GA such that T’ ¼ ’0 and T! ¼ !0.(2) Every logical bit can be mapped to every other logicalbit by some reversible transformation.

Quantum theory is obviously bit symmetric: every pairof orthogonal pure states can be mapped to every other bysome unitary. It is even more symmetric than this: analo-gous statements hold for triples, quadruples, etc., of or-thogonal pure states. As a less trivial example, considerstate spaces that are regular polygons with n vertices. Itturns out that these state spaces are bit symmetric if andonly if n is odd. In Fig. 1, this is illustrated for n ¼ 4 andn ¼ 5, i.e., for the square and the pentagon.

Classical probability theory is bit symmetric as well: then-outcome state space is the set of probability distributionsðp1; . . . ; pnÞ, Pipi ¼ 1, pi � 0. Geometrically, this con-vex set is a simplex, and the pure states are of the formð0; . . . ; 0; 1; 0; . . . ; 0Þ (full weight on one outcome). Thereversible transformations are the permutations of the nentries, which can map every pair of pure states to everyother. In fact, these ‘‘transpositions’’ generate the fullgroup of permutations.

As the last example illustrates, bit symmetry is an im-portant and basic computational primitive. In the context ofquantum computation, it implies that any ‘‘entangled’’logical bit that appears in a computation on many qubitscan in principle be mapped to the first qubit (awaiting afinal measurement) without destroying coherence. In gen-eral theories, bit symmetry means that yes-no questionswhich can be answered perfectly by (irreversible) mea-surements may in principle also be asked ‘‘coherently’’ andbe part of a larger reversible computation. In physicalterms, it means that the state of any natural two-levelsystem can be transferred to any other two-level systemby a suitable reversible interaction. One may argue that thetime evolution of the Universe would be severely con-strained if this property did not hold.

Main Result.—Now we prove our main theorem:Theorem 1. If a state space is bit symmetric, then it isalso self-dual.

Moreover, the corresponding inner product can bechosen to be non-negative on all states, invariant underall reversible transformations, and to satisfy h!;!i ¼ 1 forall pure states ! and h!;’i ¼ 0 if ! and ’ are perfectlydistinguishable.

Remark.—In quantum theory, h�; �i is theHilbert-Schmidtinner product between self-adjoint matrices: hX; Yi ¼TrðXYÞ; invariance means that hUXUy;UYUyi¼hX;Yifor all unitaries U. In all bit-symmetric theories, if one of! and ’ is pure, then h!;’i ¼ 0 implies that ! and ’ areperfectly distinguishable. However, we were not able toprove that the same holds true in general if both are mixed.

Proof.—If ! 2 �A is any pure state, then there is al-ways another pure state ’ that is perfectly distinguishablefrom! (unless the state space contains only a single point).Thus, bit symmetry implies transitivity: to every pair ofpure states !, c , there is a reversible transformation T 2GA such that T! ¼ c . This allows us to define a maxi-mally mixed state �A as �A :¼ R

T2GAT!dT, where ! 2

�A is any pure state. Because of transitivity, �A does notdepend on the choice of !. For every state !, define itsBloch vector ! :¼ !��A. Then we can decompose the

space A into A ¼ A R ��A, where A is the set of allpoints x 2 A with uðxÞ ¼ 0, with u the unit on A. If ! is a

state, then its Bloch vector ! is an element of A.Since reversible transformations preserve normalization,

they leave the subspace A invariant. According to grouprepresentation theory [13], there is an inner product ð�; �Þ onA such that ðTx; TyÞ ¼ ðx; yÞ for all T 2 GA and x, y 2 A.We may scale this product by an arbitrary positive factorsuch that ð!; !Þ ¼ 1 for all pure states! (they all have thesame inner product due to transitivity).Define c :¼ min!;’2�A

ð!; ’Þ � ð�A; �AÞ ¼ 0 to be the

minimal inner product between the Bloch vectors of anytwo states. Our next step is to prove the following state-ments: (i) For all! and ’, we have c � ð!; ’Þ � 1, wherec < 0. (ii) If ! is pure and ’ is arbitrary and ð!; ’Þ ¼ c,then ! and ’ are perfectly distinguishable. (iii) If !and ’ are arbitrary perfectly distinguishable states, thenð!; ’Þ ¼ c.To this end, define a linear map E!:A ! R for every

pure state ! by the linear extension of

E!ð’Þ :¼ ð!; ’Þ � c

1� cð’ 2 �AÞ:

Since c � 0, this is well defined, and since ð!; ’Þ � c, wehave E!ð’Þ � 0 for all ’ 2 �A. Because of convexity of

the norm k ! k� ffiffiffiffiffiffiffiffiffiffiffiffiffiffih!; !ip, all mixed states! satisfy k ! k

� 1, with equality for the pure states. Thus, the Cauchy-Schwarz inequality implies ð!; ’Þ �k ! k � k ’ k� 1;hence, E!ð’Þ � 1 for all ’ 2 �A. In other words, forevery pure state !, the map E! is a proper effect. Nowsuppose that ! 2 �A is pure and ’ 2 �A is arbitrary, andð!; ’Þ ¼ c. Then E!ð’Þ ¼ 0 and E!ð!Þ ¼ 1, hence’ and! are perfectly distinguishable. This proves (ii). Moreover,if c ¼ 0, we would have ð!; �AÞ ¼ 0 ¼ c, and so ! and�A would be perfectly distinguishable, which is impos-sible. Hence c < 0, proving (i).Choose !, ’ 2 �A such that ð!; ’Þ ¼ c. We can de-

compose! and’ into pure states!i and’j:! ¼ Pi�i!i,

’ ¼ Pj�j’j with �i,�j > 0. Since c ¼ P

ij�i�jð!i; ’jÞ,and c is the minimal possible value, every addend musthave this value due to convexity, so ð!i; ’jÞ ¼ c for all i, j.

Thus !i and ’j are pure and perfectly distinguishable.

Choose some pair of indices i, j. Now if !0 and ’0 areanother pair of pure and perfectly distinguishable states,

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there is a reversible transformation T such that T!i ¼ !0and T’j ¼ ’0, hence ð!0; ’0Þ ¼ ðT!i; T’jÞ ¼ ð!i; ’jÞ ¼c. That is, every pair of perfectly distinguishable purestates has the inner product c between its Bloch vectors.Now suppose that ! and ’ are arbitrary perfectly distin-guishable states. Decomposing them as above, it followsthat every !i is perfectly distinguishable from every ’j,

hence ð!; ’Þ ¼ Pij�i�jð!i; ’jÞ ¼ c. This proves state-

ment (iii).Let E be any effect such that Rþ

0 � E is an exposed ray

of A�þ. That is, there is some x 2 A with the followingproperty:

F 2 A�þ; FðxÞ ¼ 0 ) F ¼ �E for some � � 0: (3)

The point x defines a supporting hyperplane of A�þ, touch-ing it in the ray generated by E. Thus, either FðxÞ � 0 forall F 2 A�þ, or FðxÞ � 0 for all F 2 A�þ. In the last case,we can redefine x � ð�xÞ, such that FðxÞ � 0 for all F 2A�þ, or, in other words, x 2 ðA�þÞ� ¼ Aþ. Since x � 0, wehave uðxÞ � 0, and ! :¼ x=uðxÞ defines a state ! 2 �A

which depends on E, and will be mixed in general.Set � :¼ max’2�A

Eð’Þ> 0 and F :¼ E=�, then

Fð!Þ ¼ 0, and the set of states ’ with Fð’Þ ¼ 1 is anonempty face of �A. Let !

0 be any extremal point ofthat face, then it is a pure state which is by constructionperfectly distinguishable from !. Hence ð!; !0Þ ¼ c, andso E!0 ðxÞ ¼ uðxÞE!0 ð!Þ ¼ 0. Because of (3), it followsthat there is some � � 0 such that E!0 ¼ �E. We havethus shown that every ray-exposed effect is of the form�0E!0 for some �0 > 0 and pure state !0. According toStraszewicz’s theorem [14], the exposed rays are dense inthe set of extremal rays; hence, every ray-extremal effect isof this form.

Now we extend ð�; �Þ to an inner product h�; �i on all of A.If x, y 2 A, use the decomposition x ¼ x0�

A þ xwith x 2A (and similarly for y) and define hx; yi :¼ �x0y0 þ ð1��Þðx; yÞ, where � :¼ �c=ð1� cÞ 2 ð0; 1Þ, since c < 0. Itis easy to check that this is an inner product, satisfying allstatements of the theorem. We can now identify linear

functionals L:A ! R with vectors ~L 2 A via LðxÞ ¼h ~L; xi. Every ray-extremal effect is of the form E!ð’Þ ¼h!;’i for some pure state !, hence ~E! ¼ !. Thus, in thisidentification, all extremal rays of A�þ are contained in Aþ.Since they generate the full cone A�þ, we have A�þ � Aþ.On the other hand, consider an extremal ray of Aþ; it isspanned by some pure state!. By construction, h!;’i � 0for all !, ’ 2 �A; hence, the corresponding effect E! iscontained in A�þ. Thus, Aþ � A�þ. In summary, we getAþ ¼ A�þ under the inner product h�; �i—that is, A isself-dual. h

In low dimensions, bit-symmetric state spaces are rare.Using the classification of transitive state spaces in [15], itfollows that the only bit-symmetric two-dimensional statespaces are the unit disc and the regular polygons with anodd number of vertices. In three dimensions, there is only

the unit ball (representing a qubit) and the unique regularself-dual polytope, the tetrahedron (representing a classical4-level system). For a different set of postulates leading toself-duality, see [16].Nonlocality.—Given two state spaces A and B, we can

consider the set of all joint states (that is, correlations) onAB which are consistent with the no-signalling principle[3]; this is called the maximal tensor product A �max B of Aand B. Explicitly, �AB is the set of all ! 2 A � B withuA � uBð!Þ ¼ 1 and EA � EBð!Þ � 0 for all EA 2 A�þ,EB 2 B�þ.If A andB are the square state space (2), then A �max B is

called the ‘‘no-signalling polytope.’’ It contains so-calledPopescu-Rohrlich boxes [2] which violate the Bell-CHSHinequality [8] by more than any quantum state. It has beenasked why quantum theory does not allow for such ‘‘maxi-mally nonlocal’’ states. The following theorem generalizesthe results in [17]:Theorem 2. The maximal tensor product A �max B of two

state spaces can only be bit symmetric if it does not containany entangled states at all.Proof.—From the definition of �AB, it follows that all

extremal rays of the effect cone ðABÞ�þ are of the formEA � EB. If �AB is bit symmetric, then it is self-dual;hence, all pure states (generating the state cone) are prod-uct states. Since all states are mixtures of those, they mustbe unentangled. hIf A and B are classical nA- and nB-level systems, then

A �max B is a classical nAnB-level system. It is bit sym-metric, but does not contain any entangled states. On theother hand, any bit-symmetric composition AB of two statespaces A and B which does contain entangled states (suchas quantum theory) must be a proper subset of A �max B—there are at least some maximally nonlocal states ofA �max B which AB cannot contain.While the omission of some states of A �max B does not

in itself necessarily reduce the amount of nonlocality in a

theory [18], this result still gives a hint that bit symmetry

might introduce constraints on the amount of Bell inequal-

ity violations. This conjecture is further substantiated by

the findings in [8], where it was shown that a class of

composites of regular n-gons as in Fig. 1 satisfies the

Tsirelson bound if and only if n is odd; i.e., the theory is

locally bit symmetric.Conclusions.—We have shown that self-duality, one of

the defining features of quantum theory [16], follows fromthe computational primitive of bit symmetry. Thus, thepower of reversible computation (or, equivalently, timeevolution) severely constrains the statistical behavior ofany physical theory.We have also proven that bit symmetryrestricts the set of allowed bipartite states, leaving theinteresting open problem of quantifying the consequencesfor violations of Bell inequalities.We would like to thank Christian Gogolin, Peter Janotta,

Lluıs Masanes, Jonathan Oppenheim, Tony Short, and

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Stephanie Wehner for discussions. Research at PerimeterInstitute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontariothrough the Ministry of Research and Innovation.

[1] Here we only consider finite-dimensional systems, wherethere is no distinction between bounded and trace-classoperators.

[2] J. Barrett, Phys. Rev. A 75, 032304 (2007).[3] H. Barnum, J. Barrett, M. Leifer, and A. Wilce, Phys. Rev.

Lett. 99, 240501 (2007).[4] G. Mackey, Mathematical Foundations of Quantum

Mechanics (Addison-Wesley, Reading, MA, 1963).[5] A. S. Holevo, Probabilistic and Statistical Aspects of

Quantum Theory (North-Holland, New York, 1980).[6] L. Hardy, arXiv:quant-ph/0101012.[7] C. D. Aliprantis, R. Tourky, Cones and Duality (American

Mathematical Society, Providence, 2007).[8] P. Janotta, C. Gogolin, J. Barrett, and N. Brunner, New J.

Phys. 13, 063024 (2011).

[9] In the relevant literature, this is usually called strong self-duality, as opposed to a certain weaker form of self-duality. However, since we do not study this weaker notionof self-duality in this Letter, we drop the prefix strong.

[10] M. P. Muller, O. C. O. Dahlsten, and V. Vedral,arXiv:1107.6029.

[11] A face of �A is a convex subset B � �A such that ½x; y 2�A; � > 0; �xþ ð1� �Þy 2 B� ) x; y 2 B.

[12] Technically, when we talk about a logical bit, we alsoassume a fixed choice of perfectly distinguishable purestates ’, ! in that face, analogous to a choice of ‘‘basisstates’’ in quantum theory.

[13] B. Simon, Representations of Finite and Compact Groups(American Mathematical Society, Providence, 1996).

[14] R. Webster, Convexity (Oxford University Press, NewYork, 1994).

[15] G. Kimura and K. Nuida, arXiv:1012.5350.[16] A. Wilce, arXiv:0912.5530.[17] D. Gross, M. Muller, R. Colbeck, and O. C.O. Dahlsten,

Phys. Rev. Lett. 104, 080402 (2010).[18] H. Barnum, S. Beigi, S. Boixo, M.B. Elliott, and S.

Wehner, Phys. Rev. Lett. 104, 140401 (2010).

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