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Information causality and its tests for quantum communications. I- Ching Yu Host : Prof. Chi-Yee Cheung Collaborators: Prof. Feng -Li Lin (NTNU) Prof. Li-Yi Hsu (CYCU). Outline-I. Information Causality (IC) and quantum correlations - PowerPoint PPT Presentation
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INFORMATION CAUSALITY AND ITS TESTS FOR QUANTUMCOMMUNICATIONS
I- Ching YuHost : Prof. Chi-Yee Cheung Collaborators: Prof. Feng-Li Lin (NTNU) Prof. Li-Yi Hsu (CYCU)
Outline-I
1. Information Causality (IC) and quantum correlations Quantum non-locality No-signaling theory Information Causality (IC)
2. IC and the signal decay theorem IC and signal decay theorem The generalized Tsirelson-type inequality
Outline-II
3. Testing IC for general quantum communication protocols Feasibility for maximizing mutual information by convex
optimization? Solutions Solution (i): Solution (ii):
4. The conclusion
Quantum non-locality-IThe violation of the Bell-type inequality
The CHSH inequalityThe measurement scenario
x y
x,y=0,1A , B 1, 1
Quantum non-locality-II The local hidden variable theory:
0 1 0 1
0 0 0 1 1 0 1 1
0,0 0,1 1,0 1,1
x,y x y x y
A ,A , B , B {1, 1}
A B + A B + A B - A B
C C C C 2
C Pr(A B | x, y) Pr(A B | x, y)
If A0, A1, B0, B1=1, -1, then Cxy=<Ax By>; CHSH=(A0 + A1) B0 + (A0 − A1) B1; |<CHSH>| ≤ 2
Quantum non-locality-III The maximal amount of quantum violation- Tsirelson
bound
Since
0 0 0 1 0 1 1 1A B + A B + A B - A B 2 2
x y
0 0 0 1 0 1 1 1
A B [1, 1]
A B + A B + A B - A B 4
Why Quantum mechanics cannot be more nonlocal?
No-signaling theory-I The speed of the propagating information
cannot be faster than the light speed
To be specific, despite of any non-local correlations previously shared between Alice and Bob, Alice cannot signal to the distant Bob by her choice of inputs due to the no-signaling theory.
x yPr(A , B | x, y)
No-signaling theory-I I Does the no-signaling theory limit the
quantum non-locality? The PR-box:
x y
0,0 0,1 1,0 1,1
x,y=0,1A , B 0,1
C C C C 4
Information Causality-I What is Information Causality (IC)?In the communication protocol, the information
gain cannot exceed the amount of classical communication.
Information Causality-IIThe Random Access Code (RAC) protocol
•Alice prepares a data base { } in secret.•She sends Bob a bit •Bob decode Alice’s bit ay by •Bob is successful only ifi.e.,
0 1, 0,1a a
0 xa A
yB
x yA B xy
yB 0
0
0 0 1( )
x ya A B
a xya a a y
0
1
01
y ay a
IC says total mutual information between Bob’s guess bit β and Alice’s database is bounded by 1, i.e., 1
0
( ; | ) 1k
ii
I I a y i
x y
0 1
0 1
The PR-box :Pr(A B xy | x, y) 1 x, y
I( ;a | b 0) I( ;a | b 1) 1I( ;a | b 0) I( ;a | b 1) 2 1
IC is violated!
Information Causality-IIIFor binary quantum system
with two measurement settings per side
IC is satisfied by quantum mechanics. IC is violated by PR-box. The Tsirelson bound is consistent with IC.
IC could be the physical principle to distinguish quantum correlations from the non-quantum (non-local) correlations.
Information Causality and signal decay theory
MULTI-SETTING RAC PROTOCOL
Alice encodes her database by x(i+1)=a0+ai, i=0,…,k-1.
Bob encodes his given bit b as a k-1 bits string y.
NOISE PARAMETER Bob’s success probability to guess is
Define the coding noise parameter to be
The result:
The noise parameter and the CHSH inequality
0 1 0,0 0,1 1,0 1,11 (C C C C )2
ba1P Pr[ ] Pr( | , )2
b b x yx
a y A B xy x y
2 Pr[ ] 1 y ba y
,1 ( 1)2
xyy x y
x
C
IC and signal propagation
Binary symmetric channel The signal decay theorem
2( ; )( ; ) ( ; ) , ( ; )
I X ZI X Z I X YI X Y
2( ; | )i yI a y i
IC yields:
2 1.y
y
I
Binary symmetric channel
] W. Evans and L. J. Schulman, Proceedings of the 34th Annual Symposium on Foundations of Computer Science, 594 (1993).
Y Z
2 Pr[ ] 1 y ba y b
Generalized Tsirelson-type inequalities
Multi-setting Tsirelson-type inequalities Using the Cauchy-Schwarz inequality, we
can obtain
When k=2,
{ }y
y
k ,
{ , }
1 ( 1)dim{ }
x yx y
x y
C kx
1,
{ , }
( 1) 2x y kx y
x y
C k
1,
{ , }
0,0 0,1 1,0 1,1
( 1) 2
2 2
x y kx y
x y
C k
C C C C
The Tsirelson inequality
Checking the bound by semidefinite programing (SDP)
SDP can solve the problem of optimizing a linear function which subject to the constraint that the combination of symmetric matrices is positive semidefinite.
We use the same method proposed by Stephanie Wehner to calculate the quantum bound .
S. Wehner, Phys. Rev. A 73, 022110 (2006). It is consistent with the bound from IC !!
TESTING IC FOR GENERAL QUANTUM COMMUNICATION PROTOCOLS
More general quantum communication protocols and IC
For multi-level quantum communication protocols, IC is satisfied by quantum correlation? saturated?
FEASIBILITY FOR MAXIMIZING MUTUAL INFORMATION BY CONVEX OPTIMIZATION?
We use the convex optimization to maximize the mutual information (I) over Alice’s input probabilities and quantum joint probability from NS-box.
Minimizing a convex function with the equality or inequality constraints is called convex optimization.
The solutions We could find a concave function: the Bell-type function
which is monotonically increasing to the mutual information (I) and maximize it over all quantum joint probability of NS-box and Alice’s input probabilities.
Brutal force : without relying on convex optimization. We have to pick up all the sets of quantum and input marginal probability and then evaluate the corresponding mutual information (I).
x yPr(A , B | x, y)
iPr(a )
The Bell-type function -I
If , Bob can guess perfectly.
The symmetric channel
Multi-level RAC protocol
The signal decay theorem for di-nary channel
i i 0x a a
0xA a
yB
ia {0,1,...,d 1}
2 22
( ; ) ( ; | ) log( ; )
i y
I X Z I a y i dI X Y
(d 1) 1 1Pr(Z i | Y i) , Pr(Z i | Y i)d d
22 2IC: (log ) log .
yy
I d dy x
B -A =x y
ba
The Bell-type function -I I Using the Cauchy-Schwarz inequality, we can
obtain
If , and is uniform, we then prove the mutual information (I) is monotonically increasing with the noise parameter .
{ }
yy
k
y
iPr(a )
Finding the quantum bound and Maximal mutual information
Quantum mechanics satisfies IC
Using the quantum constraints of the joint probabilities of NS-box proposed by
One can write down the constraints in convex optimization problem and find the quantum bound of the Bell-type function.
Moreover one can calculate the associated mutual information.
Result: The associated maximal mutual information is less than the bound from information causality.
The solutions We could find a concave function which is
monotonically increasing to the mutual information (I) and then evaluate the corresponding mutual information (I).
For example: the object of quantum non-locality.
Brutal force : without relying on convex optimization. We have to pick up all the sets of quantum correlation and input marginal probability and then evaluate the corresponding mutual information (I).
x yPr(A , B | x, y)
iPr(a )
Testing IC for different cases Fixed input: symmetric channels with i.i.d.
and uniform input marginal probabilities Fixed joint probability: with non-uniform input
marginal probabilities. The most general case.
The condition for d=2 k=2 quantum correlations The necessary and sufficient condition for correlation
functions
Symmetric channels with i.i.d. and uniform input marginal probabilities
The non-locality is characterized by the CHSH function.
The red part can be achieved also by sharing the local correlation.
The maximal mutual information for the local or quantum correlations is bound by 1. IC is saturated.
When
1. The mutual information is not monotonically related to the quantum non-locality.
2. The more quantum non-locality may not always yield the more mutual information.
Non-locality
CHSH=max quantum non-localityI 0.8CHSH=marginally non-localI=1
0,0 0,1 1,0 1,1
0,0 0,1 1,0 1,1
0,0 0,1 1,0 1,1
0,0 0,1 1,0 1,1
C C C C 2
C C C C 2
C C C C 2
C C C C 2
Case with non-uniform input marginal probabilities
Symmetric channels
Asymmetric channels
i i
0 0 1
1 1 0
P( =0|a =0,b=i)=P( =1|a =1,b=i)I( ;a |b=0) only depends on a not aI( ;a |b=1) only depends on a not a
i i
0 0 1
1 0 1
0 1 0 1
P( =0|a =0,b=i) P( =1|a =1,b=i)I( ;a |b=0) depends both a and aI( ;a |b=1) depends both a and aI( ;a |b=0)+I( ;a |b=1) I( ;a |b=0)+I( ;a |b=1) (symmetric case)
The most general channels By partitioning the defining domains of the
probabilities into 100 points.
We find IC is saturated.
0 1max I( ;a |b=0)+I( ;a |b=1)=1
The conclusion We combine IC and the signal decay theorem and
then obtain a series of Tsirelson-type inequalities for two-level and bi-partite quantum systems.
For the quantum communication protocols discussed in our work, the IC is never violated. Thus, IC is supported and could be treated as a physical principle to single out quantum mechanics.
We also find that the IC is saturated not for the case with the associated Tsirelson bound but for the case saturating the local bound of the CHSH inequality. Sharing more non-local correlation does not imply the better performance in our communication protocols.
Thanks for your listening
The hierarchical quantum constraints
How to know the given joint probabilitiescould be reproduced by quantum system?A: Unless they satisfy a hierarchical quantum
constraints.
The quantum constraints of joint probabilities come from the property of projection operators.
Hermiticity: Orthogonality: Completeness: Commutativity:
x yPr(A , B | x, y)
x yx y A BPr(A , B | x,y) Tr(E E )
x x y y
† †A A B BE E , E E
x x x x xA A ' A ,A ' A x xE E E if A ,A ' have same input x
x y
x y
A BA for same x B for same y
E 1, E 1
x yA B[E ,E ]=0
The hierarchical quantum constraints The constraint becomes stronger than the
previous step of the hierarchical constraint.
Q1Q2Q3
From noisy communication to noisy computation
von Neumann suggested that the error of the computation should be treated by thermodynamical method as the treatment for the communication in Shannon's work. This means that, for the noisy computation, one should use some information-theoretic methods related to the noisy communication.
Finding the quantum bound and Maximal mutual information
The first step of the hierarchical constraints
The second step of the hierarchical constraints
Quantum correlations satisfy IC
Symmetric channels with i.i.d. and uniform input marginal probabilities
The top region
Non-locality Non-locality
Case with non-uniform input marginal probabilities
Symmetric Symmetric and
0max I=1, at Pr(a )=0.5
0 1max I 0.8, at Pr(a )=Pr(a )=0.5
Case with non-uniform input marginal probabilities- asymmetric channel
i 0 1I depends on both Pr(a ) and Pr(a )
max I max I (symmetric channel)