I NFORMATION 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)

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

3. Testing IC for general quantum communication protocols Feasibility for maximizing mutual information by convex optimization? Solutions Solution (i): Solution (ii): 4. The conclusion

The CHSH inequality The measurement scenario

The local hidden variable theory: If A0, A1, B0, B1=1, -1, then Cxy= ; CHSH=(A0 + A1) B0 + (A0 − A1) B1; | | ≤ 2

Tsirelson bound The maximal amount of quantum violation - Tsirelson bound Since Why Quantum mechanics cannot be more nonlocal?

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.

Does the no-signaling theory limit the quantum non-locality? The PR-box:

What is Information Causality (IC)? In the communication protocol, the information gain cannot exceed the amount of classical communication.

The 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 if i.e., IC says total mutual information between Bob’s guess bit β and Alice’s database is bounded by 1, i.e., IC is violated!

For 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.

M ULTI - 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.

Bob’s success probability to guess is Define the coding noise parameter to be The result: The noise parameter and the CHSH inequality

Binary symmetric channel The signal decay theorem IC yields: Binary symmetric channel ] W. Evans and L. J. Schulman, Proceedings of the 34 th Annual Symposium on Foundations of Computer Science, 594 (1993). YZ

Multi-setting Tsirelson-type inequalities Using the Cauchy-Schwarz inequality, we can obtain When k=2, The Tsirelson inequality

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, (2006).

T ESTING IC FOR GENERAL QUANTUM COMMUNICATION PROTOCOLS

For multi-level quantum communication protocols, IC is satisfied by quantum correlation? saturated?

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.

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).

If, Bob can guess perfectly. The symmetric channel Multi-level RAC protocol The signal decay theorem for di-nary channel

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.

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.

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).

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 necessary and sufficient condition for correlation functions

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

Symmetric channels Asymmetric channels

By partitioning the defining domains of the probabilities into 100 points. We find IC is saturated.

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.

How to know the given joint probabilities could 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:

The constraint becomes stronger than the previous step of the hierarchical constraint. Q1 Q2 Q3

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.

The first step of the hierarchical constraints The second step of the hierarchical constraints Quantum correlations satisfy IC

The top region Non-locality

SymmetricSymmetric and