xPr(~x) Pr(B~y− A~x = ~x · ~y|~x, ~y) − 1
d − 1 . (3.15)
As for the d = 2 case, we assume the upper bound of (3.14) is capped by the informa-tion causality to yield a quadratic constraint on the noise parameters. Again, using the Cauchy-Schwarz inequality to linearize the quadratic constraint, we find that the general-ized inequalityP
~
yξ~y ≤√
k. This inequality could be the Tsirelson-type inequality and it need to be checked. Especially, if the input marginal probabilities Pr(ai) are uniform, the bound on the object P
~
yξ~y yields a constraint on Pr(A~x, B~y|~x, ~y). Therefore, we obtain a proper object of characterizing non-locality: P
~
yξ~y with uniform Pr(ai).
Then, it is ready to ask the question: does the joint probabilities Pr(A~x, B~y|~x, ~y) giving the maximal quantum non-locality saturate the upper bound of information causality?
Next, we are going to answer this question.
3.3 Convexity and mutual information
3.3.1 Feasibility for maximizing mutual information by convex optimization?
In order to test information causality for difference quantum communication protocols, we have to maximize mutual information I over quantum channel and Alice’s input prob-ability. One way is to formulate the problem as the convex optimization programming, so that we may exploit some numerical recipes such as [63] to carry out the task.
Minimizing a function with the equality or inequality constraints is called convex op-timization. The object function could be linear or non-linear. For example, SDP is a kind of convex optimization with a linear object function. Regardless of linear or non-linear object functions, the minimization (maximization) problem requires them to be convex (concave). Thus, if we define the mutual information I as the object function for maximization in the context of information causality, we have to check the concavity of it.
A concave function f (x) (f :Rn →R) should satisfy the following condition:
f (λx1+ (1 − λ)x2) ≥ λf (x1) + (1 − λ)f (x2), (3.16) where x1 and x2 are n-dimensional real vectors, and 0 < λ < 1.
Mutual information between input X and output Z can be written as I(X; Z) = H(Z) − H(Z|X) = H(Z) −X
i
Pr(X = i)H(Z|X = i), (3.17) where H(Z) = −P
iPr(Z = i) log2Pr(Z = i) is the entropy function. We will study the convexity of I(X; Z) by varying over the marginal probability Pr(X) and the channel probability Pr(Z|X).
The following theorem is mentioned in [68]. If we fix the channel probability Pr(Z|X) in (3.17), then I(X; Z) is a concave function with respect to Pr(X). This is the usual way in obtaining the channel capacity i.e., maximizing mutual information I over input marginal probability for a fixed channel.
However, in the context of information causality, the channel probability is related to both the joint probability of the NS-box and the input marginal probability. This means that the above twos will be correlated if we fix the channel probability. This cannot fit to our setup in which we aim to maximize the mutual information I by varying over the joint probability of NS-box and the input marginal probability. For example, in d = 2 and k = 2 case, the channel probability is given by
Pr(β|ai, b = i) =
From the above, we see that the channel probability Pr(β|ai, b = i) cannot be fixed by varying over Pr(By − Ax|x, y) and Pr(ai) independently. Similarly, for higher d and k protocols, we will also have the constraints between the above three probabilities. Thus, maximizing the mutual information I by varying the NS-box in the context of information causality is quite different from the usual way of finding the channel capacity.
To achieve the goal of maximizing the mutual information I over the NS-box, we should check if it is a convex (or concave) optimization problem or not. If it is yes, then we can adopt the numerical recipe as [63] to carry out the task. Otherwise, we can either impose more constraints for our problem or just do it by brutal force. It is known that [67] one can check if maximizing function f (y1, · · · , yn) over yi’s is a concave problem or not by
For the maximization to be a concave problem, the Hessian matrix should be negative semidefinite. That is, all the odd order principal minors of H(f ) should be negative and all the even order ones should be positive. Note that each first-order principal minor of H(f ) is just the second derivative of f , i.e. ∂2f
∂y2i. So, the problem cannot be concave if
∂2f
∂y2i > 0 for some i.
With the above criterion, we can now show that the problem of maximizing I over Pr(B~y − A~x|~x, ~y) and Pr(ai) cannot be a concave problem. To do this, we rewrite the mutual information I defined in (3.1) as following:
I =
where ~x and ~y in the above are given by the encoding of the RAC protocol, namely, con-ditions of total probability. Thus we need to solve these concon-ditions such that the mutual information I is expressed as the function of independent probabilities. After that, we can evaluate the corresponding Hessian matrix to examine if the maximization of I over these probabilities is a concave problem or not.
For illustration, we first consider the d = 2 and k = 2 case. By using the relations (3.21) and the normalization conditions of total probability to implement the chain-rule while taking derivative, we arrive Obviously, (3.22) cannot always be negative. This can be seen easily if we set Pr(a0) = 1 − Pr(a1) so that the first term on the RHS of (3.22) is zero. Then, the remaining terms are non-negative definiteness. This then indicates that maximizing I over the joint probability is not a concave problem.
The check for the higher d and k cases can be done similarly, and the details can be found in the Appendix B. Again, we can set all the Pr(ai) to be uniform so that we have
d2kln 2 · ∂2I 3.3.2 Convex optimization for symmetric and isotropic channels with i.i.d.
and uniform input marginal probabilities
Recall that we would like to check if the boundaries of the information causality and the quantum non-locality agree or not. To achieve this, we may either maximizing the mutual
information I with the quantum constraint, or maximizing the quantum non-locality and then evaluate the corresponding mutual information I which can be compared with the bound of information causality. These two tasks are not equivalent but complementary.
However, unlike the first task, the second task will be concave problem as known in [56, 45]. The only question in this case is if the corresponding mutual information I is monotonically related to the quantum non-locality or not. If yes, then maximizing quantum non-locality is equivalent to maximizing the mutual information I. The answer is partially yes as we will show because this monotonic relation holds only for symmetric and isotropic channels with i.i.d. and uniform input marginal probabilities.
Assuming Alice’s input is i.i.d., we have H(β|b = i) = log2(d). Also, once the channel is symmetric, we have Pr(β = t|ai = j, b = i) = (d−1)ξd i+1 for t = j, and Pr(β = t|ai = If we also assume the channels are isotropic i.e., ξ~y = ξ, then for such a case the mutual information I can be further simplified to
I = k[log2d +(d − 1)ξ + 1
d log2((d − 1)ξ + 1
d ) + (1 − (d − 1)ξ
d ) log2(1 − ξ
d )]. (3.25) The value of ξ is in the interval [0, 1] with ξ = 0 for the completely random channel, and ξ = 1 for the noiseless one.
We can show that the mutual information I is the monotonic increasing function of the quantum non-locality parameterized by the noise parameter ξ. To do this, we calculate the first and second derivative of I with respect to ξ and obtain
dI
From the above, we see that dIdξ is always positive for ξ ∈ [0, 1]. Moreover, it is easy to see that I is minimal at ξ = 0 since ddξ2I2 = d − 1 > 0. Thus, the mutual information I is a monotonically increasing function of ξ for the symmetric and isotropic quantum channel with i.i.d. and uniform input marginal probabilities.