Information Causality and the boundary of quantum correlations . 24

or not. In their proof, they considered the simplest case, n = 1, k = 2 RAC protocol and the quantum correlations for two-inputs/two-outputs no-signaling box. They studied the no-signaling correlations by several two dimensional slices of the non-signalling polytope, that is equivalent to considering the following noisy PR-box.

P R_λ,µ= λP R + µB + (1 − λ − µ)I, (1.50) where P R is the PR-box and B could be another extremal non-local box (1.37) or local deterministic box (1.36). Thus, the noisy PR-boxes can be grouped into two families. In

[35], the Uffink’s inequality (1.49) is taken as the condition for the violation of IC. On the other hand, they imposed two quantum constrains to obtain the quantum boundary.

These constraints correspond to different families of noisy PR boxes. Finally, one may compare the condition for the violation of IC and the quantum boundary. Two results were arrived in [35]. First, the bi-partite quantum correlations satisfy IC. Second, only parts of the quantum boundary emerges from IC.

For the first family, B is one of the extremal non-local box (1.37) except the PR one. In this family, the marginal probabilities Pr(A_x|x) and Pr(B_y|y) are uniform. The correlation functions C_x,y can be rewritten as

C0,0= λ + (−1)^γµ, C0,1= λ + (−1)^β+γµ, C1,0= λ + (−1)^α+γµ, C1,1= λ + (−1)^α+β+γµ, (1.51) where α, β and γ are used to label the types of extremal non-local boxes (1.37). For fixed α, β and γ (or equivalently, a chosen extremal non-local box B), one could plug (1.51) into the condition (1.49) for the violation of IC so that IC and the noisy PR-box are related. For example, if (α, β, γ) = (0, 1, 0), one may find that IC is violated when

λ²+ µ² > 1

2. (1.52)

On the other hand, if a set of correlation functions Cx,y can be reproduced by the quantum system (quantum operators and quantum state), this set of correlation functions need to satisfy the following quantum condition,

|C_0,0C_1,0− C_0,1C_1,1| ≤ X

j=0,1

q

(1 − C_0,j² )(1 − C_1,j² ). (1.53)

The above condition was proposed by Landau [42], and an equivalent set of conditions were proposed by Masanes [43] expressed in different form. Note that, if (1.53) is satisfied by a set of correlation functions C_x,y, the corresponding marginal probabilities Pr(A_x|x) and Pr(By|y) must be uniform. As for the condition for the violation of IC, for fixed α, β and γ, one can plug (1.51) into the quantum condition (1.53) and find the associated quantum boundary. For example, when (α, β, γ) = (0, 1, 0), the boundary of quantum correlations and the condition for the violation of IC are the same. In this case, the quantum boundary emerges from IC. However, this is not always true, for example for (α, β, γ) = (1, 1, 1). There are some non-quantum correlations satisfying IC. This does

not mean IC cannot exclude these non-quantum correlations, because if one use another strategy for the protocol such as nesting some non-quantum boxes mentioned in the previous subsection, these non-quantum boxes may violate IC.

In the second family of the noisy PR-box, B is the local deterministic box considered in (1.36). Among these local boxes, one could consider the special ones which lie on the CHSH facets of the polytope of locality discussed in section 1.3.2. Therefore, the parameters of these special boxes given by (1.36) should satisfy αγ + β + δ = 0 mod 2.

As for the first family, one can use the parameters of the noisy PR-box given in (1.50) to rewrite the correlation function C_x,y. For example, if (α, β, γ, δ) = (0, 0, 0, 0), one then obtains

C_0,0= C_0,1= C_1,0 = λ + µ, C_1,1 = µ − λ. (1.54) Note that, the marginal probabilities Pr(A_x|x) and Pr(B_y|y) are no longer uniform.

Therefore, the marginal correlations C_x and C_y are not zero. The marginal correlations are the linear combinations of the marginal probabilities. The definition for marginal cor-relations is Cx = Pr(Ax = 0|x) − Pr(Ax = 1|x) and Cy = Pr(By = 0|y) − Pr(By = 1|y).

Here, the marginal correlations C_x=0 = C_x=1 = C_y=0 = C_y=1 = µ. In this case, the quantum constraint (1.53) is not suitable for picking up the quantum correlations. In-stead, one may use the following condition and its three symmetric partners by shifting the minus sign. These conditions include the marginal correlations, i.e.,

| arcsin(D_0,0) + arcsin(D_0,1) + arcsin(D_1,0) − arcsin(D_1,1)| ≤ π, (1.55) where Dx,y = √^Cx,y−CxCy

(1−Cx²)(1−Cy²). This condition was proposed in [44, 45]. Note that, (1.55) is the quantum constraint in the first step of the hierarchal semidefinite programming [45].

This means, if a set of correlation functions C_x,y, C_x and C_y satisfies (1.55), we could not make sure if it could be reproduced by a quantum system, unless it can satisfy all the quantum constraint in each step of the hierarchical semidefinite programming. Plug all the correlation functions (1.54) into the quantum constraint (1.55), one can obtain the associated quantum boundary. Similarly, plug (1.54) into the condition (1.49) for the violation of IC, one may find that IC is violated when (λ + µ)²+ λ² > ¹₂. In this case, the quantum boundary from the quantum constraint (1.55) and the condition of IC are not the same. There are some non-quantum correlations satisfying IC. As for the case in

the first family, one may use another strategy such that these non-quantum correlations will still violate IC.

However, with the above examples, one may only know that quantum correlations satisfy IC and could not make sure if IC can exclude all the non-quantum correlations.

Moreover, one may find there are some assumptions and approximations in arriving the condition (1.48) for the violation of IC. Therefore, instead of using this condition, we directly calculate the mutual information I of the quantum correlations and then compare with IC. See Chapter 3 for more details.

1.3 Signal propagating and noisy computation

Signal propagating suffers signal decay. It is a very important issue when considering communication systems. On the other hand, one could image the noisy computation as a sequence of steps [49]. Each step has an input and an output. The output has some information about the associated input, and the output will be the input of the next step. Once some step produces a random noise, one can interleave some parallel step to control the error rate of the entire computation. In this way, each step of the noisy computation can be seen as a noisy communication channel. Thus, one may expect that the issue of signal decay will also appear for the noisy computation. von Neumann was the first to be aware of this fact [47]. He suggested that the error of the computation should be treated by thermodynamical method as the treatment for the communication in Shannon’s work [46]. This means that, for the noisy computation, one should use some information-theoretic methods related to the noisy communication. After thirty years from the publication of von Neumann’s work, Pippenger took some information theoretic arguments to study the reliable noisy computation [48]. After that, Evans and Schulman proposed a new result for the efficient signal propagating and then use it to study the threshold of noisy computation [50, 51]. In this section, we want to review Evans and Schulman’s work and also compare it with the previous results obtained by von Neumann and Pippenger.