In the previous discussion we have considered the information causality using a single nonlocal NS-box. Instead, we can treat the NS-box as a non-local gate for performing the nonlocal computation, i.e., computing the function f (~x, ~y) [51]. Unlike using the same gate for the RAC, no classical communication between Alice and Bob is required to perform the nonlocal computation. In details, Alice’s and Bob’s local outputs are A~xand B~y, respectively. The computation is successful if A~x+ B~y = f (~x, ~y). The computational noise parameter is defined as
~x,~y := 2 Pr[A~x+ B~y = f (~x, ~y)|~x, ~y] − 1. (2.8) From (2.8) and (2.3) the computational noise of the gate is related to its coding noise by
ξ~y = 1 N~x
X
{~x}
~x,~y . (2.9)
Basically, computational errors inherently come from the gate noise. Information causality constraints the noisy extent of the NS-box as a gate. From this perspective, information causality is deeply connected with nonlocal computation.
Furthermore, we can combine the NS-box gates to form a more complicated circuit without worrying about the coding protocol. Then the total task function for the whole
Figure 2.1: RAC protocol for a (n, k, l)-circuit. Each vertex of the circuit corresponds to a NS-box, with its details shown in the big ellipses.
circuit will be a complicated function, i.e., a composite of task functions of all NS-boxes.
We can then try to answer the following fundamental question: could a noiseless (nonlocal) computation be simulated using a noisy nonlocal physical resource?
Specifically, we consider the so-called (n, k, l)-circuit, G, formed by cascading layers of noisy gates into a circuit in the form of a directed, acyclic tree (see Fig 1). On the top of G, there are n inputs to the NS-boxes — the leaves; at the bottom there is only one NS-box — the root. The longest path from the leaves to the root is called the depth of the circuit, denoted by l. The maximum input number of a gate in G is k. Note that, in [29] G comprises k = 2 gates and is exploited to compress n bits of ~x into one bit A~x. However, there is no restriction on the task function for each NS-box, as long as the final circuit is a consistent acyclic tree diagram.
We then use the circuit G to perform the following nonlocal computation. Alice’s
n-bit database ~a := (a0, a1, · · · , an−1) is given to the leaves of G, and a conditional input b ∈ {0, 1, · · · , n − 1} is given to the distant Bob. The previous encoding ~a → −→x and b →
~
y for the RAC protocol, is also exploited here. Alice’s output is properly encoded and then fed into the NS-box at the next layer, again with Bob’s conditional input. The same procedure is performed recursively until reaching the root, with its output as the answer to the total task function at the root.
Alternatively, Bob’s decoding gates can be thought to be noise free, and the com-putational noise is only due to Alice’s encoding gates, and vice versa. This makes it easier to understand the above procedure of noisy computation. Now we can consider the information flow of G.
Theorem 2: For a noisy, local circuit G with an arbitrary depth, the root outputs at most one-bit information.
Note that the circuit G can perform the RAC if the appropriate protocol is given at each layer and 1-bit communication is allowed for the whole process. Then, the above theorem implies that information causality holds true for the circuit G.
To prove the theorem, we will show that the mutual information between the leaves and the root of G is bounded by one. This can be done by mathematical induction as follows. We begin with a circuit of depth one, which is nothing but a single NS-box; information causality ensures the bound. We then assume that the bound holds true for a circuit of depth `. According to information causality and sub-additivity, the mutual information I`(m)between the leaves and the root obeys I`(m) ≤P
imI(Xim; Rm) = P
imI(Xim; Rm|Bob’s knowledge) ≤ 1, where the index m labels a collection of circuits of depth ` with root Rm, and the index im labels the inputs of the m-th circuit. Now, we construct a circuit of depth ` + 1 by connecting all roots Rm’s to a single NS-box whose output is R. Then, the mutual information I`+1 between leaves and root R of the final circuit should obey the subadditivity, i.e., I`+1 ≤P
m
P
imI(Xim; R). From Theorem 1, we have I(Xim; R) ≤ ξm2I(Xim; Rm) because we have a cascade of two channels: Xim ,→
Rm ,→ R where the second channel is a binary symmetric one with the noise ξm. Using this result, we have I`+1 ≤P
mξm2 P
imI(Xim; Rm) ≤P
mξm2 ≤ 1. Q.E.D.
Here, we have only considered the case in which the computational noise is isotropic to ~x, denoted by ~y. From (2.9) we have ~y = ξ~y and the information causality requires P
{~y}2~y ≤ 1. We would like to know whether the reliable computation is also constrained
by the information causality or not. To check this, we invoke the main Evans-Schulman theorem on the conditions for reliable noisy computation as follows [50, 51].
Evans-Schulman Theorem: A circuit of complete k-ary tree with depth l ( i.e., n = kl) can perform δ-reliable noisy computation only
• (i) if P
{~y}2~y > 1 then ` ≥ log(n∆)/ log(P
{~y}2~y) ,
• (ii) if P
{~y}2~y ≤ 1 then n ≤ 1/∆,
where ∆ := 1 + δ log δ + (1 − δ) log(1 − δ). The computation is called δ-reliable if the root outputs correctly with a probability 1 − δ (with δ < 1/2). This theorem provides stricter conditions than the original proposal by Von Neumann [47, 48].
By definition, smaller ~y means larger noise, and the condition (ii) is for the cases with larger noise such that only functions with a smaller number of inputs can be reliably computed. Immediately, we see that information causality implies a large computational noise for the RAC circuit such that only condition (ii) for reliable noisy computation can possibly be fulfilled. As a result, Alice’s output asymptotically becomes random because
∆ → 0 and hence δ → 12 as n → ∞. In summary, this implies that information causality prevents any physically realizable (n, k, l)-circuit from achieving reliable computations of excessively complicated functions, i.e., with either too many inputs or lengthy steps needed.
The above result applies only when classical communication between Alice and Bob is disallowed. Under such circumstances, the noise of the gate is intrinsically constrained by the underlying physical theory. Otherwise, the classical communication can be exploited to improve the reliability of the gates so that the no-go result could be lifted.