It is known that random quantum channels exhibit significant violations of multiplicativity of maximum output p-norms for any p > 1. In this work, we show that a weaker variant of multiplicativity nevertheless holds for these channels. For any constant p > 1, given a random quantum channel ( N ) (i.e. a channel whose Stinespring representation corresponds to a random subspace S), we show that with high probability the maximum output p-norm of ( N ^ n ) decays exponentially with n. The proof is based on relaxing the maximum output ∞-norm of ( N ) to the operator norm of the partial transpose of the projector onto S, then calculating upper bounds on this quantity using ideas from random matrix theory.
Some very recent work by Collins, Fukuda and Nechita @cite_6 also uses related techniques to those which we use here to prove Theorem (e.g. calculations with Weingarten functions @cite_19 , see Section below). The goal of @cite_6 was to find the state which, when input to a tensor product of two random quantum channels (either the same channel, or a channel and its conjugate), achieves minimal output entropy. A sequence of recent papers by Collins and Nechita @cite_4 @cite_3 @cite_11 @cite_8 carries out a number of interesting quantum information-theoretic calculations using Weingarten functions. This method seems to be a powerful tool which may be expected to find many other applications in quantum information.
@cite_4 @cite_8 @cite_3 @cite_6 @cite_19 @cite_11
2020956776 2020559278 2071160621 2112866726
In this paper, we present applications of the calculus developed in collins-nechita-1 , and obtain an exact formula for the moments of random quantum channels whose input is a pure state thanks to gaussianization methods. Our main application is an in-depth study of the random matrix model introduced by Hayden and Winter and used recently by Brandao, Horodecki, Fukuda and King to refine the Hastings counterexample to the additivity conjecture in Quantum Information Theory. This model is exotic from the point of view of random matrix theory, as its eigenvalues obey to two different scalings simultaneously. We study its asymptotic behavior and obtain an asymptotic expansion for its von Neumann entropy. We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs’ eigenvalues between a quantum channel and its complementary version in the case of a non-pure input. For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p > 1, the minimum output Renyi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p = 1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2. The goal of this note is to show that the analysis of the minimum output p-Renyi entropy of a typical quantum channel essentially amounts to applying Milman’s version of Dvoretzky’s theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden–Winter, disproving the additivity conjecture for the minimal output p-Renyi entropy (for p>1).