Towards a state minimizing the output entropy of a tensor product of random quantum channels

Abstract

Abstract. 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 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.1. Introduction 1.1. Background. One of the most important questions in quantum communication theory is whether a quantum channel has additive properties or not [1,4,9,10,11,12,13,14,18]. If a channel Φ is additive for the Holevo capacity χ(·) in the sense that ∃N ∈ N, ∀n N χ(Φ ⊗n ) = nχ(Φ) (1) then the classical capacity of the quantum channel equals the Holevo capacity, giving a one-shot (non-asymptotic) formula for the classical capacity C(·):where, {p i , ρ i } are ensembles and S(·) is the von Neumann entropy. The above additive property was conjectured to be true for all quantum channels until Hastings showed [13] existence of channels such thatHere, S min (·) is the minimal output entropy (MOE). Indeed, this result also gives counterexamples to (1) by [18,12,9] and as a result C(Φ) = χ(Φ) in general. Note thatwhere the minimum is take over all pure inputs (rank-one projections).One of the most important results in the additivity theory of quantum channels is Hastings' proof of the additivity of the minimal output entropy [13]. The proof contains two disjoint parts: establishing a lower bound for the minimum output entropy for one channel, which was the main 2000 Mathematics Subject Classification. Primary 15A52; Secondary 94A17, 94A40.

Low Entropy Output States for Products of Random Unitary Channels

“…One of the main results of our paper is that the typical outputs are deterministic when one takes a product of such a channel and its complex conjugate and applies it to entangled input states, in the spirit of [14,16,12]. More precisely, our main results can be stated informally as follows (for precise statements, see Theorems 5.2 and 6.7): Theorem 1.1.…”

“…In particular, the limiting eigenvalue distribution of [Φ ⊗Φ](ψ n ψ * n ), where Φ is random quantum chanel and ψ n is a generalised Bell state, was calculated in [12]. The aim of this paper is to perform similar computations for random unitary channels instead of random channels.…”

“…The method of graphical calculus was introduced in [14] and later used in [16,17,18,12] to study random matrix models in which Haar distributed unitary operators played a major role. In particular, the limiting eigenvalue distribution of [Φ ⊗Φ](ψ n ψ * n ), where Φ is random quantum chanel and ψ n is a generalised Bell state, was calculated in [12].…”

“…In the previous work in which random quantum channels were analysed using the graphical calculus method [14,16,17,12], the isometry defining the channel in the Stinespring picture was a truncation of a Haar-distributed random unitary matrix. Hence, only one random unitary operator was needed to perform the computations.…”

Revisiting Additivity Violation of Quantum Channels

“…2) Is the violation a phenomenon for bipartite systems? Through the project in [CFN12], I feel that the first question is true for the random quantum channels. For the second question, weak form of additivity is proven in [Mon13].…”