This paper studies the difficulty of discriminating between an arbitrary quantum channel and a "replacer" channel that discards its input and replaces it with a fixed state. The results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination. We show that, in this particular setting, the most general adaptive discrimination strategies provide no asymptotic advantage over non-adaptive tensor-power strategies. This conclusion follows by proving a quantum Stein's lemma for this channel discrimination setting, showing that a constant bound on the Type I error leads to the Type II error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels. The strong converse part of the lemma states that any attempt to make the Type II error decay to zero at a rate faster than the channel relative entropy implies that the Type I error necessarily converges to one. We then refine this latter result by identifying the optimal strong converse exponent for this task. As a consequence of these results, we can establish a strong converse theorem for the quantum-feedback-assisted capacity of a channel, sharpening a result due to Bowen. Furthermore, our channel discrimination result demonstrates the asymptotic optimality of a non-adaptive tensor-power strategy in the setting of quantum illumination, as was used in prior work on the topic. The sandwiched Rényi relative entropy is a key tool in our analysis. Finally, by combining our results with recent results of Hayashi and Tomamichel, we find a novel operational interpretation of the mutual information of a quantum channel N as the optimal type II error exponent when discriminating between a large number of independent instances of N and an arbitrary "worst-case" replacer channel chosen from the set of all replacer channels.

Abstract. In this work we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value ω * and the entangled value with one-way quantum communication ω qow of these games. We will show that the value ω * can be efficiently approximated up to a 4-multiplicative factor and that ω qow can be efficiently computed. We also study the behavior of these values under the parallel repetition of rank-one quantum games. We will show that ω * does not verify a perfect parallel repetition theorem and that ω qow does verify such a theorem.

Tsirelson's problem deals with how to model separate measurements in quantum mechanics. In addition to its theoretical importance, the resolution of Tsirelson's problem could have great consequences for device independent quantum key distribution and certified randomness. Unfortunately, understanding present literature on the subject requires a heavy mathematical background. In this paper, we introduce quansality, a new theoretical concept that allows to reinterpret Tsirelson's problem from a foundational point of view. Using quansality as a guide, we recover all known results on Tsirelson's problem in a clear and intuitive way.

Let G be a locally compact abelian group with dual groupĜ. The Hausdorff-Young theorem states that if f ∈ L p (G), where 1 ≤ p ≤ 2, then its Fourier transform Fp(f ) belongs to L q (Ĝ) (where 1 p + 1 q = 1) and ||Fp(f )||q ≤ ||f ||p. Kunze and Terp extended this to unimodular and locally compact groups, respectively. We further generalize this result to an arbitrary locally compact quantum group G by defining a Fourier transform Fp : Lp(G) → Lq(Ĝ) and showing that this Fourier transform satisfies the Hausdorff-Young inequality.

Several information measures have recently been defined which capture the notion of "recoverability." In particular, the fidelity of recovery quantifies how well one can recover a system A of a tripartite quantum state, defined on systems ABC, by acting on system C alone. The relative entropy of recovery is an associated measure in which the fidelity is replaced by relative entropy. In this paper, we provide concrete operational interpretations of the aforementioned recovery measures in terms of a computational decision problem and a hypothesis testing scenario. Specifically, we show that the fidelity of recovery is equal to the maximum probability with which a computationally unbounded quantum prover can convince a computationally bounded quantum verifier that a given quantum state is recoverable. The quantum interactive proof system giving this operational meaning requires four messages exchanged between the prover and verifier, but by forcing the prover to perform his actions in superposition, we construct a different proof system that requires only two messages. The result is that the associated decision problem is in QIP(2) and another argument establishes it as hard for QSZK (both classes contain problems believed to be difficult to solve for a quantum computer). We finally prove that the regularized relative entropy of recovery is equal to the optimal Type II error exponent when trying to distinguish many copies of a tripartite state from a recovered version of this state, such that the Type I error is constrained to be no larger than a constant.