Usefulness of adaptive strategies in asymptotic quantum channel discrimination

TL;DR

The paper analyzes adaptive versus non-adaptive strategies for asymptotic quantum channel discrimination, establishing when adaptiveness yields no gain and when it can offer a separation. For cq-channels, adaptive strategies with classical feed-forward do not improve the generalized Chernoff and Hoeffding exponents beyond non-adaptive tensor-product inputs, while for certain qq-channels with entanglement-breaking structures, asymptotic separations can occur, and quantum feedback can provide advantages in some cases. The authors also explore the discrimination power of a general quantum channel, showing that classical feedback alone does not enhance power while quantum feedback can, depending on the channel structure, and they illustrate their results with detailed examples on depolarizing, Pauli, and amplitude-damping channels. These findings clarify the precise roles of classical/quantum memory and feedback in quantum channel discrimination and guide the design of optimal strategies across symmetric and asymmetric regimes.

Abstract

Adaptiveness is a key principle in information processing including statistics and machine learning. We investigate the usefulness of adaptive methods in the framework of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel and observing outputs. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and non-adaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) We prove that for classical-quantum channels, adaptive and non-adaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding, Stein) settings. (2) The first separation between adaptive and non-adaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for non-adaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. (3)We prove, in some sense generalizing the previous statement, that for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior in the asymptotic limit to non-adaptive product state strategies. (4) As an application of our findings, we address the discrimination power of an arbitrary quantum channel and show that adaptive strategies with classical feedback and no quantum memory at the input do not increase the discrimination power of the channel beyond non-adaptive tensor product input strategies.

Figures (9)

• Figure 1: Adaptive strategy for cq-channel discrimination. Solid and dashed lines denote flow of classical and quantum information, respectively. The classical outputs of instruments $\{\Gamma_{k_m|\vec{x}_m,\vec{k}_{m-1}}^{(m)}\}_{k_m \in {\cal K}_m}$ are employed to decide the inputs adaptively, and leave a post-measurement state that can be accessed together with the next channel output.
• Figure 2: The most general adaptive strategy for discrimination of qq-channels, from the class $\mathbb{A}_{n}$. After the $m$-th use of the unknown channel (denoted '$?$'), the output system $B_{m}$ as well as the state on the memory, i.e. the reference system $R_{m}$, is processed by the cptp map $\mathcal{F}_{m}$, resulting in $\rho_{m+1}^{R_{m+1}A_{m+1}}$; this continues as long as $m<n$. After the $n$-th use of the channel, the state $\omega^{R_{n}B_{n}}_{n}$ is measured by a two-outcome POVM. Two variants of this strategy include restricting feed-forward information to be only classical, and additionally only allowing products state inputs; these variants are denoted by $\mathbb{A}_{n}^{c}$ and $\mathbb{A}_{n}^{c,0}$, respectively.
• Figure 3: The most general parallel strategy for discrimination of qq-channels, from the class $\mathbb{P}_{n}$. An $(n+1)$-partite state $\rho$ on $RA_{1}\ldots A_{n}$ is prepared and each system $A_i$ is fed into a separate channel input; the final measurement is performed with a two-outcome POVM on $RB_{1}\ldots B_{n}$. If we do not allow input states to be entangled among different $A$-systems or with the reference system $R$, the strategy falls into the class $\mathbb{P}_{n}^{0}$.
• Figure 4: Adaptive quantum channel discrimination with classical feed-forward and without quantum memory at the channel input, from the class $\mathbb{A}^{c,0}_n$. Solid and dashed arrow denote the flow of quantum and classical information, respectively. At step $m$, Alice sends the state $\rho_{{x}_{m}}$ which she has prepared using Bob's $m-1$ classical feed-forward informations, and sends it via either $\mathcal{M}$ or $\overline{\mathcal{M}}$ to Bob.
• Figure 5: Discrimination with a quantum channel $\mathcal{M}_o$. At step $m$, Alice prepares a state, either $\rho_{{x}_{m}}$ or $\sigma_{{x}_{m}}$, which she has prepared using Bob's $m-1$ feedbacks (dashed arrows), and sends it via the channel $\mathcal{M}_{o}$ to Bob. Bob's measurements resembles the PVM's of Section \ref{['secadaptive']}; they are used to extract classical information fed back to Alice and to prepare post-measurement states that he keeps for the next round of communication.

Theorems & Definitions (46)

• Lemma 1
• Theorem 2: Generalized Chernoff bound & Hoeffding bound
• Definition 3
• Proposition 4
• proof
• Example 5
• Example 6
• Theorem 7
• Remark 8
• Theorem 9