Asymptotic quantification of entanglement with a single copy

TL;DR

The article tackles two central entanglement tasks—testing for the presence of entanglement and distilling entanglement from noisy states—by reframing performance in terms of error exponents. It proves a generalised quantum Sanov's theorem, yielding a single-letter asymptotic exponent equal to the reverse relative entropy of entanglement $D(\\mathcal{S}_{A:B} \\| \\rho_{AB})$, computable from a single copy, and shows this exponent also governs the distillation error under non-entangling operations: $E_{d,\\rm err}(\\rho_{AB}) = \\mathrm{Sanov}(\\rho_{AB} \\| \\mathcal{S}_{A:B}) = D(\\mathcal{S}_{A:B} \\| \\rho_{AB})$. This leads to a regularisation-free, single-letter benchmark for entanglement processing with broad applicability to quantum resource theories and clarifies the deep connection between hypothesis testing and entanglement manipulation. The work uses a classical-to-quantum lifting via a blurring technique and rests on an axiomatic framework (Brandão–Plenio) augmented by a measurement-compatibility axiom, enabling a robust, non-iid analysis of composite hypotheses. Overall, the results provide a powerful, computable target for operational entanglement tasks and open avenues for extending similar single-letter characterisations to other quantum resources.

Abstract

Despite the central importance of quantum entanglement in quantum technologies, the understanding of the optimal ways to exploit it is still beyond our reach, and even measuring entanglement in an operationally meaningful way is prohibitively difficult. Here we study two fundamental tasks in the processing of entanglement: entanglement testing, which is a quantum state discrimination problem concerned with entanglement detection in the many-copy regime, and entanglement distillation, concerned with purifying entanglement from noisy entangled states. We introduce a way of benchmarking the performance of distillation that focuses on the best achievable error rather than its yield in the asymptotic limit. When the underlying set of operations used for entanglement distillation is the axiomatic class of non-entangling operations, we show that the two figures of merit for entanglement testing and distillation coincide. We solve both problems by proving a generalised quantum Sanov's theorem, enabling the exact evaluation of asymptotic error rates of composite quantum hypothesis testing. We show in particular that the asymptotic figure of merit is given by the reverse relative entropy of entanglement, a single-letter quantity that can be evaluated using only a single copy of a quantum state -- a distinct feature among measures of entanglement that quantify the optimal performance of information-theoretic tasks.

Figures (2)

• Figure 1: The set-up and figure of merit in entanglement testing. Entanglement testing is a quantum hypothesis testing problem concerned with distinguishing the case when a source is generating copies of a target entangled state $\rho_{AB}$ from the case when it malfunctions and instead produces only states $\sigma_n\in \pazocal{S}_{A^n:B^n}$ that are globally separable, i.e. exhibit no entanglement between between Alice's systems on one side and Bob's systems on the other. (a) The procedure of entanglement testing consists of making a general two-outcome quantum measurement on the overall $n$-copy system that models the output of the device. The choice of the measurement here is arbitrary and it is precisely the experimenter's task to optimise this choice. (b) Two types of error may occur: false positive, where a working device is mistaken for a faulty one, and false negative, where the opposite happens. By choosing a measurement optimally, the probability of a false negative can be constrained to be arbitrarily small while the probability of a false positive can be made to decay exponentially fast to zero. The coefficient governing this exponential behaviour, called the Sanov exponent, is a central object of interest in this work.
• Figure 2: Two ways of benchmarking entanglement distillation. Entanglement distillation is the process of converting copies of a noisy entangled quantum state $\rho_{AB}$ into fewer copies of the pure maximally entangled state $\Phi_+$. To account for physical imperfections in manipulating quantum states, the process is not required to be exact: the resulting states must approximate copies of $\Phi_+$ only to some desired degree of precision, as quantified by the distillation error $\varepsilon$. (a) Conventional approaches to distillation focus on maximising distillation yield, that is, the number of copies of $\Phi_+$ obtained per each copy of $\rho_{AB}$. The error of the procedure is irrelevant as long as it converges to zero in the asymptotic limit as the available number of copies of $\rho_{AB}$ grows to infinity --- for a fixed number of copies, the errors may be large. (b) In this paper, we instead focus on minimising the above error, potentially sacrificing some yield to obtain higher-quality entanglement. Specifically, we require that the distillation error vanish exponentially fast as the number of available copies of $\rho_{AB}$ grows, while the total number of maximally entangled states $\Phi_+$ produced in the process is still as large as desired. Accordingly, our figure of merit is not the number of copies produced but the optimal error exponent, that is, the rate of decay of the distillation error, which directly quantifies the quality of the entanglement at the output of the protocol.

Theorems & Definitions (20)

• Lemma 1
• Theorem 2
• Lemma 1
• Remark
• Remark
• Definition 1
• Definition 1: brandao_adversarial
• Lemma 1
• Definition 1: brandao_adversarial
• Theorem 1: (Generalised classical Sanov's theorem)