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arXiv:1611.09165

Optimal estimation and discrimination of excess noise in thermal and amplifier channels

Masahiro Takeoka, Mark M. Wilde

Published Nov 28, 2016• 23 citations
arXivHTMLPDF

Abstract

We determine a fundamental upper bound on the performance of any adaptive protocol for discrimination or estimation of a channel which has an unknown parameter encoded in the state of its environment. Since our approach relies on the principle of data processing, the bound applies to a variety of discrimination measures, including quantum relative entropy, hypothesis testing relative entropy, Rényi relative entropy, fidelity, and quantum Fisher information. We apply the upper bound to thermal (amplifier) channels with a known transmissivity (gain) but unknown excess noise. In these cases, we find that the upper bounds are achievable for several discrimination measures of interest, and the method for doing so is non-adaptive, employing a highly squeezed two-mode vacuum state at the input of each channel use. Estimating the excess noise of a thermal channel is of principal interest for the security of quantum key distribution, in the setting where a fiber-optic cable has a known transmissivity but a tampering eavesdropper alters the excess noise on the channel, so that estimating the excess noise as precisely as possible is desirable. Finally, we outline a practical strategy which can be used to achieve these limits.

DOI
10.48550/arxiv.1611.09165

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We determine a fundamental upper bound on the performance of any adaptive protocol for discrimination or estimation of a channel which has an unknown parameter encoded in the state of its environment. Since our approach relies on the principle of data processing, the bound applies to a variety of discrimination measures, including quantum relative entropy, hypothesis testing relative entropy, Rényi relative entropy, fidelity, and quantum Fisher information. We apply the upper bound to thermal (amplifier) channels with a known transmissivity (gain) but unknown excess noise. In these cases, we find that the upper bounds are achievable for several discrimination measures of interest, and the method for doing so is non-adaptive, employing a highly squeezed two-mode vacuum state at the input of each channel use. Estimating the excess noise of a thermal channel is of principal interest for the security of quantum key distribution, in the setting where a fiber-optic cable has a known transmissivity but a tampering eavesdropper alters the excess noise on the channel, so that estimating the excess noise as precisely as possible is desirable. Finally, we outline a practical strategy which can be used to achieve these limits.
We consider the estimation of noise parameters in a quantum channel, assuming the most general strategy allowed by quantum mechanics. This is based on the exploitation of unlimited entanglement and arbitrary quantum operations, so that the channel inputs may be interactively updated. In this general scenario, we draw a novel connection between quantum metrology and teleportation. In fact, for any teleportation-covariant channel (e.g., Pauli, erasure, or Gaussian channel), we find that adaptive noise estimation cannot beat the standard quantum limit, with the quantum Fisher information being determined by the channel's Choi matrix. As an example, we establish the ultimate precision for estimating excess noise in a thermal-loss channel, which is crucial for quantum cryptography. Because our general methodology applies to any functional that is monotonic under trace-preserving maps, it can be applied to simplify other adaptive protocols, including those for quantum channel discrimination. Setting the ultimate limits for noise estimation and discrimination paves the way for exploring the boundaries of quantum sensing, imaging, and tomography.
We establish three different upper bounds on the energy-constrained quantum and private capacities of bosonic thermal channels. The first upper which we call the bound, is obtained by decomposing a thermal channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing can be at most 1.45 bits larger than a known lower on these capacities of the thermal channel. The other two upper bounds, which we call the bound and the bound, are established using the notion of approximate degradability along with energy constraints. A comparison of these three bounds shows that the data-processing is near to a known lower for both low and high thermal noise and is very near to the ε\varepsilonε-close-degradable only for low thermal noise. Moreover, for certain parameter regimes, we show that the ε\varepsilonε-degradable is tighter than all other bounds. All three upper bounds are very near to a known lower for the case of low thermal noise and high transmissivity. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the R\'enyi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.
Discriminating between noisy quantum processes is a central primitive for quantum communication, metrology, and computing. While discrimination limits for finite-dimensional channels are well understood, the continuous-variable setting, particularly under experimentally relevant energy constraints, remains significantly less developed. In this work, we establish an energy-constrained chain rule for the Belavkin-Staszewski channel divergence, which yields a fundamental upper bound on the error exponents achievable by fully adaptive, energy-constrained quantum channel discrimination protocols. We then derive efficiently computable bounds on asymmetric error exponents for energy-constrained discrimination of bosonic dephasing and loss-dephasing channels. Specifically, we show that three operationally relevant quantities -- the measured relative entropy, the Umegaki relative entropy, and the geometric Renyi divergence -- admit semidefinite program (SDP) formulations when the input energy is bounded and the Hilbert space is suitably truncated. Applying these tools, we demonstrate that optimal probes for these channels under energy constraints are Fock-diagonal, and we also enable numerically precise evaluation of bounds on achievable error exponents across discrimination strategies ranging from separable to fully adaptive. The resulting SDPs provide practical benchmarks for quantum-limited sensing in low-energy bosonic platforms.
There have been several upper bounds on the quantum capacity of the single-mode Gaussian channels with thermal noise, such as thermal attenuator and amplifier. We consider a class of attenuator and amplifier with more general noises, including squeezing or even non-Gaussian cases. We introduce new upper bounds on the energy-constrained quantum capacity of those channels by using the conditional quantum entropy power inequality. Also, we obtain lower bounds for the same channels by means of Gaussian optimizer with fixed input entropy. They give narrow bounds when the transmissivity is near unity and the energy of input state is low.
The goal of quantum channel discrimination and estimation is to determine the identity of an unknown channel from a discrete or continuous set, respectively. The query complexity of these tasks is equal to the minimum number of times one must call an unknown channel to identify it within a desired threshold on the error probability. In this paper, we establish lower bounds on the query complexities of channel discrimination and estimation, in both the parallel and adaptive access models. We do so by establishing new or applying known upper bounds on the squared Bures distance and symmetric logarithmic derivative Fisher information of channels. Phrasing our statements and proofs in terms of isometric extensions of quantum channels allows us to give conceptually simple proofs for both novel and known bounds. We also provide alternative proofs for several established results in an effort to present a consistent and unified framework for quantum channel discrimination and estimation, which we believe will be helpful in addressing future questions in the field.
Protocols for discriminating between a pair of channels or for estimating a channel parameter can often be aided by adaptivity or by entanglement between the probe states. This can make it difficult to bound the best possible performance for such protocols. In this Letter, we introduce a quantity that we call the relative fidelity of a given pair of channels and a pair of input states to those channels. Constraining the allowed input states to all pairs of states whose fidelity is greater than some minimum "input fidelity" and minimizing this quantity over the valid pairs of states, we get the minimum relative fidelity for that input fidelity constraint. We are then able to lower bound the fidelity between the possible output states of any protocol acting on one of two possible channels in terms of the minimum relative fidelity. This allows us to bound the performance of the most general, adaptive discrimination and parameter estimation protocols. By finding a continuity bound for the relative fidelity, we also provide a simple confirmation that the quantum Fisher information (QFI) of the output of an N-use protocol is no more than N^{2} times the one-shot QFI.