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arXiv:1506.00819 · New Journal of Physics

Fidelity and Fisher information on quantum channels

Haidong Yuan, Chi‐Hang Fred Fung

Published Nov 28, 2017• 55 citations
arXivHTMLPDF

Abstract

The fidelity function for quantum states has been widely used in quantum information science and frequently arises in the quantification of optimal performances for the estimation and distinguishing of quantum states. A fidelity function for quantum channels is expected to have the same wide applications in quantum information science. In this paper we propose a fidelity function for quantum channels and show that various distance measures on quantum channels can be obtained from this fidelity function; for example, the Bures angle and the Bures distance can be extended to quantum channels via this fidelity function. We then show that the distances between quantum channels lead naturally to a quantum channel Fisher information which quantifies the ultimate precision limit in quantum metrology; the ultimate precision limit can thus be seen as a manifestation of the distances between quantum channels. We also show that the fidelity of quantum channels provides a unified framework for perfect quantum channel discrimination and quantum metrology. In particular, we show that the minimum number of uses needed for perfect channel discrimination is exactly the counterpart of the precision limit in quantum metrology, and various useful lower bounds for the minimum number of uses needed for perfect channel discrimination can be obtained via this connection.

DOI
10.1088/1367-2630/aa874c

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The fidelity function for quantum states has been widely used in quantum information science and frequently arises in the quantification of optimal performances for the estimation and distinguishing of quantum states. A fidelity function for quantum channels is expected to have the same wide applications in quantum information science. In this paper we propose a fidelity function for quantum channels and show that various distance measures on quantum channels can be obtained from this fidelity function; for example, the Bures angle and the Bures distance can be extended to quantum channels via this fidelity function. We then show that the distances between quantum channels lead naturally to a quantum channel Fisher information which quantifies the ultimate precision limit in quantum metrology; the ultimate precision limit can thus be seen as a manifestation of the distances between quantum channels. We also show that the fidelity of quantum channels provides a unified framework for perfect quantum channel discrimination and quantum metrology. In particular, we show that the minimum number of uses needed for perfect channel discrimination is exactly the counterpart of the precision limit in quantum metrology, and various useful lower bounds for the minimum number of uses needed for perfect channel discrimination can be obtained via this connection.
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.
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.
We consider the problem of estimating a classical parameter encoded in a quantum channel, assuming the most general strategy allowed by quantum mechanics. This strategy is based on the exploitation of an unlimited amount of pre-shared entanglement plus the use of adaptive probings, where the input of the channel is interactively updated during the protocol. We show that, for the wide class of teleportation-stretchable channels in finite dimension, including all Pauli channels and erasure channels, the quantum Fisher information cannot exceed an ultimate bound given by the Choi matrix of the encoding channel. We also extend our methods and results to quantum channel discrimination, finding a corresponding ultimate bound for the minimum error probability. Thus, our findings establish the ultimate precision limits that are achievable in quantum metrology and quantum discrimination for the most basic models of discrete-variable quantum channels.
We define a local Riemannian metric tensor in the manifold of Gaussian channels and the distance that it induces. We adopt an information-geometric approach and define a metric derived from the Bures-Fisher metric for quantum states. The resulting metric inherits several desirable properties from the Bures-Fisher metric and is operationally motivated by distinguishability considerations: It serves as an upper bound to the attainable quantum Fisher information for the channel parameters using Gaussian states, under generic constraints on the physically available resources. Our approach naturally includes the use of entangled Gaussian probe states. We prove that the metric enjoys some desirable properties like stability and covariance. As a by-product, we also obtain some general results in Gaussian channel estimation that are the continuous-variable analogs of previously known results in finite dimensions. We prove that optimal probe states are always pure and bounded in the number of ancillary modes, even in the presence of constraints on the reduced state input in the channel. This has experimental and computational implications. It limits the complexity of optimal experimental setups for channel estimation and reduces the computational requirements for the evaluation of the metric: Indeed, we construct a converging algorithm for its computation. We provide explicit formulas for computing the multiparametric quantum Fisher information for dissipative channels probed with arbitrary Gaussian states and provide the optimal observables for the estimation of the channel parameters (e.g., bath couplings, squeezing, and temperature).
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. Joint work with Zixin Huang, Johannes Jakob Meyer, and Theshani Nuradha.
Abstract Quantum mechanics establishes the ultimate limit to the scaling of the precision on any parameter, by identifying optimal probe states and measurements. While this paradigm is, at least in principle, adequate for the metrology of quantum channels involving the estimation of phase and loss parameters, we show that estimating the loss parameters associated with a quantum channel and a realistic quantum detector are fundamentally different. While Fock states are provably optimal for the former, we identify a crossover in the nature of the optimal probe state for estimating detector imperfections as a function of the loss parameter using Fisher information as a benchmark. We provide theoretical results for on-off and homodyne detectors, the most widely used detectors in quantum photonics technologies, when using Fock states and coherent states as probes.