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arXiv:2012.11104 · Physical Review A

Optimal single-shot discrimination of optical modes

Ignatius William Primaatmaja, Asaph Ho, Valerio Scarani

Published May 10, 2021• 3 citations
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Abstract

Retrieving classical information encoded in optical modes is at the heart of many quantum information processing tasks, especially in the field of quantum communication and sensing. Yet, despite its importance, the fundamental limits of optical mode discrimination have been studied only in a few specific examples. Here we present a toolbox to find the optimal discrimination of any set of optical modes. The toolbox uses linear and semidefinite programming techniques, which provide rigorous (not heuristic) bounds, and which can be efficiently solved on standard computers. We study both probabilistic and unambiguous single-shot discrimination in two scenarios: the channel-discrimination scenario, typical of metrology, in which the verifier holds the light source and can set up a reference frame for the phase; and the source-discrimination scenario, more frequent in cryptography, in which the verifier only sees states that are diagonal in the photon-number basis. Our techniques are illustrated with several examples. Among the results, we find that, for many sets of modes, the optimal state for mode discrimination is a superposition or mixture of at most two number states; but this is not general, and we also exhibit counterexamples.

DOI
10.1103/physreva.103.052410

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Retrieving classical information encoded in optical modes is at the heart of many quantum information processing tasks, especially in the field of quantum communication and sensing. Yet, despite its importance, the fundamental limits of optical mode discrimination have been studied only in a few specific examples. Here we present a toolbox to find the optimal discrimination of any set of optical modes. The toolbox uses linear and semidefinite programming techniques, which provide rigorous (not heuristic) bounds, and which can be efficiently solved on standard computers. We study both probabilistic and unambiguous single-shot discrimination in two scenarios: the channel-discrimination scenario, typical of metrology, in which the verifier holds the light source and can set up a reference frame for the phase; and the source-discrimination scenario, more frequent in cryptography, in which the verifier only sees states that are diagonal in the photon-number basis. Our techniques are illustrated with several examples. Among the results, we find that, for many sets of modes, the optimal state for mode discrimination is a superposition or mixture of at most two number states; but this is not general, and we also exhibit counterexamples.
Quantum information processing using linear optics is challenging due to the limited set of deterministic operations achievable without using complicated resource-intensive methods. While techniques such as the use of ancillary photons can enhance the information processing capabilities of linear optical systems, they are technologically demanding. Therefore, determining the constraints posed by linear optics and optimizing linear optical operations for specific tasks under those constraints, without the use of ancillae, can facilitate their potential implementation. Here, we consider the task of unambiguously discriminating between Bell-like states using linear optics and without the use of ancillary photons. This is a basic problem relevant in diverse settings, for example, in the measurement of the output of an entangling quantum circuit or for entanglement swapping at a quantum repeater station. While it is known that exact Bell states of two qubits can be discriminated with an optimal success probability of 50%, we find, surprisingly, that for Bell-like states the optimal probability can be only 25%. We analyze a set of Bell-like states in terms of their distinguishability, entanglement as measured by concurrence, and parameters of the beam-splitter network used for unambiguous discrimination. Further, we provide the linear optical configuration comprising single-photon detectors and beam splitters with input-state-dependent parameters that achieves optimal discrimination in the Bell-like case.
Energy is arguably the main resource in classical physics; coherence is the prototypical quantum resource. In this paper, we study the interplay between these two resources in the task of optimal mode discrimination. This task, motivated by cryptography and metrology, consists of finding in which mode a signal has been encoded, for a given finite set of possible modes. After recalling that the task can be cast as a semi-definite program, we solve completely the case of two-mode discrimination under a hard energy constraint n≤Nn\leq Nn≤N. Specifically, we identify the region of modes overlap, for which the maximal energy state (the Fock state with N excitations) is the best choice for discrimination. For other overlaps, having access to coherence leads to better discrimination.
Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are a prime candidate as information carriers for optical communications. A large body of literature exists on quantum-limited parameter estimation and discrimination for coherent states. However, very little is known about practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and establish a theory of unambiguous discrimination of coherent states, with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, un-excited auxiliary input modes, and on-off photon detection. Our results indicate that these currently-available optical components are near optimal for unambiguous discrimination of multiple coherent states in a constellation.
Estimation of the properties of a physical system with minimal uncertainty is a central task in quantum metrology. Optical phase estimation is at the center of many metrological tasks where the value of a physical parameter is mapped to the phase of an electromagnetic field and single-shot measurements of this phase are necessary. While there are measurements able to estimate the phase of light in a single shot with small uncertainties, demonstrations of near-optimal single-shot measurements for an unknown phase of a coherent state remain elusive. Here, we propose and demonstrate strategies for single-shot measurements for ab initio phase estimation of coherent states that surpass the sensitivity limit of heterodyne measurement and approach the Cramer-Rao lower bound for coherent states. These single-shot estimation strategies are based on real-time optimization of coherent displacement operations, single photon counting with photon number resolution, and fast feedback. We show that our demonstration of these optimized estimation strategies surpasses the heterodyne limit for a wide range of optical powers without correcting for detection efficiency with a moderate number of adaptive measurement steps. This is, to our knowledge, the most sensitive single-shot measurement of an unknown phase encoded in optical coherent states.
Quantum state discrimination is a fundamental primitive in quantum information processing, underpinning tasks in quantum communication, sensing, and learning. We consider the general Bayes framework, as introduced by Helstrom, for state discrimination when, instead of a classical description of the candidate states, one has access to their \emph{source code}: the quantum circuit that prepares them. We show that the semidefinite program (SDP) for the discrimination problem can be reformulated in terms of the Gram matrix of these states, reducing the SDP variable dimensions from dLdLdL to NLNLNL, where ddd is the Hilbert space dimension, NNN is the number of candidate states, and LLL is the number of possible guesses. Importantly, we further introduce a quantum pre-processing procedure which efficiently constructs the reduced semidefinite program from the source code, enabling our method to operate directly on quantum data. We consider two applications. First, we characterize the optimal identifications for quantum changepoint problems under several reward structures, including multiple-changepoint settings that were previously computationally inaccessible. Second, we consider a quantum error classification problem and show how our reduction makes it tractable for systems of hundreds of qubits.
In this work, we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage of entanglement. We quantify the distance between two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance between the measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product, the program finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.