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arXiv:1212.2048 · Physical Review A

Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit

Ranjith Nair, Saikat Guha, Si-Hui Tan

Published Mar 12, 2014• 40 citations
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Abstract

We derive the quantum limit on the error probability exponent for discriminating any $M$ multimode coherent states of light and show that it is four times that of an ideal heterodyne receiver for the same signal set. We then propose a receiver that achieves the quantum limit using auxiliary coherent-state fields, beam splitters, and single-photon detectors. The performance of the receiver is compared to standard measurements for various imaging and communication tasks. A related receiver for discriminating arbitrary multicopy pure quantum states is shown to achieve the $M$-ary quantum Chernoff exponent and does so using only local operations and classical communication.

DOI
10.1103/physreva.89.032318

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We derive the quantum limit on the error probability exponent for discriminating any MMM multimode coherent states of light and show that it is four times that of an ideal heterodyne receiver for the same signal set. We then propose a receiver that achieves the quantum limit using auxiliary coherent-state fields, beam splitters, and single-photon detectors. The performance of the receiver is compared to standard measurements for various imaging and communication tasks. A related receiver for discriminating arbitrary multicopy pure quantum states is shown to achieve the MMM-ary quantum Chernoff exponent and does so using only local operations and classical communication.
Discriminating coherent states of light is an important instance of quantum state discrimination that is central to all applications of laser light. We obtain the ultimate quantum limit on the error probability exponent for discriminating among any M multimode coherent-state signals via the recently developed theory of the quantum Chernoff exponent in M-ary multi-copy state discrimination. A receiver called the Sequential Waveform Nulling (SWN) receiver is proposed for discriminating an arbitrary coherent-state ensemble using only auxiliary coherent-state fields, beam splitters, and non-number-resolving single photon detectors. An explicit error probability analysis of the SWN receiver is used to show that it achieves the quantum limit on the error probability exponent, which is shown to be a factor of four greater than the error probability exponent of an ideal heterodyne-detection receiver on the same ensemble. Apart from being of fundamental interest, these results are relevant to communication, sensing, imaging, and quantum information processing systems that use laser light.
We demonstrate an optical receiver that achieves the quantum Chernoff bound for discriminating coherent states from thermal states in the multicopy scenario. In contrast, we find that repeated use of the receiver approaching the Helstrom bound for single-copy measurement is suboptimal in this multicopy case. Furthermore, for a large class of multicopy discrimination tasks between a pure and a mixed state, we prove that any Helstrom-bound achieving single-copy receiver is suboptimal by a factor of at least 2 in error-probability exponent compared to the multicopy quantum Chernoff bound. This behavior has a classical analog in the performance gap between soft-decision and hard-decision receivers for detecting a multicopy signal embedded in white Gaussian noise.
Laser light is widely used for communication and sensing applications, so the optimal discrimination of coherent states---the quantum states of light emitted by an ideal laser---has immense practical importance. Due to fundamental limits imposed by quantum mechanics, such discrimination has a finite minimum probability of error. While concrete optical circuits for the optimal discrimination between two coherent states are well known, the generalization to larger sets of coherent states has been challenging. In this paper, we show how to achieve optimal discrimination of any set of coherent states using a resource-efficient quantum computer. Our construction leverages a recent result on discriminating multicopy quantum hypotheses [Blume-Kohout, Croke, and Zwolak, arXiv:1201.6625]. As illustrative examples, we analyze the performance of discriminating a ternary alphabet and show how the quantum circuit of a receiver designed to discriminate a binary alphabet can be reused in discriminating multimode hypotheses. Finally, we show that our result can be used to achieve the quantum limit on the rate of classical information transmission on a lossy optical channel, which is known to exceed the Shannon rate of all conventional optical receivers.
Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are prime candidates as information carriers for optical communications. A large body of literature exists on their quantum-limited estimation and discrimination. However, very little is known about the practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and outline a theory of USD with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, auxiliary vacuum modes, and on-off photon detection. Our results indicate that, in some regimes, these currently-available optical components are typically sufficient to achieve near-optimal unambiguous discrimination of multiple, multimode coherent states.
The quantum receiver is an important tool for overcoming the standard quantum limit (SQL) of discrimination errors in optical communication. We theoretically study the quantum receivers for discriminating ternary and quaternary phase-shift-keying coherent states in terms of average error rate and mutual information. Our receiver consists of on-off--type photon detectors and displacement operations without electrical feedforward operations. We show that for the ternary signals, the receiver shows a reasonable gain from the SQL even without feedforward. This scheme is realizable with the currently available technology. For the quaternary signals feedforward operation is crucial to overcome the SQL with imperfect devices. We also analytically examine the asymptotic limit of the performance of the proposed receiver with respect to the number of feedforward steps.
Laser-light (coherent-state) modulation is sufficient to achieve the ultimate (Holevo) capacity of classical communication over a lossy and noisy optical channel, but requires a receiver that jointly detects long modulated code words with highly nonlinear quantum operations, which are near-impossible to realize using current technology. We analyze the capacity of the lossy-noisy optical channel when the transmitter uses coherent-state modulation but the receiver is restricted to a general quantum-limited Gaussian receiver, i.e., one that may involve arbitrary combinations of Gaussian operations [passive linear optics: beam splitters and phase shifters; second-order nonlinear optics (or active linear optics): squeezers, along with homodyne or heterodyne detection measurements] and any amount of classical feedforward within the receiver. Under these assumptions, we show that the Gaussian receiver that attains the maximum mutual information is either homodyne detection, heterodyne detection, or time sharing between the two, depending upon the received power level. In other words, our result shows that to exceed the theoretical limit of conventional coherent optical communication, one has to incorporate non-Gaussian, i.e., third- or higher-order nonlinear operations in the receiver. Finally we compare our Gaussian receiver limit with experimentally feasible non-Gaussian receivers and show that in the regime of low received photon flux, it is possible to overcome the Gaussian receiver limit by relatively simple non-Gaussian receivers based on photon counting.