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arXiv:2009.13201 · Physical Review Research

Optimal quantum-programmable projective measurements with coherent states

Niraj Kumar, Ulysse Chabaud, Elham Kashefi, Damian Markham, Eleni Diamanti

Published Oct 14, 2021• 6 citations
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Abstract

We consider a device which can be programed using coherent states of light to approximate a given projective measurement on an input coherent state. We provide and discuss three practical implementations of this programmable projective measurement device with linear optics, involving only balanced beam splitters and single photon threshold detectors. The three schemes optimally approximate any projective measurement onto a program coherent state. We further extend these to the case where there are no assumptions on the input state. In this setting, we show that our scheme enables an efficient verification of an unbounded untrusted source with only local coherent states, balanced beam splitters, and threshold detectors. Exploiting the link between programmable measurements and generalized swap test, we show as a direct application that our schemes provide an asymptotically quadratic improvement in existing quantum fingerprinting protocol to approximate the Euclidean distance between two unit vectors.

DOI
10.1103/physrevresearch.3.043035

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We consider a device which can be programed using coherent states of light to approximate a given projective measurement on an input coherent state. We provide and discuss three practical implementations of this programmable projective measurement device with linear optics, involving only balanced beam splitters and single photon threshold detectors. The three schemes optimally approximate any projective measurement onto a program coherent state. We further extend these to the case where there are no assumptions on the input state. In this setting, we show that our scheme enables an efficient verification of an unbounded untrusted source with only local coherent states, balanced beam splitters, and threshold detectors. Exploiting the link between programmable measurements and generalized swap test, we show as a direct application that our schemes provide an asymptotically quadratic improvement in existing quantum fingerprinting protocol to approximate the Euclidean distance between two unit vectors.
We present a scheme for a universal device which can be programed by quantum states to approximate a chosen projective measurement to a given precision. Our scheme can be viewed as an extension of the swap test to the instance where one state is supplied many times. As such, it has many potential applications given the variety of quantum information tasks which make use of the swap test. In particular, we show that our scheme is optimal for state discrimination under the one-sided error requirement, and optimally approximates any projective measurement. Furthermore, we propose a practical implementation of our scheme with passive linear optics, which involves a simple interferometer composed only of balanced beam splitters.
We investigate the implementation of a measurement for binary optical coherent states that minimizes the error probability with a fixed rate of an inconclusive result. As a special case, the optimal measurement without inconclusive results, which is expressed as a projective measurement, can be implemented using only a beam splitter, a local coherent light source, a photon detector, and a feedback circuit. We find that the optimal measurement for binary optical coherent states with any probability of an inconclusive result can be implemented by the same optical components, even though the measurement is generally not projective.
The possibility of implementing a given photonic projective measurement with linear optics and photon detectors is discussed. This problem can be viewed as a single-shot discrimination of orthogonal pure quantum states. It is particularly shown that any two orthogonal states can be perfectly discriminated using only linear optics, photon counting, coherent ancillary states, and feedforward. This means that one can construct any binary projective measurement with these means, but without any nonclassical ancillary state. The statement holds in the asymptotic limit of a large number of these physical resources. To extend this result, we also address the problem of discriminating a simple set of three orthogonal states.
We present a protocol for quantum fingerprinting that is ready to be implemented with current technology and is robust to experimental errors. The basis of our scheme is an implementation of the signal states in terms of a coherent state in a superposition of time-bin modes. Experimentally, this requires only the ability to prepare coherent states of low amplitude and to interfere them in a balanced beam splitter. The states used in the protocol are arbitrarily close in trace distance to states of O(log2n)O({log}_{2}n)O(log2​n) qubits, thus exhibiting an exponential separation in abstract communication complexity compared to the classical case. The protocol uses a number of optical modes that is proportional to the size nnn of the input bit strings but a total mean photon number that is constant and independent of nnn. Given the expended resources, our protocol achieves a task that is provably impossible using classical communication only. In fact, even in the presence of realistic experimental errors and loss, we show that there exist a large range of input sizes for which our quantum protocol transmits an amount of information that can be more than two orders of magnitude smaller than a classical fingerprinting protocol.
Multiport interferometers with NNN ports are basic devices in both classical and quantum photonics. Ideally, they implement a linear unitary transformation between the input and output electric field vectors with NNN components, each associated with a spatial mode of classical coherent light or a single photon. Standard designs for a fully reconfigurable universal multiport interferometer are given by the Reck or the Clements schemes. In this work, we introduce routing schemes to implement a generic unitary transformation on classical coherent light or single photons using linear or tree geometries via multiple projective measurements on a single detector with the minimum number of components. Then, we generalize this result to the case of any multi-photon state for scattershot boson sampling experiments with multi-routing schemes. Finally, we test the robustness of routing schemes compared to universal schemes with respect to losses and phase noise.