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arXiv:quant-ph/0304057 · Physical Review A

Simple criteria for the implementation of projective measurements with linear optics

Peter van Loock, Norbert Lütkenhaus

Published Jan 9, 2004• 39 citations
arXivHTMLPDF

Abstract

We derive a set of criteria to decide whether a given projection measurement can be, in principle, exactly implemented solely by means of linear optics. The derivation can be adapted to various detection methods, including photon counting and homodyne detection. These criteria enable one to obtain no-go theorems easily for the exact distinguishability of orthogonal quantum states with linear optics, including the use of auxiliary photons and conditional dynamics.

DOI
10.1103/physreva.69.012302

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We derive a set of criteria to decide whether a given projection measurement can be, in principle, exactly implemented solely by means of linear optics. The derivation can be adapted to various detection methods, including photon counting and homodyne detection. These criteria enable one to obtain no-go theorems easily for the exact distinguishability of orthogonal quantum states with linear optics, including the use of auxiliary photons and conditional dynamics.
We present a set of criteria to decide whether a given projection measurement can be, in principle, exactly implemented by quantum optical means when only quadratic interaction Hamiltonians are available. These criteria hence apply to any implementation that is based on a linear Bogoliubov transformation of the optical modes, i.e., an arbitrary combination of linear optical elements (multi‐port interferometers), multi‐mode squeezers, and phase‐space displacements. The derivation can be adapted to different detection methods, including photon counting and homodyne detection. Our criteria enable one to obtain easily No‐Go theorems for the exact distinguishability of orthogonal quantum states using linear Bogoliubov transformations, auxiliary photons, and conditional dynamics.
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 investigate the possibility of implementing a given projection measurement using linear optics and arbitrarily fast feedforward based on the continuous detection of photons. In particular, we systematically derive the so-called Dolinar scheme that achieves the minimum-error discrimination of binary coherent states. Moreover, we show that the Dolinar-type approach can also be applied to projection measurements in the regime of photonic-qubit signals. Our results demonstrate that for implementing a projection measurement with linear optics, in principle, unit success probability may be approached even without the use of expensive entangled auxiliary states, as they are needed in all known (near-)deterministic linear-optics proposals.
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 state- ment 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.
A major challenge in photonic quantum technologies is developing strategies to prepare suitable discrete-variable quantum states using simple input states, linear optics, and auxiliary photon measurements to identify successful outcomes. Fundamentally, this challenge arises from the lack of strong non-linearities on the single-photon level, meaning that photonic state preparation based on linear optics cannot benefit from the deterministic gate-based approach available to other physical platforms. Instead, the preparation of quantum states can be probabilistically implemented using single photons, linear-optical networks, and photon detection. However, determining whether an input state can be transformed into a target state using a specific measurement pattern - a problem that can be mapped to deciding the feasibility of a system of polynomial equations - is a complex problem in general. To solve it, we apply the Nullstellensatz Linear Algebra algorithm from algebraic geometry to quantum state generation; this can provide definitive no-go results by proving infeasibility when the state preparation task in question has no solution. We demonstrate this capability to validate and establish lower bounds on the physical resource requirements for the realization of several ubiquitous optical states and gates.
We discuss the problem of implementing generalized measurements [positive operator-valued measures (POVMs)] with linear optics, either based upon a static linear array or including conditional dynamics. In our approach, a given POVM shall be identified as a solution to an optimization problem for a chosen cost function. We formulate a general principle: the implementation is only possible if a linear-optics circuit exists for which the quantum mechanical optimum (minimum) is still attainable after dephasing the corresponding quantum states. The general principle enables us, for instance, to derive a set of necessary conditions for the linear-optics implementation of the POVM that realizes the quantum mechanically optimal unambiguous discrimination of two pure nonorthogonal states. This extends our previous results on projection measurements and the exact discrimination of orthogonal states.