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Physical Review Letters

Complete Characterization of Arbitrary Quantum Measurement Processes

Alfredo Luis, Luis Lorenzo Sánchez-Soto

Published Nov 1, 1999• 120 citations
Publisher

Abstract

We examine two simple and feasible practical schemes allowing the complete determination of any quantum measuring arrangement. This is illustrated with the example of parity measurement.
DOI
10.1103/physrevlett.83.3573

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We present a simulated model to study the characterization of arbitrary unitary matrices implemented in an NxN integrated photonic Mach–Zehnder interferometer (MZI) mesh. We extend the measurement method of Hoch et al., which itself generalizes Liang et al., for the practical characterization of an arbitrary unitary transformation, U, implemented on such devices as an equivalent real bordered form Urb. The measurement scheme and reconstruction shown here require fewer and simpler measurements than Liang et al. and are more complete than Hoch et al. Completing the method of Hoch using our hybrid greedy-Tabu and Breadth First Search (BFS) algorithm makes our scheme practical in the sense that only simple classical resources are needed. This allows for practical characterization of the phases of Urb outside of a laboratory setting. We examine the performance of our reconstruction under simulated measurement noise and its scaling in time as a function of system size N.
Following the approach to quantum measurement, proposed in an earlier paper, in which the apparatus is treated as a purely classical instrument to be described by the laws of classical mechanics we examine some simple experiments for the purpose of illustration. The Stern-Gerlach experiment is examined in detail, within our approach, and we see explicitly how the relevant information is transferred to the apparatus variables. We also examine a magnetic interferometer arrangement and a crossed-field arrangement. Consideration of these experiments leads us to the inclusion of an extra procedural step in our approach. This extra step is necessary so that an observer may measure the classical apparatus variables at will after the interaction is completed.
The quantum properties of quantum measurements are indispensable resources in quantum information processing and have drawn extensive research interest. The conventional approach to revealing quantum properties relies on the reconstruction of entire measurement operators by quantum detector tomography. However, many specific properties can be determined by a part of the matrix components of the measurement operators, which makes it possible to simplify the characterization process. We propose a general framework to directly obtain individual matrix elements of the measurement operators by sequentially measuring two noncompatible observables. This method allows us to circumvent the complete tomography of the quantum measurement and extract the required information. We experimentally implement this scheme to monitor the coherent evolution of a general quantum measurement by determining the off-diagonal matrix elements. The investigation of the measurement precision indicates the good feasibility of our protocol for arbitrary quantum measurements. Our results pave the way for revealing the quantum properties of quantum measurements by selectively determining the matrix components of the measurement operators.