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arXiv:quant-ph/0510086 · Journal of Physics A Mathematical and General

Optimal pointers for joint measurement of σxand σzvia homodyne detection

Bas Janssens, Luc Bouten

Published Mar 1, 2006• 4 citations
arXivHTMLPDF

Abstract

We study a model of a qubit in interaction with the electromagnetic field. By means of homodyne detection, the field-quadrature A t + A * t is observed continuously in time. Due to the interaction, information about the initial state of the qubit is transferred to the field, thus influencing the homodyne measurement results. We construct random variables (pointers) on the probability space of homodyne measurement outcomes having distributions close to the initial distributions of σ x and σ z . Using variational calculus, we find the pointers that are optimal. These optimal pointers are very close to hitting the bound imposed by Heisenberg's uncertainty relation on joint measurement of two non-commuting observables. We close the paper by giving the probability densities of the pointers.

DOI
10.1088/0305-4470/39/11/013

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We study a model of a qubit in interaction with the electromagnetic field. By means of homodyne detection, the field-quadrature A t + A * t is observed continuously in time. Due to the interaction, information about the initial state of the qubit is transferred to the field, thus influencing the homodyne measurement results. We construct random variables (pointers) on the probability space of homodyne measurement outcomes having distributions close to the initial distributions of σ x and σ z . Using variational calculus, we find the pointers that are optimal. These optimal pointers are very close to hitting the bound imposed by Heisenberg's uncertainty relation on joint measurement of two non-commuting observables. We close the paper by giving the probability densities of the pointers.
A fundamental feature of quantum mechanics is that there are observables which can be measured jointly only when some noise is added to them. Their sharp versions are said to be incompatible. In this work we investigate time-continuous joint qubit observables induced by a indirect time-continuous measurements. In particular we study a paradigmatic situation where a qubit is interacting with a mode of light in a cavity and the light escaping the cavity is continuously monitored. We find that the properties of the qubit observables can be tuned by changing the type of the monitoring scheme or by tuning the initial state of the cavity. We observe that homodyning two orthogonal quadratures produces an optimal pair of biased jointly measurable qubit observables.
A fundamental feature of quantum mechanics is that there are observable pairs that cannot be measured jointly, such as observables corresponding to position and momentum or spin direction measurements. However, unsharp versions of non-jointly measurable observables may become jointly measurable. In this study, we investigate the joint measurability of time-continuous observables emerging from indirect time-continuous measurements. In particular, we study a paradigmatic situation where a qubit is interacting with a mode of light in a cavity, and the light escaping the cavity is continuously monitored. We find that the properties of the observables can be tuned by changing the type of the monitoring scheme or by tuning the initial state of the cavity. In particular, we demonstrate that heterodyne measurements are a joint measurement of a noisy homodyne measurement of a pair of canonical quadratures. Moreover, we investigate the purity of the induced qubit observables as a function of the noise.
Homodyne detection is a typical continuous-variable measurement scheme, ubiquitously used in quantum optics and quantum information. A conventional homodyne detector performs a projective measurement onto a quadrature of the electromagnetic field mode, thus yielding a continuously distributed measurement outcome. The outcome distribution is the marginal of the Wigner function of the state. Homodyne detection also forms the basis of heterodyne, or “dual homodyne” detection, which corresponds to measuring the Q function of the state.
We discuss a balanced homodyne detection scheme with imperfect detectors in the framework of the operational approach to quantum measurement. We show that a realistic homodyne measurement is described by a family of operational observables that depends on the experimental setup, rather than a single field quadrature operator. We find an explicit form of this family, which fully characterizes the experimental device and is independent of a specific state of the measured system. We also derive operational homodyne observables for the setup with a random phase, which has been recently applied in an ultrafast measurement of the photon statistics of a pulsed diode laser. The operational formulation directly gives the relation between the detected noise and the intrinsic quantum fluctuations of the measured field. We demonstrate this with two examples: the operational uncertainty relation for the field quadratures, and the homodyne detection of suppressed fluctuations in photon statistics.
Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which we apply to arbitrary finite collections of projective observables on a finite dimensional Hilbert space. The quantification of errors is based on an arbitrary cost function, which assigns a penalty to getting result x rather than y, for any pair ( x , y ) . This induces a notion of optimal transport cost for a pair of probability distributions, and we include an Appendix with a short summary of optimal transport theory as needed in our context. There are then different ways to form an overall figure of merit from the comparison of distributions. We consider three, which are related to different physical testing scenarios. The most thorough test compares the transport distances between the marginals of a joint measurement and the reference observables for every input state. Less demanding is a test just on the states for which a “true value” is known in the sense that the reference observable yields a definite outcome. Finally, we can measure a deviation as a single expectation value by comparing the two observables on the two parts of a maximally-entangled state. All three error quantities have the property that they vanish if and only if the tested observable is equal to the reference. The theory is illustrated with some characteristic examples.