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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum information via novel measurements

Stephen Mark Barnett

Published Dec 15, 1997• 16 citations
Publisher

Abstract

The state projection associated with the familiar (von Neumann) measurement of an observable does not describe the most general means of interrogating a quantum system. Two problems requiring a more general description are the discrimination between non–orthogonal quantum states and the simultaneous measurement of incompatible observables. We discuss both of these problems within the context of measurements of optical polarization. We illustrate the technique of field measurement by projection synthesis by describing a possible means for determining the optical phase probability distribution.

DOI
10.1098/rsta.1997.0126

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The state projection associated with the familiar (von Neumann) measurement of an observable does not describe the most general means of interrogating a quantum system. Two problems requiring a more general description are the discrimination between non–orthogonal quantum states and the simultaneous measurement of incompatible observables. We discuss both of these problems within the context of measurements of optical polarization. We illustrate the technique of field measurement by projection synthesis by describing a possible means for determining the optical phase probability distribution.
Since the state of a quantum system is not an observable, it cannot be determined completely, in principle, if no a priori information is available. In this case, one needs to resort to various estimation strategies. The situation is completely different if a priori knowledge is available. Even unambigous discrimination is possible if the system is always prepared in a state that is selected from a known set of quantum states that are not orthogonal but linearly independent. This type of discrimination strategy has a higher probability of success by using generalised measurements than by using standard von Neumann measurements. We discuss here possible optical realizations, via generalized interferometers and single photon states, for the implementation of strategies that maximize the success rate. The implementations are based on exact analytical results. We also discuss possible applications for entanglement enhancement and quantum communication protocols.
We present an extension of the projection synthesis technique which allows, at least in principle, the determination of the probability distribution of any physical observable of a quantized optical field mode. This involves directly measuring a quantity proportional to the squared modulus of the projection of an input pure state onto the eigenstate associated with the observable to be measured. It is found that we need to synthesize a specific reference state conditioned upon the property under investigation. A general expression for these reference states is given. We provide the specific reference states needed to synthesize a measurement of canonical phase. A variant on homodyne detection is investigated using projection synthesis and we form the probability operator measure (POM) required to characterize the measurement. The POM is not that which is formed in conventional balanced homodyne detection. As an application, we utilize these results to form the POM and a quasi probability distribution function associated with a measurement scheme used to distinguish between nonorthogonal coherent states.
Extracting information from a quantum system inevitably requires the performance of a measurement, and it is no surprise that the theory of measurement plays a central role in our subject. The physical nature of the measurement process remains one of the great philosophical problems in the formulation of quantum theory. Fortunately, however, it is sufficient for us to take a pragmatic view by asking what measurements are possible and how the theory describes them, without addressing the physical mechanism of the measurement process. This is the approach we shall adopt. We shall find that it leads us to a powerful and general description of both the probabilities associated with measurement outcomes and the manner in which the observation transforms the quantum state of the measured system. The simplest form of measurement was given a mathematical formulation by von Neumann, and we shall refer to measurements of this type as von Neumann measurements or projective measurements. It is this description of measurements that is usually introduced in elementary quantum theory courses. We start with an observable quantity A represented by a Hermitian operator Â, the eigenvalues of which are the possible results of the measurement of A. The relationship between the operator, its eigenstates {|λnñ}, and its (real) eigenvalues {λn} is expressed by the eigenvalue equation . . . Â |λn_ = λn|λn_. (4.1) . . .
We present the results of generalized measurements of optical polarization designed to provide one of three or four distinct outcomes. This has allowed us to discriminate between non-orthogonal polarization states with an error probability that is close to the minimum allowed by quantum theory. The sets of states we have prepared and measured are (i) two non-orthogonal states of linear polarisation, (ii) the trine, consisting of three states of linear polarization separated by 60º and, (iii) the tetrad, consisting of four polarization states (two linear and two elliptical) arranged in a tetrahedron on the Poincaré sphere. This has enabled us to realize the generalized measurements required for (i) optimal unambiguous discrimination between two non-orthogonal polarization states, (ii) discrimination between the three trine states and four tetrad states with minimum probability of error and, (iii) maximizing the mutual information associated with our trine and tetrad measurements and showing that it exceeds the value assocated with the best possible von Neumann measurement. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We present an extension of the projection synthesis technique which allows us, at least in principle, to determine the probability distribution for any physical observable associated with a quantized optical field mode. This probability can be inferred from the photodetection statistics obtained by causing the field mode to be measured to interfere with a second mode prepared in a suitable reference state. We give an explicit expression for the required reference state. Weak-field homodyne detection using photon counting is, perhaps, the simplest system to which projection synthesis can be applied. We find the complete set of synthesized projectors for this system and show that they form a probability operator measure. We apply this set of projectors to study the discrimination between coherent states and to measure a quasiprobability distribution.