Characterization of observables in quantum mechanics by some properties of their expected values

Jean-Marie Barbaroux, Alain Joye

Eliot Bolduc, Genevieve Gariepy, Jonathan Leach

In quantum mechanics, predictions are made by way of calculating expectation values of observables, which take the form of Hermitian operators. Non-Hermitian operators, however, are not necessarily devoid of physical significance, and they can play a crucial role in the characterization of quantum states. Here we show that the expectation values of a particular set of non-Hermitian matrices, which we call column operators, directly yield the complex coefficients of a quantum state vector. We provide a definition of the state vector in terms of measurable quantities by decomposing these column operators into observables. The technique we propose renders very-large-scale quantum states significantly more accessible in the laboratory, as we demonstrate by experimentally characterizing a 100,000-dimensional entangled state. This represents an improvement of two orders of magnitude with respect to previous phase-and-amplitude characterizations of discrete entangled states.

Marco Gramegna, Marco Genovese, G. Brida +10 more

In quantum mechanics, the eigenvalues and their corresponding probabilities specify the expectation value of a physical observable, which is known to be a statistical property related to large ensembles of particles. In contrast to this paradigm, we demonstrate a unique method allowing to extract the expectation value of a single particle, namely, the polarisation of a single protected photon, with a single experiment. This is the first realisation of quantum protective measurements.

Ryszard Jajte, B. Nowak

K.Kong Wang, Tony Jackson

Shan Geo

Protective measurement, in the language of standard quantum mechanics, is a method to measure the expectation value of an observable on a single quantum system (Aharonov and Vaidman, 1993; Aharonov, Anandan and Vaidman, 1993). For a conventional impulsive measurement, the state of the measured system is strongly entangled with the state of the measuring device during the measurement, and the measurement result is one of the eigenvalues of the measured observable. By contrast, during a protective measurement, the measured state is protected by an appropriate procedure so that it neither changes nor becomes entangled with the state of the measuring device appreciably. In this way, such protective measurements can measure the expectation values of observables on a single quantum system, and in particular, the wave function of the system can also be measured as expectation values of certain observables. It is expected that protective measurements can be performed in the near future with the rapid development of weak measurement technologies (e.g. Kocsis et al., 2011; Lundeen et al., 2011). In this chapter, we will give a clear introduction to protective measurement in quantum mechanics. Standard quantum mechanics and impulsive measurement The standard formulation of quantum mechanics, which was first developed by Dirac (1930) and von Neumann (1955), is based on the following basic principles. 1 Physical states The state of a physical system is represented by a normalized wave function or unit vector | ψ ( t )〉 in a Hilbert space. The Hilbert space is complete in the sense that every possible physical state can be represented by a state vector in the space. 2 Physical properties Every measurable property or observable of a physical system is represented by a Hermitian operator on the Hilbert space associated with the system. A physical system has a determinate value for an observable if and only if it is in an eigenstate of the observable (this is often called the eigenvalue–eigenstate link).