Measurements Of Non-commuting Observables Research Articles

Simultaneous measurement of non-commuting observables in entangled systems

In an EPR setup, simultaneous measurement of two non-commuting observables is considered at station A and, possibly after a time delay, at station B. At each station one apparatus measures the z-component of a spin, while a second one measures the x-component of the same spin simultaneously. The dynamics of the measurements is solved explicitly. It is shown that no violations of Bell inequalities occur in Bell-like correlation experiments, and that, in fact, experiments of this type are subject to even stronger constraints. This is caused by the mutual influence of the two apparatuses coupled to the same spin, an effect which occurs at both stations.

A scheme for performing strong and weak sequential measurements of non-commuting observables

Quantum systems usually travel a multitude of different paths when evolving through time from an initial to a final state. In general, the possible paths will depend on the future and past boundary conditions, as well as the system’s dynamics. We present a gedanken experiment where a single system apparently follows mutually exclusive paths simultaneously, each with probability one, depending on which measurement was performed. This experiment involves the measurement of observables that do not correspond to Hermitian operators. Our main result is a scheme for measuring these operators. The scheme is based on the erasure protocol [Brodutch and Cohen (Phys. Rev. Lett. 116:070404, 2016)] and allows a wide range of sequential measurements at both the weak and strong limits. At the weak limit the back action of the measurement cannot be used to account for the surprising behaviour and the resulting weak values provide a consistent yet strange account of the system’s past.

Holographic thought experiments

The Hamiltonian of classical anti\char21{}de Sitter gravity is a pure boundary term on-shell. If this remains true in nonperturbative quantum gravity then (i) boundary observables will evolve unitarily in time and (ii) the algebra of boundary observables is the same at all times. In particular, information available at the boundary at any one time ${t}_{1}$ remains available at any other time ${t}_{2}$. Since there is also a sense in which the equations of motion propagate information into the bulk, these observations raise what may appear to be potential paradoxes concerning simultaneous (or spacelike separated) measurements of noncommuting observables, one at the asymptotic boundary and one in the interior. We argue that such potentially paradoxical settings always involve a breakdown of semiclassical gravity. In particular, we present evidence that making accurate holographic measurements over short time scales radically alters the familiar notion of causality. We also describe certain less intrinsically paradoxical settings which illustrate the above boundary unitarity and render the notion more concrete.

Joint measurements of spin, operational locality, and uncertainty

Joint, or simultaneous, measurements of non-commuting observables are possible within quantum mechanics, if one accepts an increase in the variances of the jointly measured observables. In this paper, we discuss joint measurements of a spin 1/2 particle along any two directions. Starting from an operational locality principle, it is shown how to obtain a bound on how sharp the joint measurement can be. We give a direct interpretation of this bound in terms of an uncertainty relation.

Noisy simultaneous measurement of noncommuting observables in eight- and twelve-port homodyne detection

Classically eight- and twelve-port homodyne detectors can be used to measure the Stokes parameters of a two-mode field simultaneously. We analyse these schemes in the quantum case as approximate or noisy joint measurements of the noncommuting Stokes operators. The statistics of the measurement are described in terms of these operators and noise. The twelve-port detector is also examined as allowing a noisy simultaneous measurement of the quadratures and intensity of a one-mode field. The vectors governing the statistics of the measurement are found. A simple procedure for obtaining the Wigner function emerges.

Imprecise measurements and non-locality in quantum mechanics

It has been recently suggested that (imprecise) measurements of non-commuting observables on correlated quantum systems may lead to observable non-local effects. The inapplicability of several “impossibility proofs” to this case is noted; however, a suitable result of Kraus is pointed out, which shows the untenability of such a scheme.

Simultaneous measurements and squeezed states of light

The relationship between performing simultaneous measurements of noncommuting observables and observing squeezed states of light is discussed.

?Simultaneous measurement? from the standpoint of quantum estimation theory

The purpose of the simultaneous measurement of noncommuting quantum observables can be viewed as the joint estimation of parameters of the density operator of the quantum system. Joint estimation involves the application of a multiply parameterized operator-valued measure. An example related to the simultaneous estimation of the position and velocity of a particle is given. Conceptual difficulties attending simultaneous measurement of noncommuting observables are avoided by this formation.