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arXiv:1605.08922 · Physical Review A

Simple procedure for phase-space measurement and entanglement validation

R. P. Rundle, P. W. Mills, Todd Tilma, John H. Samson, M. J. Everitt

Published Aug 10, 2017• 71 citations
arXivHTMLPDF

Abstract

It has recently been shown that it is possible to represent the complete quantum state of any system as a phase-space quasiprobability distribution (Wigner function) [Phys. Rev. Lett. 117, 180401 (2016)]. Such functions take the form of expectation values of an observable that has a direct analogy to displaced parity operators. In this work we give a procedure for the measurement of the Wigner function that should be applicable to any quantum system. We have applied our procedure to IBM's Quantum Experience five-qubit quantum processor to demonstrate that we can measure and generate the Wigner functions of two different Bell states as well as the five-qubit Greenberger--Horne--Zeilinger state. Because Wigner functions for spin systems are not unique, we define, compare, and contrast two distinct examples. We show how the use of these Wigner functions leads to an optimal method for quantum state analysis especially in the situation where specific characteristic features are of particular interest (such as for spin Schr\"odinger cat states). Furthermore we show that this analysis leads to straightforward, and potentially very efficient, entanglement test and state characterization methods.

DOI
10.1103/physreva.96.022117

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It has recently been shown that it is possible to represent the complete quantum state of any system as a phase-space quasiprobability distribution (Wigner function) [Phys. Rev. Lett. 117, 180401 (2016)]. Such functions take the form of expectation values of an observable that has a direct analogy to displaced parity operators. In this work we give a procedure for the measurement of the Wigner function that should be applicable to any quantum system. We have applied our procedure to IBM's Quantum Experience five-qubit quantum processor to demonstrate that we can measure and generate the Wigner functions of two different Bell states as well as the five-qubit Greenberger--Horne--Zeilinger state. Because Wigner functions for spin systems are not unique, we define, compare, and contrast two distinct examples. We show how the use of these Wigner functions leads to an optimal method for quantum state analysis especially in the situation where specific characteristic features are of particular interest (such as for spin Schr"odinger cat states). Furthermore we show that this analysis leads to straightforward, and potentially very efficient, entanglement test and state characterization methods.
We measure complete and continuous Wigner functions of a two-level cesium atom in both a nearly pure state and highly mixed states. We apply the method [T. Tilma et al., Phys. Rev. Lett. 117, 180401 (2016)] of strictly constructing continuous Wigner functions for qubit or spin systems. We find that the Wigner function of all pure states of a qubit has negative regions and the negativity completely vanishes when the purity of an arbitrary mixed state is less than 23\frac{2}{3}32​. We experimentally demonstrate these findings using a single cesium atom confined in an optical dipole trap, which undergoes a nearly pure dephasing process. Our method can be applied straightforwardly to multi-atom systems for measuring the Wigner function of their collective spin state.
Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which can be proved in the form of the Fourier-Weyl relation between density matrices and characteristic functions. This relationship constitutes the bridge between phase space and Hilbert space descriptions, which is useful in several applications in the context of quantum information. This chapter shows that the marginal Wigner function along any phase space direction describes the probability distribution of the quantum measurement of the associated quadrature operator. Hence, as far as Gaussian states are concerned, the Wigner function provides one with a local, 'realistic' model to describe quadrature measurements. If one restricts to quadrature measurements, such systems may be mimicked by multivariate classical Gaussian distributions and will never show any signature of quantum non-locality, such as a violation of Bell or CHSH inequalities.
Starting from Wigner's definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables ( ) , q p of phase space and using the known relation to the parity operator.One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics.For the coherent states we show that their Fourier transforms are again coherent states.We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries.It is not well suited for the calculation of expectation values.A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new.We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult.The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values.By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics.Both considered examples, the square well and the coherent phase states, belong to systems with ( ) 1,1 SU symmetry with the same index 1 2 k = of unitary irreducible representations.
Abstract Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867 ), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206 ) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531 ). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.