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Physical Review A

Quasiprobability distributions of negative binomial states

Richard D’Souza, Adya Prasad Mishra

Published May 1, 1992• 4 citations
Publisher

Abstract

We present the s-parametrized quasiprobability distributions for the negative binomial states. Marked changes in the quasiprobability distributions W(\ensuremath{\alpha},\ensuremath{\epsilon},s) are exhibited by states that are close to the random-phase coherent state (\ensuremath{\epsilon}=0), as the parameter s is varied continuously from s=-1, corresponding to the Q function, to s=1, corresponding to the P function. However, states close to the thermal-noise state (\ensuremath{\epsilon}=1) exhibit the usual Gaussian-like distributions for all values of s.

DOI
10.1103/physreva.45.6925

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We present the s-parametrized quasiprobability distributions for the negative binomial states. Marked changes in the quasiprobability distributions W(\ensuremath{\alpha},\ensuremath{\epsilon},s) are exhibited by states that are close to the random-phase coherent state (\ensuremath{\epsilon}=0), as the parameter s is varied continuously from s=-1, corresponding to the Q function, to s=1, corresponding to the P function. However, states close to the thermal-noise state (\ensuremath{\epsilon}=1) exhibit the usual Gaussian-like distributions for all values of s.
The Glauber-Sudarshan P representation of negative binomial states is presented and its properties are discussed. It is shown that for the limiting cases the states approach the random-phase coherent state and the thermal noise state. Thus these states may allow a unified description of coherent and thermal states of the field. The two-dimensional representation of the P function is plotted on the complex-\ensuremath{\alpha} plane. The probability density function of waiting-time intervals between the arrivals of successive photons on a photodetector irradiated by a beam of the negative binomial light is also investigated.
Properties of the radiation fields in negative binomial states are investigated. The density matrix of the field is related to the density matrix ${\mathrm{\ensuremath{\rho}}}_{\mathit{c}}$ of the chaotic field via as{\mathit{a}}^{\mathit{s}}as${\mathrm{\ensuremath{\rho}}}_{\mathit{c}}$${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathit{s}}$. Various quasiprobability distributions and the thermofield representation for negative binomial states of the field are derived. The production of negative binomial distribution in a number of nonlinear processes is demonstrated.
Negativity in a quasiprobability representation is typically interpreted as an indication of nonclassical behavior. However, this does not preclude states that are non-negative from exhibiting phenomena typically associated with quantum mechanics---the single qubit stabilizer states have non-negative Wigner functions and yet play a fundamental role in many quantum information tasks. We seek to determine what other sets of quantum states and measurements of a qubit can be non-negative in a quasiprobability distribution, and to identify nontrivial groups of unitary transformations that permute the states in such a set. These sets of states and measurements are analogous to the single qubit stabilizer states. We show that no quasiprobability representation of a qubit can be non-negative for more than two bases in any plane of the Bloch sphere. Furthermore, there is a unique set of four bases that can be non-negative in an arbitrary quasiprobability representation of a qubit. We provide an exhaustive list of the sets of single qubit bases that are non-negative in some quasiprobability distribution and are also closed under a group of unitary transformations. This list includes two nontrivial families of three bases that both include the single qubit stabilizer states as a special case. For qudits, we prove that there can be no more than 2d2{2}^{{d}^{2}}2d2 states in non-negative bases of a ddd-dimensional Hilbert space in any quasiprobability representation. Furthermore, these bases must satisfy certain symmetry constraints, corresponding to requiring the bases to be sufficiently different from each other.