Quasiprobability Representations Research Articles

Erratum: Non-negative subtheories and quasiprobability representations of qubits [Phys. Rev. A85, 062121 (2012)

Erratum: Non-negative subtheories and quasiprobability representations of qubits [Phys. Rev. A<b>85</b>, 062121 (2012)

Non-negative subtheories and quasiprobability representations of qubits

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 for a qubit can be non-negative in a quasiprobability representation, and to identify nontrivial unitary groups 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 four bases and that the non-negative bases in any quasiprobability representation must satisfy certain symmetry constraints. We provide an exhaustive list of the sets of single qubit bases that are non-negative in some quasiprobability representation and are also permuted by a nontrivial unitary group. This list includes two families of three bases that both include the single qubit stabilizer states as a special case and a family of four bases whose symmetry group is the Pauli group. For higher dimensions, we prove that there can be no more than 2^{d^2} states in non-negative bases of a d-dimensional Hilbert space in any quasiprobability representation. Furthermore, these bases must satisfy certain symmetry constraints, corresponding to requiring the bases to be sufficiently complementary to each other.

Necessity of negativity in quantum theory

A unification of the set of quasiprobability representations using the mathematical theory of frames was recently developed for quantum systems with finite-dimensional Hilbert spaces, in which it was proven that such representations require negative probability in either the states or the effects. In this article we extend those results to Hilbert spaces of infinite dimension, for which the celebrated Wigner function is a special case. Hence, this article presents a unified framework for describing the set of possible quasiprobability representations of quantum theory, and a proof that the presence of negativity is a necessary feature of such representations.

Framed Hilbert space: hanging the quasi-probability pictures of quantum theory

Building on earlier work, we further develop a formalism based on the mathematical theory of frames that defines a set of possible phase-space or quasi-probability representations of finite-dimensional quantum systems. We prove that an alternate approach to defining a set of quasi-probability representations, based on a more natural generalization of a classical representation, is equivalent to our earlier approach based on frames, and therefore is also subject to our no-go theorem for a non-negative representation. Furthermore, we clarify the relationship between the contextuality of quantum theory and the necessity of negativity in quasi-probability representations and discuss their relevance as criteria for non-classicality. We also provide a comprehensive overview of known quasi-probability representations and their expression within the frame formalism.

Representation of entanglement by negative quasiprobabilities

Any bipartite quantum state has quasi-probability representations in terms of separable states. For entangled states these quasi-probabilities necessarily exhibit negativities. Based on the general structure of composite quantum states, one may reconstruct such quasi-propabilities from experimental data. Because of ambiguity, the quasi-probabilities obtained by the bare reconstruction are insufficient to identify entanglement. An optimization procedure is introduced to derive quasi-probabilities with a minimal amount of negativity. Negativities of optimized quasi-probabilities unambiguously prove entanglement, their positivity proves separability.

Quantum noise in optical fibers I Stochastic equations

We analyze the quantum dynamics of radiation propagating in a single-mode optical fiber with dispersion, nonlinearity, and Raman coupling to thermal phonons. We start from a fundamental Hamiltonian that includes the principal known nonlinear effects and quantum-noise sources, including linear gain and loss. Both Markovian and frequency-dependent, non-Markovian reservoirs are treated. This treatment allows quantum Langevin equations, which have a classical form except for additional quantum-noise terms, to be calculated. In practical calculations, it is more useful to transform to Wigner or +P quasi-probability operator representations. These transformations result in stochastic equations that can be analyzed by use of perturbation theory or exact numerical techniques. The results have applications to fiber-optics communications, networking, and sensor technology.

Entropic measure of wave-packet spreading and ionization in laser-driven atoms.

We consider wave-packet spreading and ionization of a one-dimensional model hydrogen atom interacting with a strong high-frequency laser field in the stabilization domain. We demonstrate the effect of the pulse shape on the dynamics in phase space and visualize our results with both the Wigner and the Q function phase-space quasiprobability representation. The Wehrl entropy is introduced and utilized as a convenient one-parameter measure of wave-packet spreading, phase-space localization, and ionization. @S1050- 2947~96!02007-0# We therefore also consider a second phase-space distribution function, the Husimi or Q representation @10#, which is al- ways positive. The Husimi distribution can be used to calcu- late an information-theoretic measure of the number of par- ticipating quasiclassical Gaussian wave packets. This measure is the Wehrl entropy @11# and, apart from interfer- ence terms, is determined by the number of Gaussian coher- ent states to ''tile'' its phase-space contour. The Wehrl en- tropy has been investigated and utilized by a group of authors @12#. It has recently been studied within a wider class of entropies that are based on a comparison of the wave function with an arbitrary basis of states in phase space @13#. This Wehrl entropy is directly related to the uncertainty area of the Q function in phase space and it is appropriate as a measure of wave packet spreading and ionization. We will use the Wehrl entropy to study the effect of these phenomena in the stabilization domain using both an adiabatic sin 2 and a nonadiabatic trapezoidal pulse. In the next section we will describe our numerical model, based on the solution of the time-dependent Schrodinger equation in one dimension. We will then investigate the sta- bilization dynamics for two different pulse shapes ~one adia- batic, the other with a fast turn-on! using the Wigner func- tion, before turning our attention to the Q function and finally to the Wehrl entropy.

Quantum optical master equations: The one-atom laser.

We present a detailed numerical study of the one-atom laser, that is, a single two-level atom interacting with one lasing mode, whereby both the atom and the photon field are coupled to reservoirs. The stationary as well as the dynamical properties of the model are calculated directly from the quantum master equation with the help of two numerical methods. These numerical methods do not need any quasi-probability representation and they do not require approximations. We find that the one-atom laser exhibits most of the typical features of a normal laser. In the region far below threshold some aspects, among them the linewidth, are changed due to eigenvalues of the master equation with imaginary parts