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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Reconstruction of quantum states from propensities

Alfred Wünsche, Vladimír Bužek

Published Aug 1, 1997• 22 citations
Publisher

Abstract

We focus our attention on the problem of reconstruction of density operators of quantum states from propensities, i.e. generalized quasiprobability density distributions obtained by quantum filtering. We consider propensities obtained by filtering with pure Gaussian as well as non-Gaussian states and we present two examples: when the filter is in a squeezed coherent state and in a Fock state, respectively. We also show that even in the case of filtering with statistical mixtures a complete reconstruction of a density operator of the measured quantum state can be performed.

DOI
10.1088/1355-5111/9/4/012

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We focus our attention on the problem of reconstruction of density operators of quantum states from propensities, i.e. generalized quasiprobability density distributions obtained by quantum filtering. We consider propensities obtained by filtering with pure Gaussian as well as non-Gaussian states and we present two examples: when the filter is in a squeezed coherent state and in a Fock state, respectively. We also show that even in the case of filtering with statistical mixtures a complete reconstruction of a density operator of the measured quantum state can be performed.
The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator. While the former aspect is a main focus of current research, the latter decomposition property has been studied less frequently. In this contribution, we introduce a method for the generalized expansion of quantum states in terms of so-called quasistates. In contrast to the quasiprobability decomposition through nonclassical distributions and pure-state operators, our technique results in classical probabilities and nonpositive semidefinite operators, defining the notion of quasistates, that carry the information about the nonclassical characteristics of the system. Therefore, our method presents a complementary approach to prominent quasiprobability representations. We explore the usefulness of our technique with several examples, including applications for quantum state reconstruction and the representation of nonclassical light. In addition, using our framework, we also demonstrate that inseparable quantum correlations can be described in terms of classical joint probabilities and tensor-product quasistates for an unambiguous identification of quantum entanglement.
We derive an expression for a density operator estimated via Bayesian quantum inference in the limit of an infinite number of measurements. This expression is derived under the assumption that the reconstructed system is in a pure state. In this case the estimation corresponds to an averaging over a generalized microcanonical ensemble of pure states satisfying a set of constraints imposed by the measured mean values of the observables under consideration. We show that via the ``purification'' ansatz, statistical mixtures can also be consistently reconstructed via the quantum Bayesian inference scheme. In this case the estimation corresponds to averaging over the generalized canonical ensemble of states satisfying the given constraints, and the reconstructed density operator maximizes the von Neumann entropy (i.e., this density operator is equal to the generalized canonical density operator which follows from the Jaynes principle of maximum entropy). We study in detail the reconstruction of the spin-1/2 density operator and discuss the logical connection between the three reconstruction schemes, i.e., (1) quantum Bayesian inference, (2) reconstruction via the Jaynes principle of maximum entropy, and (3) discrete quantum tomography.
We present a Q function of a state of a quantum-mechanical system in a finite-dimensional Hilbert space. This discrete Q function is defined with the help of the W\'odkiewicz concept of propensities, i.e., we define the Q function as a discrete convolution of two Wigner functions based on Wootter's formalism, one of the state itself and one of the filter state. The discrete Q function takes nonnegative values in all ``points'' of the discrete phase space and is normalized and it is possible to reconstruct from it the density operator of the state under consideration. We analyze Q-function graphs for several states of interest.
The first part is a systematical representation of the Wigner quasiprobability in representations by canonical variables and pairs of complex conjugate variables with particular attention to the reconstruction of the density operator from the Wigner quasiprobability to show their equivalence. All formulae are derived from basic postulates of quantum mechanics. Problems of operator orderings are touched. The second part considers the integration over the amplitude in the Wigner quasiprobability and its specialization for coherent states and for even and odd coherent states and provides phase distributions which are presented in graphics that are new in their kind. It is demonstrated that the Heitler number phase uncertainty relation is incorrect. The problem of the transition from quantum to classical distribution functions by ℏ→0 is discussed with the conjecture that the inertial point of the distributions plays a basic role.
The algebraic quantification of nonclassicality, which naturally arises from the quantum superposition principle, is related to properties of regular nonclassicality quasiprobabilities. The latter are obtained by non-Gaussian filtering of the Glauber-Sudarshan PPP function. They yield lower bounds for the degree of nonclassicality. We also derive bounds for convex combinations of Gaussian states for certifying quantum non-Gaussianity directly from the experimentally accessible nonclassicality quasiprobabilities. Other quantum-state representations, such as sss-parametrized quasiprobabilities, insufficiently indicate or even fail to directly uncover detailed information on the properties of quantum states. As an example, our approach is applied to multi-photon-added squeezed vacuum states.
Quantum tomography is a cornerstone of quantum information science, enabling the reconstruction of states and channels from experimental data. Here we introduce a new paradigm, temporal state tomography (TST), for reconstructing quantum processes across multiple times. Our approach is based on temporal quasiprobability distributions (TQDs), which, in the informationally complete setting, provide a complete description of multi-time quantum processes and uniquely determine temporal states. We formulate TST as a unified framework for reconstructing both density operators and quantum channels within a single scheme. We show that any TQD can be obtained via classical post-processing of measurement outcomes generated by a fixed set of quantum instruments, thereby establishing a direct operational route to accessing TQDs experimentally. For informationally complete TQDs, the associated temporal state can be reconstructed via a temporal Bloch-type representation. Leveraging this correspondence, we derive the sample complexity of TST, thereby quantifying its statistical efficiency.