Efficient measurements, purification, and bounds on the mutual information

When a measurement is made on a quantum system in which classical information is encoded, the measurement reduces the observers average Shannon entropy for the encoding ensemble. This reduction, being the {\em mutual information}, is always non-negative. For efficient measurements the state is also purified; that is, on average, the observers von Neumann entropy for the state of the system is also reduced by a non-negative amount. Here we point out that by re-writing a bound derived by Hall [Phys. Rev. A {\bf 55}, 100 (1997)], which is dual to the Holevo bound, one finds that for efficient measurements, the mutual information is bounded by the reduction in the von Neumann entropy. We also show that this result, which provides a physical interpretation for Hall's bound, may be derived directly from the Schumacher-Westmoreland-Wootters theorem [Phys. Rev. Lett. {\bf 76}, 3452 (1996)]. We discuss these bounds, and their relationship to another bound, valid for efficient measurements on pure state ensembles, which involves the subentropy.Comment: 4 pages, Revtex4. v3: rewritten and reinterpreted somewha

Experimentally feasible measures of distance between quantum operations

We present two measures of distance between quantum processes based on the superfidelity, introduced recently to provide an upper bound for quantum fidelity. We show that the introduced measures partially fulfill the requirements for distance measure between quantum processes. We also argue that they can be especially useful as diagnostic measures to get preliminary knowledge about imperfections in an experimental setup. In particular we provide quantum circuit which can be used to measure the superfidelity between quantum processes. As the behavior of the superfidelity between quantum processes is crucial for the properties of the introduced measures, we study its behavior for several families of quantum channels. We calculate superfidelity between arbitrary one-qubit channels using affine parametrization and superfidelity between generalized Pauli channels in arbitrary dimensions. Statistical behavior of the proposed quantities for the ensembles of quantum operations in low dimensions indicates that the proposed measures can be indeed used to distinguish quantum processes.Comment: 9 pages, 4 figure

Collective vs local measurements in qubit mixed state estimation

We discuss the problem of estimating a general (mixed) qubit state. We give the optimal guess that can be inferred from any given set of measurements. For collective measurements and for a large number $N$ of copies, we show that the error in the estimation goes as 1/N. For local measurements we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that standard tomographic techniques lead to an error proportional to $1/N^{1/4}$, while with our optimal data processing it is proportional to $1/N^{3/4}$.Comment: 4 pages, 1 figure, minor style changes, refs. adde

Quantum inference of states and processes

The maximum-likelihood principle unifies inference of quantum states and processes from experimental noisy data. Particularly, a generic quantum process may be estimated simultaneously with unknown quantum probe states provided that measurements on probe and transformed probe states are available. Drawbacks of various approximate treatments are considered.Comment: 7 pages, 4 figure

General method for deforming quantum dynamics into classical dynamics while keeping Latin small letter h with stroke fixed

Using Weinberg's generalized nonlinear quantum dynamics [Ann. Phys. (N.Y.) 194, 336 (1989)], we propose a simple way to form an interpolative dynamical system that joins the classical and quantum regimes. By altering a dimensionless control parameter λ[0,1], one can smoothly metamorphose quantum evolution into classical evolution, keeping the Hilbert-space dimension and physical constants unchanged. The method suggests an approach to studies of dynamical chaos suppression in quantized classically chaotic systems

Linear Quantum Theory and Its Possible Nonlinear Generalizations

We show that the Schrödinger equation may be derived as a consequence of three postulates: 1) the hamiltonian formalism 2) a conformal structure 3) a projective structure. These suffice to deduce the geometrical structure of Hilbert space also. Furthermore, the quantum mechanical action principle, and unitary propagator, are reduced to special cases of classical results. Thereafter we explore the relaxation of these postulates. Of the possible generalizations only one appears physically fruitful. This is obtained from a simple gauge freedom of the standard theory. The result is a nonlinear quantum theory that has been the subject of recent interest. We consider the empirical status of this theory, and the problem of its interpretation. It is suggested that the only remaining option is to consider nonperturbative nonlinearities in the context of the quantum measurement problem. This is advanced as a principle of constraint

Information theory and optimal phase measurement

Using a single mode analysis, we assess the utility of Shannon information as a robust performance criterion for phase detection. To provide a sharp test, we apply it to the Shapiro, Shepard and Wong optimal phase state. This has well-known pathologies, and is thus a good hurdle. Numerical studies show that the SSW-state information gain converges rapidly to the finite value ˜ 0.966 bits at infinite photon number. For large photon numbers N the truncated phase state scheme yields ˜ log2(2N + 1) 1.220 bits whereas the coherent state scheme returns ˜ log2 N1/21.604 bits. Numerical studies confirm these asymptotic formulae and reveal an interesting crossover regime where the coherent state scheme proves superior to the truncated phase state scheme at low photon numbers N ≤ 10. The anomalous behaviour of the SSW-state is studied with the aid of a simple scaling argument

A Generalization of Dirichlet′s Integral

Using the Weyl integral transform representation of fractional calculus, we generalize Dirichlet′s iterated integral identity to include functions whose dummy argument is an arbitrary positive linear combination