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Many problems in theoretical physics are very frequently dealt with non-Hermitian operators. Recently there has been a lot of interest in non-Hermitian operators with real spectra. In this paper, by using the inverse problem for quantum systems, we show that, on finite-dimensional Hilbert spaces, all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product. In particular this allows the use of standard methods of quantum mechanics to analyze non-Hermitian operators with real spectra.

Motivated by the existence of bi-Hamiltonian classical systems and the correspondence principle, in this paper we analyze the problem of finding Hermitian scalar products which turn a given flow on a Hilbert space into a unitary one. We show how different invariant Hermitian scalar products give rise to different descriptions of a quantumsystem in the Ehrenfest and Heisenberg picture.

A matrix model to describe quantitatively certain physical properties of quantum systems of two-band eigenvalue spectrum is proposed. This formulation is a generalization of Wannier-type representations and is applicable to various situations where physical properties are defined in tensor form.

The Harmonic Balance Technique (HBT) is used to analyze the steady state performance of a two-state quantum system interacting with a classical sinusoidal electromagnetic wave and with a thermal bath at a fixed temperature. The linear time-variant differential equations describing such a system can be solved to any number of harmonics and the results can be compared with those obtained with the classical RWA approximation, thus emphasizing the validity limits of the latter.

In this paper we consider a large system of bosons or fermions. We start with an initial datum which is compatible with the Bose–Einstein, respectively Fermi–Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time t, agrees with the analogous expansion for the solution to the Uehling–Uhlenbeck equation.

This paper follows the same spirit as the companion work,² where the authors investigated the weak-coupling limit for particles obeying the Maxwell–Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.

We revise the 'no-signaling' condition for the supraluminal communication between two spatially separated finite quantum systems of arbitrary dimensions, thus generalizing a similar preceding approach for two-qubits: non-linear evolution does not necessarily imply the possibility of supraluminal communication between any sort of finite quantum systems.

We studied in this paper the bipartite quantum systems in finite Hilbert space. We applied the measurement on the first subsystem only and studied how this measurement is affecting the entropy of the second subsystem. The analytic work carried out in this paper shows that the post measurement state is factorizable when measurement is applied on the first subsystem. This result shows that the disorder of the system is reduced and hence the entropy of the whole system is reduced as expected. However, it is also shown in this work that the entropy of the second subsystem of the post-measurement state could be greater than that of the original state. Increasing entropy after measurement means that the disorder in the second subsystem is increased. The information we gain from measuring the first subsystem reduces our information about the second subsystem rather than increasing it or at least keeping it the same as expected. This means that more factors may be required in order to calculate the probability of finding the second subsystem in a certain state.

The transition time between states plays an important role in designing quantum devices as they are very sensitive to environmental influences. Decoherence phenomenon is responsible for possible destructions of the entanglement that is a fundamental requirement to implement quantum information processing systems. If the time between states is minimized, the decoherence effects can be reduced, thus, it is advantageous to the designer to develop expressions for time performance measures. Quantum speed limit (QSL) problem has been studied from the theoretical point of view, providing general results. Considering the implementation of quantum control systems, as the decoherence phenomenon is unavoidable, it is important to apply these general results to particular cases, developing expressions and performance measures, to assist control engineering designers. Here, a minimum time performance measure is defined for quantum control problems, for time-independent or time-dependent Hamiltonians, and applied to some practical examples, providing hints that may be useful for researchers pursuing optimization strategies for quantum control systems.

We detect a certain pattern of behavior of separability probabilities $p(r_A,r_B)$ for two-qubit systems endowed with Hilbert–Schmidt (HS), and more generally, random induced measures, where $r_A$ and $r_B$ are the Bloch radii ($0\le r_A,r_B\le 1$) of the qubit reduced states ($A,B$). We observe a relative repulsion of radii effect, that is $p(r_A,r_A)<p(r_A,1-r_A)$, except for rather narrow “crossover” intervals $[{\tilde{r}}_A,\frac{1}{2}]$. Among the seven specific cases we study are, firstly, the “toy” seven-dimensional $X$-states model and, then, the fifteen-dimensional two-qubit states obtained by tracing over the pure states in $4\times K$-dimensions, for $K=3,4,5$, with $K=4$ corresponding to HS (flat/Euclidean) measure. We also examine the real (two-rebit) $K=4$, the $X$-states $K=5$, and Bures (minimal monotone)–for which no nontrivial crossover behavior is observed–instances. In the two $X$-states cases, we derive analytical results; for $K=3,4$, we propose formulas that well-fit our numerical results; and for the other scenarios, rely presently upon large numerical analyses. The separability probability crossover regions found expand in length (lower ${\tilde{r}}_A$) as $K$ increases. This report continues our efforts [P. B. Slater, arXiv:1506.08739] to extend the recent work of [S. Milz and W. T. Strunz, J. Phys. A48 (2015) 035306.] from a univariate ($r_A$) framework — in which they found separability probabilities to hold constant with $r_A$ — to a bivariate ($r_A,r_B$) one. We also analyze the two-qutrit and qubit–qutrit counterparts reported in Quantum Inform. Process. 15 (2016) 3745 in this context, and study two-qubit separability probabilities of the form $p(r_A,\frac{1}{2})$. A physics.stack.exchange link to a contribution by Mark Fischler addressing, in considerable detail, the construction of suitable bivariate distributions is indicated at the end of the paper.

Dynamical high-dimensional quantum states can be tracked and manipulated in many cases. Using a new theoretical framework approach of manipulating quantum systems, we will show how one can manipulate and introduce parameters that allow tracking and descriptive insight in the dynamics of states. Using quantum topology and other novel mathematical representations, we will show how quantum states behave in critical points when the shift of probability distribution introduces changes.

We present, in a pedagogical style, many instances of reduction procedures appearing in a variety of physical situations, both classical and quantum. We concentrate on the essential aspects of any reduction procedure, both in the algebraic and geometrical setting, elucidating the analogies and the differences between the classical and the quantum situations.

We have studied quantum systems on finite-dimensional Hilbert spaces and found that all these systems are connected through local transformations. Actually, we have shown that these transformations give rise to a gauge group that connects the Hamiltonian operators associated with each quantum system. This bridge allows us to connect different quantum systems, in such a way that studying one of them allows to understand the other through a gauge transformation. Furthermore, we included the case where the Hamiltonian operator can be time-dependent. An application for this construction will be achieved in the theory of control quantum systems.

We consider a non-resonant bilinear Schrödinger equation with discrete spectrum driven by a scalar control. We prove that this system can approximately track any given trajectory, up to the phase of the coordinates, with arbitrary small controls. The result is valid both for bounded and unbounded Schrödinger operators. The method used relies on finite-dimensional control techniques applied to Lie groups.