Finite Quantum Systems Research Articles

On Entropy Production of Repeated Quantum Measurements I. General Theory

We study entropy production (EP) in processes involving repeated quantum measurements of finite quantum systems. Adopting a dynamical system approach, we develop a thermodynamic formalism for the EP and study fine aspects of irreversibility related to the hypothesis testing of the arrow of time. Under a suitable chaoticity assumption, we establish a Large Deviation Principle and a Fluctuation Theorem for the EP.

Global Stabilization of Mixed-states for Stochastic Quantum Systems via Switching Control

The global stabilization of mixed states for finite dimensional stochastic quantum systems with non-regular measurement operator and non-diagonal free Hamiltonian is investigated in this paper. A two-part switching control strategy is proposed, in which the constant control is used to steer the system state to enter the convergence domain, while the control law which is designed based on Lyapunov stability theorem is used to attract the system state in convergence domain to the target state. The convergence of the switching control is strictly proved. Moreover, the numerical experiments on a three dimensional stochastic quantum system are implemented to demonstrate the effectiveness of the control proposed.

On the equivalence of separability and extendability of quantum states

Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.

Adaptive quantum state tomography via linear regression estimation: Theory and two-qubit experiment

Adaptive techniques have great potential for wide application in enhancing the precision of quantum parameter estimation. We present an adaptive quantum state tomography protocol for finite dimensional quantum systems and experimentally implement the adaptive tomography protocol on two-qubit systems. In this adaptive quantum state tomography protocol, an adaptive measurement strategy and a recursive linear regression estimation algorithm are performed. Numerical results show that our adaptive quantum state tomography protocol can outperform tomography protocols using mutually unbiased bases and the two-stage mutually unbiased bases adaptive strategy, even with the simplest product measurements. When nonlocal measurements are available, our adaptive quantum state tomography can beat the Gill–Massar bound for a wide range of quantum states with a modest number of copies. We use only the simplest product measurements to implement two-qubit tomography experiments. In the experiments, we use error-compensation techniques to tackle systematic error due to misalignments and imperfection of wave plates, and achieve about a 100-fold reduction of the systematic error. The experimental results demonstrate that the improvement of adaptive quantum state tomography over nonadaptive tomography is significant for states with a high level of purity. Our results also show that this adaptive tomography method is particularly effective for the reconstruction of maximally entangled states, which are important resources in quantum information.

Control landscapes are almost always trap free: a geometric assessment

A proof is presented that almost all closed, finite dimensional quantum systems have trap free (i.e. free from local optima) landscapes for a large and physically general class of circumstances, which includes qubit evolutions in quantum computing. This result offers an explanation for why gradient-based methods succeed so frequently in quantum control. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator, and thus in the case of observables as well. Singular controls have been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause control optimization progress to halt, and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity (full rank) assumption of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This mild condition is shown to be sufficient for a quantum system’s landscape to be trap free. The control landscape is shown to be trap free for all but a null set of Hamiltonians using a geometric technique based on the parametric transversality theorem. Numerical evidence confirming this analysis is also presented. This new result is the analogue of the work of Altifini, wherein it was shown that controllability holds for all but a null set of quantum systems in the dipole approximation. These collective results indicate that the availability of adequate control resources remains the most physically relevant issue for achieving high fidelity control performance while also avoiding landscape traps.

Measuring higher-dimensional entanglement

We study local-realistic inequalities, Bell-type inequalities, for bipartite pure states of finite dimensional quantum systems -- qudits. There are a number of proposed Bell-type inequalities for such systems. Our interest is in relating the value of Bell-type inequality function with a measure of entanglement. Interestingly, we find that one of these inequalities, the Son-Lee-Kim inequality, can be used to measure entanglement of a pure bipartite qudit state and a class of mixed two-qudit states. Unlike the majority of earlier schemes in this direction, where number of observables needed to characterize the entanglement increases with the dimension of the subsystems, this method needs only four observables. We also discuss the experimental feasibility of this scheme. It turns out that current experimental set ups can be used to measure the entanglement using our scheme.

Relations between dissipated work in non-equilibrium process and the family of Rényi divergences

In this paper, we establish a general relation which directly links the dissipated work done on a system driven arbitrarily far from equilibrium, a fundamental quantity in thermodynamics, and the family of Rényi divergences between two states along the forward and reversed dynamics, a fundamental concept in information theory. Specifically, we find that the generating function of the dissipated work under an arbitrary time-dependent driving is related to the family of Rényi divergences between a non-equilibrium state along the forward process and a non-equilibrium state along its time-reversed process. This relation is a consequence of the principle of conservation of information and time reversal symmetry and is universally applicable to both finite classical system and finite quantum system under arbitrary driving process. The significance of the relation between the generating function of dissipated work and the family of Rényi divergences are two fold. On the one hand, the relation establishes that the macroscopic entropy production and its fluctuations are determined by the family of Rényi divergences, a measure of distinguishability of two states, between a microscopic process and its time reversal. On the other hand, this relation tells us that we can extract the family of Renyi divergences from the work measurement in a microscopic process. For classical systems the work measurement is straightforward, from which the family of Rényi divergences can be obtained; for quantum systems under time-dependent driving the characteristic function of work distributions can be measured from Ramsey interferences of a single spin, then we can extract the family of Renyi divergences from Ramsey interferences of a single spin.

Dressed coherent states in finite quantum systems: A cooperative game theory approach

A quantum system with variables in Z(d) is considered. Coherent density matrices and coherent projectors of rank n are introduced, and their properties (e.g., the resolution of the identity) are discussed. Cooperative game theory and in particular the Shapley methodology, is used to renormalize coherent states, into a particular type of coherent density matrices (dressed coherent states). The Q-function of a Hermitian operator, is then renormalized into a physical analogue of the Shapley values. Both the Q-function and the Shapley values, are used to study the relocation of a Hamiltonian in phase space as the coupling constant varies, and its effect on the ground state of the system. The formalism is also generalized for any total set of states, for which we have no resolution of the identity. The dressing formalism leads to density matrices that resolve the identity, and makes them practically useful.

Symmetry broken and restored coupled-cluster theory: II. Global gauge symmetry and particle number

We have recently extended many-body perturbation theory (MBPT) and coupled-cluster theory performed on top of a Slater determinant breaking rotational symmetry to allow for the restoration of the angular momentum at any truncation order (Duguet 2015 J. Phys. G: Nucl. Part. Phys. 42 025107). Following a similar route, we presently extend Bogoliubov MBPT and Bogoliubov coupled cluster theory performed on top of a Bogoliubov reference state breaking global gauge symmetry to allow for the restoration of the particle number at any truncation order. Eventually, formalisms can be merged to handle SU(2) and U(1) symmetries at the same time. The long-term goal relates to the ab initio description of near-degenerate finite quantum systems with an open-shell character.

A collisional extension of time-dependent Hartree–Fock

We propose a collisional extension of time-dependent mean-field theories on the basis of a recently proposed stochastic extension of mean-field dynamics (stochastic time-dependent Hartree–Fock, STDHF). The latter theory is unfortunately too involved to envision practical applications in realistic systems in the near future and is thus bound to model systems. It is also hard to explore moderate to low energies with STDHF, because of vanishing transition probabilities that are impossible to sample properly. For such moderately excited situations covering small fluctuations, we compactify sampling by employing the same average mean field for all STDHF trajectories. The new approach, coined average STDHF (ASTDHF), ignores the fluctuations of the mean field but still accounts correctly for the collisional correlations responsible for dissipative features on top of mean-field dynamics. We detail the main features of the new approach in relation to existing equations, in particular quantum kinetic theories. The new theory is directly connected to STDHF, both formally and practically. We thus discuss in detail how the two approaches are related to each other. We apply the new scheme to illustrative examples taking as benchmark STDHF dynamics in 1D. ASTDHF provides results that are in remarkable agreement with the more elaborate STDHF. It makes it a promising approach to deal with dissipative dynamics in finite quantum systems, because of its moderate cost allowing applications in realistic systems and the possibility of exploring any excitation energy range where collisional correlations are expected to play a role.

Global stabilization control of stochastic quantum systems

The global stabilization control of arbitrary eigenstates for finite dimensional stochastic quantum systems with non-diagonal free Hamiltonian and non-regular measurement operator is studied in this paper. We propose a switching feedback control law, in which a constant control is used to steer the system state to a convergence domain, and another control law designed based on Lyapunov stability theorem, is used to attract the states in the convergence domain to the desired target state. The convergence to an arbitrary target eigenstate from any initial state is strictly proved. Moreover, numerical simulation experiments on a three-dimensional stochastic quantum system are implemented to demonstrate the effectiveness of the proposed control.

Projection operator based expansion of the evolution operator

The not necessarily unitary evolution operator of a finite dimensional quantum system is studied with the help of a projection operators technique. Applying this approach to the Schrödinger equation allows the derivation of an alternative expression for the evolution operator, which differs from the traditional chronological exponent. An appropriate choice of projection operators results in the possibility of studying the diagonal and non-diagonal elements of the evolution operator separately. The suggested expression implies a particular form of perturbation expansion, which leads to a new formula for the short time dynamics. The new kind of perturbation expansion can be used to improve the accuracy of the usual chronological exponent significantly. The evolution operator for any arbitrary time can be efficiently recovered using the semigroup properties. The method is illustrated by two examples, namely the dynamics of a three-level system in two nonresonant laser fields and the calculation of the partition function of a finite XY-spin chain.

Transitional steady states of exchange dynamics between finite quantum systems.

We examine energy and particle exchange between finite-sized quantum systems and find a new form of nonequilibrium state. The exchange rate undergoes stepwise evolution in time, and its magnitude and sign dramatically change according to system size differences. The origin lies in interference effects contributed by multiply scattered waves at system boundaries. Although such characteristics are utterly different from those of true steady state for infinite systems, Onsager's reciprocal relation remains universally valid.

Transparent boundary conditions for time-dependent electron transport in the [formula omitted]-matrix method with applications to nanostructured interfaces

Transparent boundary conditions for the time-dependent Schrödinger equation are implemented using the R-matrix method. The employed scattering formalism is suitable for describing open quantum systems and provides the framework for the time-dependent coherent transport. Transmission and reflection of wave functions at the edges of a finite quantum system are essential for an accurate and efficient description of the time-dependent processes on large time scales. We detail the computational method and point out the numerical advantages stemming from the open system approach based on the R-matrix formalism. The approach is used here to describe time-dependent transport across nanostructured interfaces relevant for photovoltaic applications.

Time-dependent density functional theory with twist-averaged boundary conditions

Time-dependent density functional theory is widely used to describe excitations of many-fermion systems. In its many applications, 3D coordinate-space representation is used, and infinite-domain calculations are limited to a finite volume represented by a box. For finite quantum systems (atoms, molecules, nuclei), the commonly used periodic or reflecting boundary conditions introduce spurious quantization of the continuum states and artificial reflections from boundary; hence, an incorrect treatment of evaporated particles. These artifacts can be practically cured by introducing absorbing boundary conditions (ABC) through an absorbing potential in a certain boundary region sufficiently far from the described system. But also the calculations of infinite matter (crystal electrons, quantum fluids, neutron star crust) suffer artifacts from a finite computational box. In this regime, twist- averaged boundary conditions (TABC) have been used successfully to diminish the finite-volume effects. In this work, we extend TABC to time-dependent framework and apply it to resolve the box artifacts for finite quantum systems using as test case small- and large-amplitude nuclear vibrations. We demonstrate that by using such a method, one can reduce finite volume effects drastically without adding any additional parameters. While they are almost equivalent in the linear regime, TABC and ABC differ in the nonlinear regime in their treatment of evaporated particles.

Joint probability distributions for projection probabilities of random orthonormal states

The quantum chaos conjecture applied to a finite dimensional quantum system implies that such a system has eigenstates that show similar statistical properties as the column vectors of random orthogonal or unitary matrices. Here, we consider the different probabilities for obtaining a specific outcome in a projective measurement, provided the system is in one of its eigenstates. We then give analytic expressions for the joint probability density for these probabilities, with respect to the ensemble of random matrices. In the case of the unitary group, our results can be applied, also, to the phenomenon of universal conductance fluctuations, where the same mathematical quantities describe partial conductances in a two-terminal mesoscopic scattering problem with a finite number of modes in each terminal.

Transport in inhomogeneous quantum critical fluids and in the Dirac fluid in graphene

We develop a general hydrodynamic framework for computing direct current thermal and electric transport in a strongly interacting finite temperature quantum system near a Lorentz-invariant quantum critical point. Our framework is non-perturbative in the strength of long wavelength fluctuations in the background charge density of the electronic fluid, and requires the rate of electron-electron scattering to be faster than the rate of electron-impurity scattering. We use this formalism to compute transport coefficients in the Dirac fluid in clean samples of graphene near the charge neutrality point, and find results insensitive to long range Coulomb interactions. Numerical results are compared to recent experimental data on thermal and electrical conductivity in the Dirac fluid in graphene and substantially improved quantitative agreement over existing hydrodynamic theories is found. We comment on the interplay between the Dirac fluid and acoustic and optical phonons, and qualitatively explain experimentally observed effects. Our work paves the way for quantitative contact between experimentally realized condensed matter systems and the wide body of high energy inspired theories on transport in interacting many-body quantum systems.

Quantum measurement in coherence-vector representation

We consider the quantum measurements on a finite quantum system in coherence-vector representation. In this representation, all the density operators of an N-level (N ⩾ 2) quantum system constitute a convex set M(N) embedded in an (N2 − 1)-dimensional Euclidean space \(\mathbb{R}^{N^2 - 1}\), and we find that an orthogonal measurement is an (N − 1)-dimensional projector operator on \(\mathbb{R}^{N^2 - 1}\). The states unchanged by an orthogonal measurement form an (N − 1)-dimensional simplex, and in the case when N is prime or power of prime, the space of the density operator is a direct sum of (N + 1) such simplices. The mathematical description of quantum measurement is plain in this representation, and this may have further applications in quantum information processing.

Bethe Ansatz, Galois symmetries, and finite quantum systems

Bethe Ansatz, Galois symmetries, and finite quantum systems

Minimalistic analytical approach to non-Markovian open quantum systems

The dynamics of finite dimension open quantum systems is studied with the help of the simplest possible form of projection operators, namely the ones which project only onto one-dimensional subspaces. The simplicity of the action of the projection operators always leads to an analytical solution of the dynamical master equation, even in the non-Markovian case, in any perturbative order. The analytical solution correctly reproduces the short-time dynamics, and can be used to recursively recover the dynamics for an arbitrary time interval with arbitrary precision. The necessary number of relevant degrees of freedom to completely characterise an open quantum system is , where n is the dimension of the Hilbert space of the open system. The method is illustrated by two examples, the relaxation of a qubit in a thermal bath and the dynamics of two interacting qubits in a common environment.