Finite-dimensional Quantum Systems Research Articles

Directed-Completeness of Quantum Statistical Experiments in the Randomization Order

A parametrized family of normal states on a von Neumann algebra is called a statistical experiment, which generalizes the corresponding concepts in classical statistics and finite-dimensional quantum systems. We introduce randomization preorder and equivalence relations for statistical experiments with a fixed parameter set and for normal channels with a fixed input space by post-processing completely positive channels. In this paper, we prove that the set of equivalence classes of statistical experiments or those of normal channels is an upper and lower directed-complete partially ordered set with respect to the randomization order, i.e. any increasing or decreasing net of statistical experiments or channels has its supremum or infimum in the randomization order. We also show that if the outcome space of each statistical experiment or channel of a randomization-monotone net is commutative, the outcome space of the supremum or infimum can also be taken to be commutative. We consider two examples of homogeneous Markov processes of channels on infinite-dimensional separable Hilbert spaces, namely block-diagonalization with irrational translation and ideal quantum linear amplifier channels, and explicitly derive their infima. Throughout the paper, the concept of channel conjugation is used to obtain results for decreasing channels from those for increasing channels.

Phase-space-simulation method for quantum computation with magic states on qubits

We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et al., New J. Phys. 14, 113011 (2012)] applying only to odd dimension. Our approach is more general than previous ones based on mixtures of stabilizer states. Namely, all mixtures of stabilizer states can be efficiently simulated, but for any number of qubits there also exist efficiently simulable states outside the stabilizer polytope. Further, our simulation method extends to negative quasiprobability distributions, where it provides amplitude estimation. The simulation cost is then proportional to a robustness measure squared. For all quantum states, this robustness is smaller than or equal to robustness of magic.

On Howland time-independent formulation of CP-divisible quantum evolutions

We extend Howland time-independent formalism to the case of completely positive and trace preserving dynamics of finite-dimensional open quantum systems governed by periodic, time-dependent Lindbladian in Weak Coupling Limit, expanding our result from previous papers. We propose the Bochner space of periodic, square integrable matrix-valued functions, as well as its tensor product representation, as the generalized space of states within the time-independent formalism. We examine some densely defined operators on this space, together with their Fourier-like expansions and address some problems related to their convergence by employing general results on Banach space-valued Fourier series, such as the generalized Carleson–Hunt theorem. We formulate Markovian dynamics in the generalized space of states by constructing appropriate time-independent Lindbladian in standard (Lindblad–Gorini–Kossakowski–Sudarshan) form, as well as one-parameter semigroup of bounded evolution maps. We show their similarity with Markovian generators and dynamical maps defined on matrix space, i.e. the generator still possesses a standard form (extended by closed perturbation) and the resulting semigroup is also completely positive, trace preserving and a contraction.

Non-Hermitian Hamiltonians and no-go theorems in quantum information

Recently, apparent nonphysical implications of non-Hermitian quantum mechanics (NHQM) have been discussed in the literature. In particular, the apparent violation of the no-signaling theorem, discrimination of nonorthogonal states, and the increase of quantum entanglement by local operations were reported, and therefore NHQM was not considered as a fundamental theory. Here we show that these and other no-go principles (including the no-cloning and no-deleting theorems) of conventional quantum mechanics still hold in finite-dimensional non-Hermitian quantum systems, including parity-time symmetric and pseudo-Hermitian cases, if its formalism is properly applied. We have developed a modified formulation of NHQM based on the geometry of Hilbert spaces which is consistent with the conventional quantum mechanics for Hermitian systems. Using this formulation the validity of these principles can be shown in a simple and uniform approach.

An overview of quantum cellular automata

Quantum cellular automata are arrays of identical finite-dimensional quantum systems, evolving in discrete-time steps by iterating a unitary operator G. Moreover the global evolution G is required to be causal (it propagates information at a bounded speed) and translation-invariant (it acts everywhere the same). Quantum cellular automata provide a model/architecture for distributed quantum computation. More generally, they encompass most of discrete-space discrete-time quantum theory. We give an overview of their theory, with particular focus on structure results; computability and universality results; and quantum simulation results.

Algorithms for quantum control without discontinuities: Application to the simultaneous control of two qubits

We propose a technique to design control algorithms for a class of finite dimensional quantum systems so that the control law does not present discontinuities. The class of models considered admits a group of symmetries, which allows us to reduce the problem of control to a quotient space where the control system is “fully actuated.” As a result, we can prescribe a desired trajectory, which is, to some extent, arbitrary, and derive the corresponding control. We illustrate this technique with examples and focus on the application to the simultaneous control of two non-interacting spin 12 particles with different gyromagnetic ratios in zero field nuclear magnetic resonance (NMR). Our method provides a flexible toolbox for the design of control algorithms to drive the state of finite dimensional quantum systems to any desired final configuration, with smooth controls.

Attaining the Ultimate Precision Limit in Quantum State Estimation

We derive a bound on the precision of state estimation for finite dimensional quantum systems and prove its attainability in the generic case where the spectrum is non-degenerate. Our results hold under an assumption called local asymptotic covariance, which is weaker than unbiasedness or local unbiasedness. The derivation is based on an analysis of the limiting distribution of the estimator’s deviation from the true value of the parameter, and takes advantage of quantum local asymptotic normality, a useful asymptotic characterization of identically prepared states in terms of Gaussian states. We first prove our results for the mean square error of a special class of models, called D-invariant, and then extend the results to arbitrary models, generic cost functions, and global state estimation, where the unknown parameter is not restricted to a local neighbourhood of the true value. The extension includes a treatment of nuisance parameters, i.e. parameters that are not of interest to the experimenter but nevertheless affect the precision of the estimation. As an illustration of the general approach, we provide the optimal estimation strategies for the joint measurement of two qubit observables, for the estimation of qubit states in the presence of amplitude damping noise, and for noisy multiphase estimation.

Minimum-uncertainty states and completeness of non-negative quasiprobability of finite-dimensional quantum systems

We construct minimum-uncertainty states and a non-negative quasi probability distribution for quantum systems on a finite-dimensional space. We reexamine the theorem of Massar and Spindel for the uncertainty relationof the two unitary operators related by the discrete Fourier transformation. It is shown that some assumptions in their proof can be justified by the use of the Perron-Frobenius theorem. The minimum-uncertainty states are the ones that saturate this uncertainty inequality. The continuum limit is closely analyzed by introducing a scale factor in the limiting scheme. Using the minimum-uncertainty states, we construct a non-negative quasi probability distribution. Its marginal distributions are smeared out. However, we show that this quasi probability is optimal in the sense that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance in the phase space is assumed. Generally, it is desirable that the quasi probability is complete, i.e., it contains full information of the state. We show that the obtained quasi probability is indeed complete if the dimension of the state space is odd, whereas it is not if the dimension is even.

Enabling Computation of Correlation Bounds for Finite-Dimensional Quantum Systems via Symmetrization.

We present a technique for reducing the computational requirements by several orders of magnitude in the evaluation of semidefinite relaxations for bounding the set of quantum correlations arising from finite-dimensional Hilbert spaces. The technique, which we make publicly available through a user-friendly software package, relies on the exploitation of symmetries present in the optimization problem to reduce the number of variables and the block sizes in semidefinite relaxations. It is widely applicable in problems encountered in quantum information theory and enables computations that were previously too demanding. We demonstrate its advantages and general applicability in several physical problems. In particular, we use it to robustly certify the nonprojectiveness of high-dimensional measurements in a black-box scenario based on self-tests of d-dimensional symmetric informationally complete positive-operator-valued measurements.

Time evolution of coupled spin systems in a generalized Wigner representation

Phase-space representations as given by Wigner functions are a powerful tool for representing the quantum state and characterizing its time evolution in the case of infinite-dimensional quantum systems and have been widely used in quantum optics and beyond. Continuous phase spaces have also been studied for finite-dimensional quantum systems such as spin systems. However, much less is known for finite-dimensional, coupled systems, and we present a complete theory of Wigner functions for this case. In particular, we provide a self-contained Wigner formalism for describing and predicting the time evolution of coupled spins which lends itself to visualizing the high-dimensional structure of multi-partite quantum states. We completely treat the case of an arbitrary number of coupled spins 1∕2, thereby establishing the equation of motion using Wigner functions. The explicit form of the time evolution is then calculated for up to three spins 1∕2. The underlying physical principles of our Wigner representations for coupled spin systems are illustrated with multiple examples which are easily translatable to other experimental scenarios.

Towards space from Hilbert space: finding lattice structure in finite-dimensional quantum systems

Field theories place one or more degrees of freedom at every point in space. Hilbert spaces describing quantum field theories, or their finite-dimensional discretizations on lattices, therefore have large amounts of structure: they are isomorphic to the tensor product of a smaller Hilbert space for each lattice site or point in space. Local field theories respecting this structure have interactions which preferentially couple nearby points. The emergence of classicality through decoherence relies on this framework of tensor-product decomposition and local interactions. We explore the emergence of such lattice structure from Hilbert-space considerations alone. We point out that the vast majority of finite-dimensional Hilbert spaces cannot be isomorphic to the tensor product of Hilbert-space subfactors that describes a lattice theory. A generic Hilbert space can only be split into a direct sum corresponding to a basis of state vectors spanning the Hilbert space; we consider setups in which the direct sum is naturally decomposed into two pieces. We define a notion of direct-sum locality which characterizes states and decompositions compatible with Hamiltonian time evolution. We illustrate these notions for a toy model that is the finite-dimensional discretization of the quantum-mechanical double-well potential. We discuss their relevance in cosmology and field theory, especially for theories which describe a landscape of vacua with different spacetime geometries.

Fast adiabatic evolution by oscillating initial Hamiltonians

We propose a method to produce fast transitionless dynamics for finite-dimensional quantum systems without requiring additional Hamiltonian components not included in the initial control setup, remaining close to the true adiabatic path at all times. The strategy is based on the introduction of an effective counterdiabatic scheme: a correcting Hamiltonian is constructed which approximatively cancels nonadiabatic effects, inducing an evolution tracking the adiabatic states closely. This can be absorbed into the initial Hamiltonian by adding a fast oscillation in the control parameters. We show that a consistent speedup can be achieved without requiring strong control Hamiltonians, using it both as a stand-alone shortcut-to-adiabaticity and as a weak correcting field. A number of examples are treated, dealing with quantum state transfer in avoided-crossing problems and entanglement creation.

Maximum coherence under noisy channels

For the well-known relative entropy of coherence (REC), a quantum state attains maximum coherence only in optimal bases, which are mutually unbiased bases to the eigenvectors of the given state. In this paper, we explore the conditions of obtaining the maximum of REC under noisy channels and explain the geometric significance of optimal basis form the viewpoint of the Bloch sphere. We also show that the eigenvectors of the state will remain constant throughout the whole process of evolution, as long as the effects of channels induce a synchronous change for the components of the initial state Bloch vector. Furthermore, as a special example, we investigate the dynamics of the maximum REC for any finite dimensional single quantum system under a depolarizing (DP) channel, and obtain an analytical form of maximum coherence for a d-dimensional maximally coherent state. Meanwhile, a concise analytical expression of maximum coherence for two classes of an N-qubit system under the DP channel is also provided.

Self-averaging of random quantum dynamics

Stochastic dynamics of a quantum system driven by $N$ statistically independent random sudden quenches in a fixed time interval is studied. We reveal that with growing $N$ the system approaches a deterministic limit indicating self-averaging with respect to its temporal unitary evolution. This phenomenon is quantified by the variance of the unitary matrix governing the time evolution of a finite dimensional quantum system which according to an asymptotic analysis decreases at least as $1/N$. For a special class of protocols (when the averaged Hamiltonian commutes at different times), we prove that for finite $N$ the distance (according to the Frobenius norm) between the averaged unitary evolution operator generated by the Hamiltonian $H$ and the unitary evolution operator generated by the averaged Hamiltonian $\langle H \rangle$ scales as $1/N$. Numerical simulations enlarge this result to a broader class of the non-commuting protocols.

Antisymmetric Wilson loops in mathcal{N}=4 SYM: from exact results to non-planar corrections

We consider the vacuum expectation values of 1/2-BPS circular Wilson loops in mathcal{N}=4 super Yang-Mills theory in the totally antisymmetric representation of the gauge group U(N) or SU(N). Localization and matrix model techniques provide exact, but rather formal, expressions for these expectation values. In this paper we show how to extract the leading and sub-leading behavior in a 1/N expansion with fixed ’t Hooft coupling starting from these exact results. This is done by exploiting the relation between the generating function of antisymmetric Wilson loops and a finite-dimensional quantum system known as the truncated harmonic oscillator. Sum and integral representations for the 1/N terms are provided.

Quantum control through measurement feedback

Measurement combined with feedback that aims to restore a presumed pre-measurement quantum state will yield this state after a few measurement-feedback cycles even if the actual state of the system initially had no resemblance to the presumed state. Here we introduce this mechanism of {\it self-fulfilling prophecy} and show that it can be used to prepare finite-dimensional quantum systems in target states or target dynamics. Using two-level systems as an example we demonstrate that self-fulfilling prophecy protects the system against noise and tolerates imprecision of feedback up to the level of the measurement strength. By means of unsharp measurements the system can be driven deterministically into arbitrary, smooth quantum trajectories.

Path probabilities for consecutive measurements, and certain “quantum paradoxes”

We consider a finite-dimensional quantum system, making a transition between known initial and final states. The outcomes of several accurate measurements, which could be made in the interim, define virtual paths, each endowed with a probability amplitude. If the measurements are actually made, the paths, which may now be called “real”, acquire also the probabilities, related to the frequencies, with which a path is seen to be travelled in a series of identical trials. Different sets of measurements, made on the same system, can produce different, or incompatible, statistical ensembles, whose conflicting attributes may, although by no means should, appear “paradoxical”. We describe in detail the ensembles, resulting from intermediate measurements of mutually commuting, or non-commuting, operators, in terms of the real paths produced. In the same manner, we analyse the Hardy’s and the “three box” paradoxes, the photon’s past in an interferometer, the “quantum Cheshire cat” experiment, as well as the closely related subject of “interaction-free measurements”. It is shown that, in all these cases, inaccurate “weak measurements” produce no real paths, and yield only limited information about the virtual paths’ probability amplitudes.

Revised Geometric Measure of Entanglement in Infinite Dimensional Multipartite Quantum Systems

In Cao and Wang (J. Phys.: Math. Theor. 40, 3507–3542, 2007), the revised geometric measure of entanglement (RGME) for states in finite dimensional bipartite quantum systems was proposed. Furthermore, in Cao and Wang (Commun. Theor. Phys. 51(4), 613–620, 2009), the authors obtained the revised geometry measure of entanglement for multipartite states including three-qubit GHZ state, W state, and the generalized Smolin state in the presence of noise and the two-mode squeezed thermal state, and defined the Gaussian geometric entanglement measure. In this paper, we generalize the RGME to infinite dimensional multipartite quantum systems, and prove that this measure satisfies some necessary properties as a well-defined entanglement measure, including monotonicity under local operations and classical communications.

Conditional quantum entropy power inequality for d-level quantum systems

We propose an extension of the quantum entropy power inequality for finite dimensional quantum systems, and prove a conditional quantum entropy power inequality by using the majorization relation as well as the concavity of entropic functions also given by Audenaert et al (2016 J. Math. Phys. 57 052202). Here, we make particular use of the fact that a specific local measurement after a partial swap operation (or partial swap quantum channel) acting only on finite dimensional bipartite subsystems does not affect the majorization relation for the conditional output states when a separable ancillary subsystem is involved. We expect our conditional quantum entropy power inequality to be useful, and applicable in bounding and analyzing several capacity problems for quantum channels.

Alternating Projections Methods for Discrete-Time Stabilization of Quantum States

We study sequences (both cyclic and randomized) of idempotent completely positive trace-preserving quantum maps, and show how they asymptotically converge to the intersection of their fixed point sets via alternating projection methods, highlighting the robustness features of the protocol against randomization. The general results are then specialized to stabilizing entangled states in finite-dimensional multipartite quantum systems subject to locality constraints, a problem of key interest for quantum information applications.