Ensemble Of Quantum States Research Articles

Algorithmic Pseudorandomness in Quantum Setups

Many experimental setups in quantum physics use pseudorandomness in places where the theory requires randomness. In this Letter we show that the use of pseudorandomness instead of proper randomness in quantum setups has potentially observable consequences. First, we present a new loophole for Bell-like experiments: if some of the parties choose their measurements pseudorandomly, then the computational resources of the local model have to be limited in order to have a proper observation of nonlocality. Second, we show that no amount of pseudorandomness is enough to produce a mixed state by computably choosing pure states from some basis.

Certifying single-system steering for quantum-information processing

Einstein-Podolsky-Rosen (EPR) steering describes how different ensembles of quantum states can be remotely prepared by measuring one particle of an entangled pair. Here, we investigate quantum steering for single quantum d-dimensional systems (qudits) and devise efficient conditions to certify the steerability therein, which are applicable both to single-system steering and EPR steering. In the single-system case our steering conditions enable the unambiguous ruling-out of generic classical means of mimicking steering. Ruling out false-steering scenarios has implications for securing channels against both cloning-based individual attack and coherent attacks when implementing quantum key distribution using qudits. We also show that these steering conditions also have applications in quantum computation, in that they can serve as an efficient criterion for the evaluation of quantum logic gates of arbitrary size. Finally, we describe how the non-local EPR variant of these conditions also function as tools for identifying faithful one-way quantum computation, secure entanglement-based quantum communication, and genuine multipartite EPR steering.

A projected gradient method for optimization over density matrices

An ensemble of quantum states can be described by a Hermitian, positive semidefinite and unit trace matrix called density matrix. Thus, the study of methods for optimizing a certain function (energy, entropy) over the set of density matrices has a direct application to important problems in quantum information and computation. We propose a projected gradient method for solving such problems. By exploiting the geometry of the feasible set, which is the intersection of the cone of Hermitian positive semidefinite matrices with the hyperplane defined by the unit trace constraint, we describe an efficient procedure to compute the projection onto this set using the Frobenius norm. Some important applications, such as quantum state tomography, are described and numerical experiments illustrate the effectiveness of the method when compared to previous methods based on fixed-point iterations or semidefinite programming.

Topology of classical molecular optimal control landscapes for multi-target objectives.

This paper considers laser-driven optimal control of an ensemble of non-interacting molecules whose dynamics lie in classical phase space. The molecules evolve independently under control to distinct final states. We consider a control landscape defined in terms of multi-target (MT) molecular states and analyze the landscape as a functional of the control field. The topology of the MT control landscape is assessed through its gradient and Hessian with respect to the control. Under particular assumptions, the MT control landscape is found to be free of traps that could hinder reaching the objective. The Hessian associated with an optimal control field is shown to have finite rank, indicating an inherent degree of robustness to control noise. Both the absence of traps and rank of the Hessian are shown to be analogous to the situation of specifying multiple targets for an ensemble of quantum states. Numerical simulations are presented to illustrate the classical landscape principles and further characterize the system behavior as the control field is optimized.

Diagonal unitary entangling gates and contradiagonal quantum states

Nonlocal properties of an ensemble of diagonal random unitary matrices of order ${N}^{2}$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $lnN$, in contrast to the $ln{N}^{2}$ behavior characteristic of random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $\ensuremath{\rho}=A{A}^{\ifmmode\dagger\else\textdagger\fi{}}/{N}^{2}$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length ${N}^{2}$ into a square matrix of order $N$. This fact provides a motivation to study the ensemble of non-Hermitian unimodular matrices $A$, with all entries of the same modulus and random phases and the ensemble of quantum states $\ensuremath{\rho}$, such that all their diagonal entries are equal to $1/N$. Such a state is contradiagonal with respect to the computational basis, in the sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse-graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the ``Borel triangle.'' This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.

Quantum discord of ensemble of quantum states

We highlight an information-theoretic meaning of quantum discord as the gap between the ac- cessible information and the Holevo bound in the framework of ensemble of quantum states. This complementary relationship implies that a large amount of pre-existing arguments about the evaluation of quantum discord can be directly applied to the accessible information and vice versa. For an ensemble of two pure qubit states, we show that one can evade the optimization problem with the help of the Koashi-Winter relation. Further, for the general case (two mixed qubit states), we recover the main results presented by Fuchs and Caves [Phys. Rev. Lett. 73, 3047 (1994)], but totally from the perspective of quantum discord. Following this line of thought, we also investigate the geometric discord as an indicator of quantumness of ensembles in detail. Finally, we give an example to elucidate the difference between quantum discord and geometric discord with respect to optimal measurement strategies.

Tight bounds on accessible information and informational power

The accessible information quantifies the amount of classical information that can be extracted from an ensemble of quantum states. Analogously, the informational power quantifies the amount of classical information that can be extracted by a quantum measurement. For both quantities, we provide upper and lower bounds that depend only on the dimension of the system, and we prove their tightness. In the case of symmetric informationally complete (SIC) ensembles and measurements, stronger bounds are provided and their tightness proved for qubits and qutrits. From our upper bounds, we notice, perhaps surprisingly, that the statistics generated by SIC ensembles or measurements in arbitrary dimension, though optimal for tomographic purposes, in fact never contain more than just one bit of information, the rest being constituted by completely random bits. On the other hand, from our lower bounds, we obtain an explicit strategy beating the so-called pretty-good one for the extraction of mutual information in the case of SIC ensembles and measurements.

Matrices of fidelities for ensembles of quantum states and the Holevo quantity

The entropy of the Gram matrix of a joint purification of an ensemble of $K$ mixed states yields an upper bound for the Holevo information $\chi$ of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain meaningful bounds for $\chi$. This is done by constructing various correlation matrices involving fidelities between every pair of states from the ensemble. For $K=3$ quantum states we design a matrix of root fidelities that is positive and the entropy of which is conjectured to upper bound $\chi$. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state correlations in relation to the Holevo quantity.

Informational power of quantum measurements

We introduce the informational power of a quantum measurement as the maximum amount of classical information that the measurement can extract from any ensemble of quantum states. We prove the additivity by showing that the informational power corresponds to the classical capacity of a quantum-classical channel. We restate the problem of evaluating the informational power as the maximization of the accessible information of a suitable ensemble. We provide a numerical algorithm to find an optimal ensemble, and quantify the informational power.

Some General Properties of Unified Entropies

Basic properties of the unified entropies are examined. The consideration is mainly restricted to the finite-dimensional quantum case. Bounds in terms of ensembles of quantum states are given. Both the continuity in Fannes' sense and stability in Lesche's sense are shown for wide ranges of parameters. In particular, uniform estimates are obtained for the quantum R\'{e}nyi entropies. Stability properties in the thermodynamic limit are discussed as well. It is shown that the unified entropies enjoy both the subadditivity and triangle inequality for a certain range of parameters. Non-decreasing of all the unified entropies under projective measurement is proved.

Quantum distributions for localized quantum states

The principle of ergodicity of the quantum theory has been used for elaboration of a new technique for numerical simulation of the Wigner function of open dissipative quantum systems. With this purpose the density matrix of a quantum system is represented via averaging over the ensemble of quantum states in time intervals instead of averaging over the ensemble of stochastic variables. It is shown that this approach leads to new approximate expressions for quantum distributions in the phase space, in particular, Wigner functions for systems localized in the region of classical phase trajectories. As an application, the Wigner functions are calculated for the process of intracavity second harmonic generation in the region of Hopf bifurcations.

Typicality in Ensembles of Quantum States: Monte Carlo Sampling versus Analytical Approximations

Random quantum states are presently of interest in the fields of quantum information theory and quantum chaos. Moreover, a detailed study of their properties can shed light on some foundational issues of the quantum statistical mechanics such as the emergence of well-defined thermal properties from the pure quantum mechanical description of large many-body systems. When dealing with an ensemble of pure quantum states, two questions naturally arise, what is the probability density function on the parameters which specify the state of the system in a given ensemble, and does there exist a "most typical" value of a function of interest in the considered ensemble? Here, two different ensembles are considered, the random pure state ensemble (RPSE) and the fixed expectation energy ensemble (FEEE). By means of a suitable parametrization of the wave function in terms of populations and phases, we focus on the probability distribution of the populations in these ensembles. A comparison is made between the distribution induced by the inherent geometry of the Hilbert Space and an approximate distribution derived by means of the minimization of the informational functional. While the latter can be analytically handled, the exact geometrical distribution is sampled by a Metropolis-Hastings algorithm. The analysis is made for an ensemble of wave functions describing an ideal system composed of n spins of 1/2 and reveals the salient differences between the geometrical and the approximate distributions. The analytical approximations are proven to be useful tools in order to obtain an ensemble-averaged quantity. In particular, we focus on the distribution of the Shannon entropy by providing an explanation of the emergence of a typical value of this quantity in the ensembles.

Distinguishability measures between ensembles of quantum states

A quantum ensemble $\{(p_x, \rho_x)\}$ is a set of quantum states each occurring randomly with a given probability. Quantum ensembles are necessary to describe situations with incomplete a priori information, such as the output of a stochastic quantum channel (generalized measurement), and play a central role in quantum communication. In this paper, we propose measures of distance and fidelity between two quantum ensembles. We consider two approaches: the first one is based on the ability to mimic one ensemble given the other one as a resource and is closely related to the Monge-Kantorovich optimal transportation problem, while the second one uses the idea of extended-Hilbert-space (EHS) representations which introduce auxiliary pointer (or flag) states. Both types of measures enjoy a number of desirable properties. The Kantorovich measures, albeit monotonic under deterministic quantum operations, are not monotonic under generalized measurements. In contrast, the EHS measures are. We present operational interpretations for both types of measures. We also show that the EHS fidelity between ensembles provides a novel interpretation of the fidelity between mixed states--the latter is equal to the maximum of the fidelity between all pure-state ensembles whose averages are equal to the mixed states being compared. We finally use the new measures to define distance and fidelity for stochastic quantum channels and positive operator-valued measures (POVMs). These quantities may be useful in the context of tomography of stochastic quantum channels and quantum detectors.

Random Measurement Bases, Quantum State Distinction and Applications to the Hidden Subgroup Problem

We show that measuring any two low rank quantum states in a random orthonormal basis gives, with high probability, two probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states. This implies that for any finite ensemble of quantum states there is a single POVM that distinguishes between every pair of states from the ensemble by at least a constant times their Frobenius distance; in fact, with high probability a random POVM, under a suitable definition of randomness, suffices. There are examples of ensembles with constant pairwise trace distance where a single POVM cannot distinguish pairs of states by much better than their Frobenius distance, including the important ensemble of coset states of hidden subgroups of the symmetric group (Moore et al., Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005). We next consider the random Fourier method for the hidden subgroup problem (HSP) which consists of Fourier sampling the coset state of the hidden subgroup using random orthonormal bases for the group representations. In cases where every representation of the group has polynomially bounded rank when averaged over the hidden subgroup, the random Fourier method gives a POVM for the HSP operating on one coset state at a time and using totally a polynomial number of coset states. In particular, we get such POVMs whenever the group and the hidden subgroup form a Gel’fand pair, e.g., Abelian, dihedral and Heisenberg groups. This gives a positive counterpart to earlier negative results about random Fourier sampling when the above rank is exponentially large (Grigni et al., Combinatorica 24(1):137–154, 2004), which happens for example in the HSP in the symmetric group.

Unambiguous comparison of ensembles of quantum states

We present a solution of the problem of the optimal unambiguous comparison of two ensembles of unknown quantum states ${|{\ensuremath{\psi}}_{1}⟩}^{\ensuremath{\bigotimes}k}$ and ${|{\ensuremath{\psi}}_{2}⟩}^{\ensuremath{\bigotimes}l}$. We consider two cases: (1) The two unknown states $|{\ensuremath{\psi}}_{1}⟩$ and $|{\ensuremath{\psi}}_{2}⟩$ are arbitrary states of qudits. (2) Alternatively, they are coherent states of a harmonic oscillator. For the case of coherent states we propose a simple experimental realization of the optimal ``comparison'' machine composed of a finite number of beam splitters and a single photodetector.

Ergodicity in computer simulation of quantum dissipative processes

The ergodicity principle in quantum theory is employed for elaboration of a new quantum trajectory technique which is used for numerical simulation of quantum dissipative systems. With this purpose the density matrix of a quantum system is represented as a sum over an ensemble of quantum states in time intervals. The method is employed for computations of a quantum anharmonic oscillator.

The fractional contributions of elementary modes to the metabolism of Escherichia coli and their estimation from reaction entropies

The metabolism of a cell can be viewed as a weighted sum of elementary modes. Due to the multiplicity of modes the identification of the individual weights represents a non-trivial problem. To enable the determination of weighting factors we have identified and implemented two gene deletions in combination with defined growth conditions that limit the metabolism from 4374 original elementary modes to 24 elementary modes for a non-PHB synthesizing control and 40 modes for a PHB synthesizing strain. These remaining modes can be further grouped into five families that have the same overall stoichiometry. Thus, the complexity of the problem is significantly reduced, and weighting factors for each family of modes could be determined from the measurement of accumulation rates of metabolites. Moreover, it is shown that individual weights are inversely correlated with the entropy generated by the operation of the used pathways defined in elementary modes. This suggests that evolution developed cellular regulatory patterns that permit diversity of pathways while favoring efficient pathways with low entropy generation. Furthermore, such correlation provides a rational way of estimating metabolic fluxes based on the thermodynamic properties of elementary modes. This is demonstrated with an example in which experimentally determined, intracellular fluxes are shown to be highly correlated with fluxes computed based on elementary modes and reaction entropies. The analysis suggests that the set of elementary modes can be interpreted analogous to a metabolic ensemble of quantum states of a macroscopic system.

Remote preparation of arbitrary ensembles and quantum bit commitment

The Hughston–Jozsa–Wootters theorem shows that any finite ensemble of quantum states can be prepared “at a distance,” and it has been used to demonstrate the insecurity of all bit commitment protocols based on finite quantum systems without superselection rules. In this paper, we prove a generalized HJW theorem for arbitrary ensembles of states on a C*-algebra. We then use this result to demonstrate the insecurity of bit commitment protocols based on infinite quantum systems, and quantum systems with Abelian superselection rules.

Geometrical entropy from loop quantum gravity

We adopt the point of view that (Riemannian) classical and (loop-based) quantum descriptions of geometry are macro- and micro-descriptions in the usual statistical mechanical sense. This gives rise to the notion of geometrical entropy, which is defined as the logarithm of the number of different quantum states which correspond to one and the same classical geometry configuration (macro-state). We apply this idea to gravitational degrees of freedom induced on an arbitrarily chosen in space 2-dimensional surface. Considering an `ensemble' of particularly simple quantum states, we show that the geometrical entropy $S(A)$ corresponding to a macro-state specified by a total area $A$ of the surface is proportional to the area $S(A)=\alpha A$, with $\alpha$ being approximately equal to $1/16\pi l_p^2$. The result holds both for case of open and closed surfaces. We discuss briefly physical motivations for our choice of the ensemble of quantum states.

Variational principles for ensembles of quantum states.

The maximum-entropy principle is used to generalize the variational principle for the ground state (Rayleigh-Ritz minimization principle) to ensembles of unequally weighted states. A hierarchy of nested inequalities for the associated ``free energies'' is deduced. Applications of these inequalities to the estimation of the highest multiplet energy and a generalization of density-functional theory to ensembles of quantum states are discussed. Previous results given in the literature are obtained as special cases of our results under isothermal and adiabatic optimizations.