Ergodic and mixing quantum channels in finite dimensions

Burgarth, Daniel; Chiribella, Giulio; Giovannetti, Vittorio; Perinotti, Paolo; Yuasa, Kazuya

doi:10.1088/1367-2630/15/7/073045

Cited by 69 publications

(91 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…(1) under the action of a quantum Liuovillian L t which exhibits an explicit temporal dependence induced by the external modulation of some control parameters, say the value of a magnetic field or the intensity of a laser which are gradually changed according to some assigned protocol. In what follows we shall assume that for all t, L t admits a unique zero (instantaneous) eigenstate ρ 0 (t) and that all the other eigenvalues have a strictly negative real part (in this case the map is said to be relaxing or mixing [2,39]). This causes the system to exponentially converge to the instantaneous steady state, for a fixed value of the external modulation:…”

Section: ρ(T) = L T [ρ(T)]mentioning

confidence: 99%

Slow Dynamics and Thermodynamics of Open Quantum Systems

2017

View full text Add to dashboard Cite

We develop a perturbation theory of quantum (and classical) master equations with slowly varying parameters, applicable to systems which are externally controlled on a time scale much longer than their characteristic relaxation time. We apply this technique to the analysis of finite-time isothermal processes in which, differently from quasi-static transformations, the state of the system is not able to continuously relax to the equilibrium ensemble. Our approach allows to formally evaluate perturbations up to arbitrary order to the work and heat exchange associated to an arbitrary process. Within first order in the perturbation expansion, we identify a general formula for the efficiency at maximum power of a finite-time Carnot engine. We also clarify under which assumptions and in which limit one can recover previous phenomenological results as, for example, the Curzon-Ahlborn efficiency.A central result in the study of open quantum systems [1, 2] is the Markovian Master Equation (MME) approach which, under realistic assumptions, describes the temporal evolution of a system of interest A induced by a weak coupling with a large external environment E. This consists in a first order linear differential equatioṅ ρ(t) = L[ρ(t)] where ρ(t) is the density matrix of A and where the generator of the dynamics is provided by a quantum Liouvillian superoperator L that can be casted in the so called Gorini-Kossakowski-Sudarshan-Lindblad form [3][4][5]. For autonomous systems the latter does not exhibit an explicit time dependence and the dynamics of A exponentially relaxes to a (typically unique) equilibrium steady state ρ 0 identified by the null eigenvector equation L[ρ 0 ] = 0. MMEs can also be employed to describe the temporal evolution of A when it is tampered by the presence of slow varying, external driving forces. Indeed, as long as these operate on a time scale which is much larger than the characteristic bath correlations times and the inverse frequencies of the system of interest, the effective coupling between A and E adapts instantaneously to the driving control, resulting on a MME governed by a time-dependent Liouvillian generator, i.e.An explicit integration of this equation is in general difficult to obtain. Yet, if the control forces are so slow that their associated time scale is also larger with respect to the relaxation time of the system induced by the interaction with E, one expects ρ(t) to approximately follow the instantaneous equilibrium state ρ 0 (t) that nullifies L t . Our aim is to estimate quantitative deviations from this ultra-slow driving regime. For this purpose we develop a perturbation theory valid in the limit of slowly varying Liouvillians L t and derive a formal solution of Eq. (1) which allows one to evaluate non-equilibrium corrections up to arbitrary order. The main motivation of our analysis is to model thermodynamic processes and cycles beyond the usual reversible limit which is strictly valid only for infinitely long quasi-static transformations. Finite-time thermodynamics [7,8]...

show abstract

Section: ρ(T) = L T [ρ(T)]mentioning

confidence: 99%

Slow Dynamics and Thermodynamics of Open Quantum Systems

2017

View full text Add to dashboard Cite

show abstract

“…If the fixed point * r is unique, the quantum channel  is called ergodic [14][15][16]. It implies N N 1 , states , 2 n N n…”

Section: Notation and Mathematical Backgroundmentioning

confidence: 99%

“…Note again that  ¢ might not be diagonalizable if some of the eigenvalues n l of  are degenerated, but it is not a problem for the convergence: see [13][14][15][16].…”

Section: Notation and Mathematical Backgroundmentioning

confidence: 99%

“…We will see that the property of the channel describing the process is important. The idea is to let the probing system forget about the past by the mixing of the channel [13][14][15][16] (the channel being intrinsically mixing or designed to be mixing), which clusters the data and allows the central limit theorem to hold for appropriately chosen functionals of the data. This sequential scheme is suited to account for estimation procedures where one aims to recover g via a sequence of weak measurements that slightly perturb the probe.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quantum estimation via sequential measurements

Burgarth

Giovannetti

et al. 2015

New J. Phys.

View full text Add to dashboard Cite

The problem of estimating a parameter of a quantum system through a series of measurements performed sequentially on a quantum probe is analyzed in the general setting where the underlying statistics is explicitly non-i.i.d. We present a generalization of the central limit theorem in the present context, which under fairly general assumptions shows that as the number N of measurement data increases the probability distribution of functionals of the data (e.g., the average of the data) through which the target parameter is estimated becomes asymptotically normal and independent of the initial state of the probe. At variance with the previous studies (Guţă M 2011 Phys. Rev. A 83 062324; van Horssen M and Guţă M 2015 J. Math. Phys. 56 022109) we take a diagrammatic approach, which allows one to compute not only the leading orders in N of the moments of the average of the data but also those of the correlations among subsequent measurement outcomes. In particular our analysis points out that the latter, which are not available in usual i.i.d. data, can be exploited in order to improve the accuracy of the parameter estimation. An explicit application of our scheme is discussed by studying how the temperature of a thermal reservoir can be estimated via sequential measurements on a quantum probe in contact with the reservoir.See figure 1(a). In this context, the ultimate limits on the attainable precision in the estimation of g, optimized with respect to the general detection strategy, can be computed, resulting in the so-called quantum Cramér-Rao bound, which exhibits the functional dependence upon g r via the quantum Fisher information. See e.g. [2,3,[7][8][9][10][11][12].In many situations of physical interest, however, the possibility of reinitializing the setup to the same state is not necessarily guaranteed. In the present study we are going to consider a different scheme, in which a single probing system undergoes multiple applications of g L while being monitored during the process without being

show abstract

“…The second condition is equivalent to the statement that the channel  has one and only one eigenvalue on the unit circle [55]. Since all the eigenvalues of a quantum channel are inside the unit circle, this implies that the eigenvalue λ with the second largest modulus satisfies the condition l < 1 | | .…”

Section: Compression Protocol For Mpss With Variable Boundary Conditionsmentioning

confidence: 99%

Quantum compression of tensor network states

2020

View full text Add to dashboard Cite

We design quantum compression algorithms for parametric families of tensor network states. We first establish an upper bound on the amount of memory needed to store an arbitrary state from a given state family. The bound is determined by the minimum cut of a suitable flow network, and is related to the flow of information from the manifold of parameters that specify the states to the physical systems in which the states are embodied. For given network topology and given edge dimensions, our upper bound is tight when all edge dimensions are powers of the same integer. When this condition is not met, the bound is optimal up to a multiplicative factor smaller than 1.585. We then provide a compression algorithm for general state families, and show that the algorithm runs in polynomial time for matrix product states.

show abstract

Ergodic and mixing quantum channels in finite dimensions

Cited by 69 publications

References 43 publications

Slow Dynamics and Thermodynamics of Open Quantum Systems

Slow Dynamics and Thermodynamics of Open Quantum Systems

Quantum estimation via sequential measurements

Quantum compression of tensor network states

Contact Info

Product

Resources

About