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arXiv:1106.5330 · Journal of Physics A Mathematical and Theoretical

Statistical distribution of the local purity in a large quantum system

Antonella De Pasquale, Paolo Facchi A, Vittorio Giovannetti, Giorgio Parisi A, Saverio Pascazio, Antonello Scardicchio

Published Dec 6, 2011• 28 citations
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Abstract

The local purity of large many-body quantum systems can be studied by following a statistical mechanical approach based on a random matrix model. Restricting the analysis to the case of global pure states, this method proved to be successful, and a full characterization of the statistical properties of the local purity was obtained by computing the partition function of the problem. Here we generalize these techniques to the case of global mixed states. In this context, by uniformly sampling the phase space of states with assigned global mixedness, we determine the exact expression of the first two moments of the local purity and a general expression for the moments of higher order. This generalizes previous results obtained for globally pure configurations. Furthermore, through the introduction of a partition function for a suitable canonical ensemble, we compute the approximate expression of the first moment of the marginal purity in the high-temperature regime. In the process, we establish a formal connection with the theory of quantum twirling maps that provides an alternative, possibly fruitful, way of performing the calculation.

DOI
10.1088/1751-8113/45/1/015308

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The local purity of large many-body quantum systems can be studied by following a statistical mechanical approach based on a random matrix model. Restricting the analysis to the case of global pure states, this method proved to be successful, and a full characterization of the statistical properties of the local purity was obtained by computing the partition function of the problem. Here we generalize these techniques to the case of global mixed states. In this context, by uniformly sampling the phase space of states with assigned global mixedness, we determine the exact expression of the first two moments of the local purity and a general expression for the moments of higher order. This generalizes previous results obtained for globally pure configurations. Furthermore, through the introduction of a partition function for a suitable canonical ensemble, we compute the approximate expression of the first moment of the marginal purity in the high-temperature regime. In the process, we establish a formal connection with the theory of quantum twirling maps that provides an alternative, possibly fruitful, way of performing the calculation.
In this thesis we study the behavior of bipartite entanglement of a large quantum system, by analyzing the distribution of the Schmidt coefficients of the reduced density matrix. Applying the general methods of classical statistical mechanics, we develop a canonical approach for the study of the distribution of these coefficients for a fixed value of the average entanglement. We introduce a partition function depending on a fictitious temperature, which localizes the measure on the set of states with higher and lower entanglement, if compared to typical (random) states with respect to the Haar measure. The purity of one subsystem, which is our entanglement measure/indicator, plays the role of energy in the partition function. This thesis consists of two parts. In the first part, we completely characterize the distribution of the purity and of the eigenvalues for pure states. The global picture unveils several locally stable solutions exchange stabilities, through the presence of first and second order phase transitions. We also detect the presence of metastable states. In the second part, we focus on mixed states. Through the same statistical approach, we determine the exact expression of the first two moments of the local purity and the high temperature expansion of the first moment. We also bridge our problem with the theory of quantum channels, more precisely we exploit the symmetry properties of the twirling transformations in order to compute the exact expression for the first moment of the local purity.
We develop a formula to evaluate the purity of a series of thermal equilibrium states that can be calculated in numerical experiments without knowing the exact form of the quantum state a priori. Canonical typicality guarantees that there are numerous microscopically different expressions of such states, which we call thermal mixed quantum (TMQ) states. Suppose that we construct a TMQ state by a mixture of Nsamp{N}_{\mathrm{samp}}Nsamp​ independent pure states. The weight of each pure state is given by its norm, and the partition function is given by the average of the norms. To qualify how efficiently the mixture is done, we introduce a quantum statistical quantity called ``normalized fluctuation of partition function (NFPF).'' For smaller NFPF, the TMQ state is closer to the equally weighted mixture of pure states, which means higher efficiency, requiring a smaller Nsamp{N}_{\mathrm{samp}}Nsamp​. The largest NFPF is realized in the Gibbs state with purity-0 and exponentially large Nsamp{N}_{\mathrm{samp}}Nsamp​, while the smallest NFPF is given for thermal pure quantum state with purity-1 and Nsamp=1{N}_{\mathrm{samp}}=1Nsamp​=1. The purity is formulated using solely the NFPF and roughly gives ${N}_{\mathrm{samp}}^{\ensuremath{-}1}$. Our analytical results are numerically tested and confirmed by the two random sampling methods built on matrix-product-state-based wave functions.
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.
We compute the distribution of the purity for random density matrices (i.e. random mixed states) in a large quantum system, distributed according to the Bures measure. The full distribution of the purity is computed using a mapping to random matrix theory and then a Coulomb gas method. We find three regimes that correspond to two phase transitions in the associated Coulomb gas. The first transition is characterized by an explosion of the third derivative on the left of the transition point. The second transition is of first order, it is characterized by the detachment of a single charge of the Coulomb gas. A key remark in this paper is that the random Bures states are closely related to the O ( n ) model for n = 1. This actually led us to study 'generalized Bures states' by keeping n general instead of specializing to n = 1.
In this paper, we extend the analysis of multipartite entanglement, based on techniques from classical statistical mechanics, to a system composed of n d-level parties (qudits). We introduce a suitable partition function at a fictitious temperature with the average local purity of the system as Hamiltonian. In particular, we analyze the high-temperature expansion of this partition function, prove the convergence of the series, and study its asymptotic behavior as d → ∞. We make use of a diagrammatic technique, classify the graphs, and study their degeneracy. We are thus able to evaluate their contributions and estimate the moments of the distribution of the local purity.
Abstract. Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or in general by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two–mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p−entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixedstate entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions. 1.