Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2202.07207 · Physical review. B./Physical review. B

Purity of thermal mixed quantum states

Atsushi Iwaki, Chisa Hotta

Published Sep 8, 2022• 10 citations
arXivHTMLPDF

Abstract

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 ${N}_{\mathrm{samp}}$ 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 ${N}_{\mathrm{samp}}$. The largest NFPF is realized in the Gibbs state with purity-0 and exponentially large ${N}_{\mathrm{samp}}$, while the smallest NFPF is given for thermal pure quantum state with purity-1 and ${N}_{\mathrm{samp}}=1$. 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.

DOI
10.1103/physrevb.106.094409

Related papers

Statistical distribution of the local purity in a large quantum system

Antonella De Pasquale, Paolo Facchi A, Vittorio Giovannetti +3 more

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.

arXiv:1106.5330•Journal of Physics A Mathematical and Theoretical•Dec 6, 2011•28 citations

Canonical Thermal Pure Quantum State

Sho Sugiura, Akira Shimizu

A thermal equilibrium state of a quantum many-body system can be represented by a typical pure state, which we call a thermal pure quantum (TPQ) state. We construct the canonical TPQ state, which corresponds to the canonical ensemble of the conventional statistical mechanics. It is related to the microcanonical TPQ state, which corresponds to the microcanonical ensemble, by simple analytic transformations. Both TPQ states give identical thermodynamic results, if both ensembles do, in the thermodynamic limit. The TPQ states corresponding to other ensembles can also be constructed. We have thus established the TPQ formulation of statistical mechanics, according to which all quantities of statistical-mechanical interest are obtained from a single realization of any TPQ state. We also show that it has great advantages in practical applications. As an illustration, we study the spin-1/2 kagome Heisenberg antiferromagnet.

arXiv:1302.3138•Physical Review Letters•Jul 1, 2013•230 citations

Thermal pure quantum states of many-particle systems

Masahiko Hyuga, Sho Sugiura, Kazumitsu Sakai +1 more

We generalize the thermal pure quantum (TPQ) formulation of statistical mechanics, in such a way that it is applicable to systems whose Hilbert space is infinite dimensional. Assuming particle systems, we construct the grand-canonical TPQ (gTPQ) state, which is the counterpart of the grand-canonical Gibbs state of the ensemble formulation. A single realization of the gTPQ state gives all quantities of statistical-mechanical interest, with exponentially small probability of error. This formulation not only sheds new light on quantum statistical mechanics but also is useful for practical computations. As an illustration, we apply it to the Hubbard model, on a one-dimensional (1D) chain and on a two-dimensional (2D) triangular lattice. For the 1D chain, our results agree well with the exact solutions over wide ranges of temperature, chemical potential, and the on-site interaction. For the 2D triangular lattice, for which exact results are unknown, we obtain reliable results over a wide range of temperature. We also find that finite-size effects are much smaller in the gTPQ state than in the canonical TPQ state. This also shows that in the ensemble formulation the grand-canonical Gibbs state of a finite-size system simulates an infinite system much better than the canonical Gibbs state.

arXiv:1405.3755•Physical Review B•Sep 29, 2014•30 citations

Purity- and entropy-bounded uncertainty relations for mixed quantum states

V. V. Dodonov

We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the `most classical' thermal states is used as a `measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose `purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the `standard purity' Tr 2 and 37% if the `superpurity' lim r →0 [Tr ( 1+1/ r )] r is chosen as a measure of deviation).

Journal of Optics B Quantum and Semiclassical Optics•Apr 5, 2002•68 citations

Formulation of statistical mechanics based on thermal pure quantum states

Sho Sugiura

Quantum statistical mechanics can be fomulated based on a single pure quantum state.In this thesis, I establish this formulation, see the properties of such a pure quantum state, and apply the formulation to a practical application.Historically, it has been found that quantum states have robust "typicality", which is the property that almost all the realizable state give equilibrium values correctly for all "mechanical variables", such as energy and a correlation function.I generally call the pure quantum states which can represent the equilibrium state "thermal pure quantum (TPQ) state".I extend the typicality so that it is applicable to "genuine thermodynamic variables", such as entropy and temperature.I introduce many kinds of the TPQ states which include a "(grand)canonical TPQ state" , which is specified by intensive variables such as temperature, and a "microcanonical TPQ state", which is easy to generate.Owing to these results, I establish the new formulation of statistical mechanics based on the TPQ states.

•Mar 24, 2015•0 citations

NEW FORMULATION OF STATISTICAL MECHANICS USING THERMAL PURE QUANTUM STATES

Sho Sugiura, Akira Shimizu

We formulate statistical mechanics based on a pure quantum state, which we call a ``thermal pure quantum (TPQ) state''. A single TPQ state gives not only equilibrium values of mechanical variables, such as magnetization and correlation functions, but also those of genuine thermodynamic variables and thermodynamic functions, such as entropy and free energy. Among many possible TPQ states, we discuss the canonical TPQ state, the TPQ state whose temperature is specified. % We also propose there are other TPQ states. In the TPQ formulation of statistical mechanics, thermal fluctuations are completely included in quantum-mechanical fluctuations. As a consequence, TPQ states have much larger quantum entanglement than the equilibrium density operators of the ensemble formulation. We also show that the TPQ formulation is very useful in practical computations, by applying the formulation to a frustrated two-dimensional quantum spin system.

•Aug 15, 2016•3 citations
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 Nsamp\ensuremath−1{N}_{\mathrm{samp}}^{\ensuremath{-}1}Nsamp\ensuremath−1​. Our analytical results are numerically tested and confirmed by the two random sampling methods built on matrix-product-state-based wave functions.
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.
A thermal equilibrium state of a quantum many-body system can be represented by a typical pure state, which we call a thermal pure quantum (TPQ) state. We construct the canonical TPQ state, which corresponds to the canonical ensemble of the conventional statistical mechanics. It is related to the microcanonical TPQ state, which corresponds to the microcanonical ensemble, by simple analytic transformations. Both TPQ states give identical thermodynamic results, if both ensembles do, in the thermodynamic limit. The TPQ states corresponding to other ensembles can also be constructed. We have thus established the TPQ formulation of statistical mechanics, according to which all quantities of statistical-mechanical interest are obtained from a single realization of any TPQ state. We also show that it has great advantages in practical applications. As an illustration, we study the spin-1/2 kagome Heisenberg antiferromagnet.
We generalize the thermal pure quantum (TPQ) formulation of statistical mechanics, in such a way that it is applicable to systems whose Hilbert space is infinite dimensional. Assuming particle systems, we construct the grand-canonical TPQ (gTPQ) state, which is the counterpart of the grand-canonical Gibbs state of the ensemble formulation. A single realization of the gTPQ state gives all quantities of statistical-mechanical interest, with exponentially small probability of error. This formulation not only sheds new light on quantum statistical mechanics but also is useful for practical computations. As an illustration, we apply it to the Hubbard model, on a one-dimensional (1D) chain and on a two-dimensional (2D) triangular lattice. For the 1D chain, our results agree well with the exact solutions over wide ranges of temperature, chemical potential, and the on-site interaction. For the 2D triangular lattice, for which exact results are unknown, we obtain reliable results over a wide range of temperature. We also find that finite-size effects are much smaller in the gTPQ state than in the canonical TPQ state. This also shows that in the ensemble formulation the grand-canonical Gibbs state of a finite-size system simulates an infinite system much better than the canonical Gibbs state.
We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the `most classical' thermal states is used as a `measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose `purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the `standard purity' Tr 2 and 37% if the `superpurity' lim r →0 [Tr ( 1+1/ r )] r is chosen as a measure of deviation).
Quantum statistical mechanics can be fomulated based on a single pure quantum state.In this thesis, I establish this formulation, see the properties of such a pure quantum state, and apply the formulation to a practical application.Historically, it has been found that quantum states have robust "typicality", which is the property that almost all the realizable state give equilibrium values correctly for all "mechanical variables", such as energy and a correlation function.I generally call the pure quantum states which can represent the equilibrium state "thermal pure quantum (TPQ) state".I extend the typicality so that it is applicable to "genuine thermodynamic variables", such as entropy and temperature.I introduce many kinds of the TPQ states which include a "(grand)canonical TPQ state" , which is specified by intensive variables such as temperature, and a "microcanonical TPQ state", which is easy to generate.Owing to these results, I establish the new formulation of statistical mechanics based on the TPQ states.
We formulate statistical mechanics based on a pure quantum state, which we call a ``thermal pure quantum (TPQ) state''. A single TPQ state gives not only equilibrium values of mechanical variables, such as magnetization and correlation functions, but also those of genuine thermodynamic variables and thermodynamic functions, such as entropy and free energy. Among many possible TPQ states, we discuss the canonical TPQ state, the TPQ state whose temperature is specified. % We also propose there are other TPQ states. In the TPQ formulation of statistical mechanics, thermal fluctuations are completely included in quantum-mechanical fluctuations. As a consequence, TPQ states have much larger quantum entanglement than the equilibrium density operators of the ensemble formulation. We also show that the TPQ formulation is very useful in practical computations, by applying the formulation to a frustrated two-dimensional quantum spin system.