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arXiv:1112.0740 · Physical Review Letters

Thermal Pure Quantum States at Finite Temperature

Sho Sugiura, Akira Shimizu

Published Jun 11, 2012• 215 citations
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Abstract

An equilibrium state can be represented by a pure quantum state, which we call a thermal pure quantum (TPQ) state. We propose a new TPQ state and a simple method of obtaining it. A single realization of the TPQ state suffices for calculating all statistical-mechanical properties, including correlation functions and genuine thermodynamic variables, of a quantum system at finite temperature.

DOI
10.1103/physrevlett.108.240401

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An equilibrium state can be represented by a pure quantum state, which we call a thermal pure quantum (TPQ) state. We propose a new TPQ state and a simple method of obtaining it. A single realization of the TPQ state suffices for calculating all statistical-mechanical properties, including correlation functions and genuine thermodynamic variables, of a quantum system at finite temperature.
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.
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.
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 propose a quantum algorithm to investigate finite-temperature properties of quantum many-body systems by implementing the microcanonical thermal pure quantum (TPQ) states [S. Sugiura and A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012)] on a quantum computer. In the formalism, the TPQ state is constructed by applying the power of a rescaled Hamiltonian to a random state. We clarify that, using techniques based on quantum signal processing, the power operation can be efficiently realized when the ground-state energy of a quantum system of interest is given with some precision. Our algorithm has an advantage in calculating thermodynamic quantities at low temperatures.
We present a quantum algorithm for the microcanonical thermal pure quantum (TPQ) method, which has an advantage in evaluating thermodynamic quantities at finite temperatures, by combining with some recently developed techniques derived from quantum singular value transformation. When the ground energy of quantum systems has already been obtained precisely, the multiple products of the Hamiltonian are efficiently realized and the TPQ states at low temperatures are systematically constructed in quantum computations.