Systematic derivation of order parameters through reduced density matrices

A systematic method for determining order parameters for quantum many-body systems on lattices is developed by utilizing reduced density matrices. This method allows one to extract the order parameter directly from the wave functions of the degenerate ground states without the aid of empirical knowledge, and thus opens a way to explore unknown exotic orders. The applicability of this method is demonstrated numerically or rigorously in models that are considered to exhibit dimer, scalar chiral, and topological orders.     

Spectral properties of cones of approximately representable reduced density matrices

We introduce a definition of the approximate spectrum of an operator which is useful in reduced density matrix theory. With precise knowledge of $\text{D}$² (the cone of representable reduced 2-densities) our approximate spectrum of any Hermitian, symmetric, one-body operator, A, agrees with the usual spectrum of operators on Fock space. The virtue of our definition is that with only approximate knowledge of $\text{D}$² we can compute the approximate spectrum for A. The approximate spectrum turns out to be a sensitive tool in assessing the quality of cones of approximately representable reduced 2-densities. Using this notion, we are able to point out a dramatic failure of one cone of approximately representable reduced 2-densities often referred to in the literature. In addition we show how reduced density matrices for excited states of systems with two-body interactions can be computed, given $\text{D}$⁴ the cone of reduced 4-densities.     

Joint product numerical range and geometry of reduced density matrices

The reduced density matrices of a many-body quantum system form a convex set, whose three-dimensional projection Θ is convex in R3. The boundary ?Θ of Θ may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces: symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti's theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range Π of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that a ruled surface emerges naturally when taking a convex hull of Π. We show that, a ruled surface on ?Θsitting in Π has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of Θ, with two boundary pieces of symmetry breaking origin separated by two gapless lines.     

Explicit approximate relation between reduced two- and one-particle density matrices

A class of approximation of the two-particle density matrix d₂ resembling the Hartree-Fock dependence on the one-particle density matrix d₁ is suggested. The integral relation between d₂ and d₁ is exactly maintained. An optimal choice is performed by means of the Pauli principle. The smallness of the error when applied to Coulomb systems, is qualitatively discussed. Extended Thomas-Fermi theory, as recently introduced by the authors is shortly outlined for the case of Coulomb interaction.     

Physical origins of ruled surfaces on the reduced density matrices geometry

The reduced density matrices(RDMs) of many-body quantum states form a convex set. The boundary of low dimensional projections of this convex set may exhibit nontrivial geometry such as ruled surfaces. In this paper, we study the physical origins of these ruled surfaces for bosonic systems. The emergence of ruled surfaces was recently proposed as signatures of symmetrybreaking phase. We show that, apart from being signatures of symmetry-breaking, ruled surfaces can also be the consequence of gapless quantum systems by demonstrating an explicit example in terms of a two-mode Ising model. Our analysis was largely simplified by the quantum de Finetti's theorem—in the limit of large system size, these RDMs are the convex set of all the symmetric separable states. To distinguish ruled surfaces originated from gapless systems from those caused by symmetrybreaking, we propose to use the finite size scaling method for the corresponding geometry. This method is then applied to the two-mode XY model, successfully identifying a ruled surface as the consequence of gapless systems.     

Only n-Qubit Greenberger-Horne-Zeilinger states are undetermined by their reduced density matrices

The generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only states of n qubits that are not uniquely determined among pure states by their reduced density matrices of n-1 qubits. Thus, among pure states, the generalized GHZ states are the only ones containing information at the n-party level. We point out a connection between local unitary stabilizer subgroups and the property of being determined by reduced density matrices.     

Hierarchy of equations for reduced density matrices in the case of thermodynamic equilibrium

A hierarchy of equations for s-particle density matrices at thermodynamic equilibrium is obtained, with the equation for the nonequilibrium density matrix as the starting point. When deducing the hierarchy the hypothesis of maximum statistical independence for the density matrices is used. The hierarchy obtained is an analogue of the classical equilibrium BBGKY hierarchy and goes over into it when . It is shown that thermodynamic quantities can be expressed in terms of functions that enter only into the first hierarchy equations. The hierarchy is analysed in detail in the case of a uniform fluid. As an example in which the equations can be solved easily enough, a hard-sphere system wherein triplet correlations are neglected is considered. Different approximations that can be used when solving the equations derived are discussed. Comparisons are made with the results of other theoretical treatments.     

Geometrical characterization of reduced density matrices reveals quantum phase transitions in many-body systems

<正>Quantum phase transitions (QPTs) play a central role for understanding many-body physics [1]. Different from classical phase transitions which are driven by thermal fluctuations, QPTs are driven by quantum fluctuations at zero temperature and can be accessed by varying some physical parameters of the many-body system. Characterizing QPTs, which normally needs complicated theoretical calculations, becomes a fundamental problem to further study quantum matters. Here a group of physicists proposed to connect the geometrical properties of reduced density matrices (RDMs) of the physical system with its quantum phase transitions [2,3]     

Atomic versus ionized states in many-particle systems and the spectra of reduced density matrices: A model study

We study the spectrum of appropriate reduced density matrices for a model consisting of one quantum particle (electron) in a classical fluid (of protons) at thermal equilibrium. The quantum and classical particles interact by a shortrange, attractive potential such that the quantum particle can form atomic bound states with a single classical particle. We consider two models for the classical component: an ideal gas and the cell model of a fluid. We find that when the system is at low density the spectrum of the electron-proton pair density matrix has, in addition to a continuous part, a discrete part that is associated with atomic bound states. In the high-density limit the discrete eigenvalues disappear in the case of the cell model, indicating the existence of pressure ionization or a Mott effect according to a general criterion for characterizing bound and ionized electron-proton pairs in a plasma proposed recently by M. Girardeau. For the ideal gas model, on the other hand, eigenvalues remain even at high density.     

Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices

In a system of n quantum particles, we define a measure of the degree of irreducible n-way correlation, by which we mean the correlation that cannot be accounted for by looking at the states of n-1 particles. In the case of almost all pure states of three qubits, we show that there is no such correlation: almost every pure state of three qubits is completely determined by its two-particle reduced density matrices.     

Coset parameterization of density matrices

We give an explicit parameterization for the state space of an n-level density matrix. The parameterization is based on the canonical coset decomposition of unitary matrices. As an application, we also calculate explicitly the Bures metric tensor over the state space of two-level quantum system. The text was submitted by the author in English.     

Invariants and nonequilibrium density matrices

A new method of calculating nonequilibrium density matrices with the aid of the quantum integrals of motion is proposed. The method is shown to be very effective in the case of systems described by means of quadratic Hamiltonians. The possibility of constructing phenomenological nonstationary Hamiltonians for a wide class of dissipative systems is discussed. The exact formulas for nonequilibrium density matrices of arbitrary quadratic systems are obtained. The quantum problem of the motion of a charged particle in uniform electric and magnetic fields in the presence of a frictional force proportional to the velocity is solved exactly by means of introducing the new phenomenological Hamiltonian.     

A flow phase diagram for helium superfluids

The existence of the flow phase diagram predicted by Volovik [JETP Lett. 78, 553 (2003)] is discussed based on the available experimental data for He II and ³He-B. The effective temperature-dependent but scale-independent Reynolds number Re_eff≡1/q≡(1?α′)/α, where α and α′ are the mutual friction parameters, and the superfluid Reynolds number characterizing the circulation of the superfluid component in units of the circulation quantum are used as the dynamic parameters. In particular, the flow diagram permits the identification of the experimentally observed turbulent states I and II in counterflowing He II with the classical and quantum turbulent regimes suggested by Volovik.     

Spectral density of mixtures of random density matrices for qubits

We derive the spectral density of the equiprobable mixture of two random density matrices of a two-level quantum system. We also work out the spectral density of mixture under the so-called quantum addition rule. We use the spectral densities to calculate the average entropy of mixtures of random density matrices, and show that the average entropy of the arithmetic-mean-state of n qubit density matrices randomly chosen from the Hilbert–Schmidt ensemble is never decreasing with the number n. We also get the exact value of the average squared fidelity. Some conjectures and open problems related to von Neumann entropy are also proposed.     

Integral equations for particle density matrices

A system of integral equations for particle density matrices is obtained. The method of expansion in one-, two-, three-particle (etc.) parameters is used. It is shown that in the classical limit the resulting equations become the system of familiar equations for particle distribution functions of classical statistical mechanics.Translated from Izvestiya VUZ. Fizika, Vol. 11, No. 8, pp. 81–86, August, 1968.