Eigenvalue Distributions of Reduced Density Matrices

Christandl, Matthias; Doran, Brent; Kousidis, Stavros; Walter, Michael

doi:10.1007/s00220-014-2144-4

Cited by 54 publications

(78 citation statements)

References 104 publications

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“…While formal (and more general) definitions will be provided later in the paper, these examples serve to introduce the notions of sharing (1-2 sharing) and joining (1-2 joining), respectively. In its most general formulation, the joinability problem is also known as the quantum marginal problem (or local consistency problem), which has been heavily investigated both from a mathematical-physics [9][10][11] and a quantum-chemistry perspective [12,13] and is known to be QMA-hard [14]. Our choice of terminology, however, facilitates a uniform language for describing the joinability/sharability scenarios.…”

Section: Introductionmentioning

confidence: 99%

Compatible quantum correlations: Extension problems for Werner and isotropic states

Johnson

Viola

2013

Phys. Rev. A

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We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global N -partite quantum state. Intuitively, we say that the multipartite state "joins" the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible N -partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in d dimensions, and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is m-n sharability of a Werner state across N subsystems, which may be seen as a variant of the N -representability problem to the case where subsystems are partitioned into two groupings of m and n parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we determine necessary and sufficient conditions for three-party joinability and 1-n sharability for arbitrary d. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed.

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Section: Introductionmentioning

confidence: 99%

Compatible quantum correlations: Extension problems for Werner and isotropic states

Johnson

Viola

2013

Phys. Rev. A

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“…where l(w) is the length of the Weyl group element w, and δ α for the Dirac measure at α; * means the convolution, the same below. Moreover, we also have the following result [8]:…”

Section: The Product Of Co-adjoint Orbitsmentioning

confidence: 62%

“…The paper is organized as follows. In Section 2, we present background tools related to this paper, and recall the results obtained in [10] by formulation used in [8]. Then, we consider the equiprobable mixture of several qubit states, i.e., we derive the spectral density of two qubit states and three qubit states (Theorem 3.4-Theorem 3.7), respectively, in Section 3.…”

Section: Introductionmentioning

confidence: 99%

Duistermaat–Heckman measure and the mixture of quantum states

Zhang

Jiang

2019

J. Phys. A: Math. Theor.

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In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum A + B of two independent random Hermitian matrices A and B. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.Mathematics Subject Classification. 22E70, 81Q10, 46L30, 15A90, 81R05

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“…Presumably, as long as the set of occupation numbers {n k } does not lie on the boundary of the N -representable region defined by the inequality constraints, the n k can be considered as linearly independent degrees of freedom. It is also worth noting that symplectic geometry has very recently been applied to this problem and similar problems [63]. At least for the first level of the hierarchy, it appears that we can indeed consider the n k as linearly independent degrees of freedom.…”

Section: B Symplectic Structure Of the Von Neumann Equationmentioning

confidence: 90%

Hamiltonian formulation of nonequilibrium quantum dynamics: Geometric structure of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy

Requist

2012

Phys. Rev. A

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Time-resolved measurement techniques are opening a window on nonequilibrium quantum phenomena that is radically different from the traditional picture in the frequency domain. The simulation and interpretation of nonequilibrium dynamics is a conspicuous challenge for theory. This paper presents an approach to quantum many-body dynamics that is based on a Hamiltonian formulation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of motion for reduced density matrices. These equations have an underlying symplectic structure, and we write them in the form of the classical Hamilton equations for canonically conjugate variables. Applying canonical perturbation theory or the Krylov-Bogoliubov averaging method to the resulting equations yields a systematic approximation scheme. The possibility of using memory-dependent functional approximations to close the Hamilton equations at a given level of the hierarchy is discussed. The geometric structure of the equations gives rise to reduced geometric phases that are observable even for noncyclic evolutions of the many-body state. The approach is applied to a finite Hubbard chain which undergoes a quench in on-site interaction energy U . Canonical perturbation theory, carried out to second order, fully captures the nontrivial real-time dynamics of the model, including resonance phenomena and the coupling of fast and slow variables.

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Eigenvalue Distributions of Reduced Density Matrices

Cited by 54 publications

References 104 publications

Compatible quantum correlations: Extension problems for Werner and isotropic states

Compatible quantum correlations: Extension problems for Werner and isotropic states

Duistermaat–Heckman measure and the mixture of quantum states

Hamiltonian formulation of nonequilibrium quantum dynamics: Geometric structure of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy

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