O. Buerschaper; J. M. Mombelli; M. Christandl; M. Aguado
2013
A hierarchy of topological tensor network states
Type
Journal Article
Author
O. Buerschaper; J. M. Mombelli; M. Christandl; M. Aguado
Year
2013
Journal
Journal of Mathematical Physics
Abstract
We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [Ann. Phys. 303, 2 (2003)]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.
The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected since at least the 1970s, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Here, we provide the first analytic analysis of the physical relevance of these constraints. We compute the natural occupation numbers for the ground states of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are almost, but not exactly, pinned to the boundary of the allowed region (quasipinned). The result suggests that the physics behind the phenomenon is richer than previously appreciated. In particular, it shows that for some models, the generalized Pauli constraints play a role for the ground state, even though they do not limit the ground-state energy. Our findings suggest a generalization of the Hartree-Fock approximation.
Multi-particle entanglement is a fundamental feature of quantum mechanics and an essential resource for quantum information processing and interferometry. Yet, our understanding of its structure is still in its infancy. A systematic classification of multiparticle entanglement is provided by the study of equivalence of entangled states under stochastic local operations and classical communication. Determining the precise entanglement class of a state in the laboratory, however, is impractical as it requires measuring a number of parameters exponential in the particle number. Here, we present a solution to the challenge of classifying multi-particle entanglement in a way that is both experimentally feasible and systematic, i.e., applicable to arbitrary quantum systems. This is achieved by associating to each class an entanglement polytope--the collection of eigenvalues of the one-body reduced density matrices of all states contained in the class. Determining whether the eigenvalues of an entangled state belong to a given entanglement polytope provides a new criterion for multiparticle entanglement. It is decidable from a linear number of locally accessible parameters and robust to experimental noise. We describe an algorithm for computing entanglement polytopes for any number of particles, both distinguishable and indistinguishable. Further, we illustrate the power of entanglement polytopes for witnessing genuine multipartite entanglement and relate them to entanglement distillation. The polytopes for experimentally relevant systems comprised of either several qubits or bosonic two-level systems are explained.
Computing Multiplicities of Lie Group Representations
M. Christandl; B. Doran; M. Walter
2012
Computing Multiplicities of Lie Group Representations
Type
Online Database
Author
M. Christandl; B. Doran; M. Walter
Year
2012
Abstract
For fixed compact connected Lie groups H \subseteq G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes. The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.
We analyse the entanglement of the antisymmetric state in dimension d × d and present two main results. First, we show that the amount of secrecy that can be extracted from the state is low, more precisely, the distillable key is bounded by O(1/d). Second, we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state: entanglement cost is larger than a constant, independent of d. The second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant. Finally, we note that the regularised relative entropy of entanglement is asymptotically continuous in the state. Elementary and advanced facts from the representation theory of the unitary group, including the concept of plethysm, play a central role in the proofs of the main results.
Eigenvalue Distributions of Reduced Density Matrices
M. Christandl; B. Doran; S. Kousidis; M. Walter
2012
Eigenvalue Distributions of Reduced Density Matrices
Type
Online Database
Author
M. Christandl; B. Doran; S. Kousidis; M. Walter
Year
2012
Abstract
Given a random quantum state of multiple (distinguishable or indistinguishable) particles, we provide an algorithm, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices, and hence some associated physical invariants of the state. As a corollary, by taking the support of this probability distribution, which is a convex polytope, we recover a complete solution to the one-body quantum marginal problem, i.e., the problem of characterizing the one-body reduced density matrices that arise from some multi-particle quantum state. In the fermionic instance of the problem, which is known as the one-body N-representability problem, the famous Pauli principle amounts to one linear inequality in the description of the convex polytope. We obtain the probability distribution by reducing to computing the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for a compact Lie group action to that for the maximal torus action; we state and prove our results in this more general symplectic setting. Our approach is in striking contrast to the existing solution to the computation of the supporting polytope by Klyachko and by Berenstein and Sjamaar, which made crucial use of non-Abelian features. In algebraic geometry, Duistermaat-Heckman measures correspond to the asymptotic distribution of multiplicities of irreducible representations in the associated coordinate ring. In the case of the one-body quantum marginal problem, these multiplicities include bounded height Kronecker and plethysm coefficients. A quantized version of the Abelianization procedure provides an efficient algorithm for their computation.
Detection of Multiparticle Entanglement: Quantifying the Search for Symmetric Extensions
F. G.S.L. Brandao; M. Christandl
2012
Detection of Multiparticle Entanglement: Quantifying the Search for Symmetric Extensions
Type
Journal Article
Author
F. G.S.L. Brandao; M. Christandl
Year
2012
Journal
Physical Review Letters
Abstract
We provide quantitative bounds on the characterization of multiparticle separable states by states that have locally symmetric extensions. The bounds are derived from two-particle bounds and relate to recent studies on quantum versions of de Finetti’s theorem. We discuss algorithmic applications of our results, in particular a quasipolynomial-time algorithm to decide whether a multiparticle quantum state is separable or entangled (for constant number of particles and constant error in the norm induced by one-way local operations and classical communication, or in the Frobenius norm). Our results provide a theoretical justification for the use of the search for symmetric extensions as a test for multiparticle entanglement.
Complete Insecurity of Quantum Protocols for Classical Two-Party Computation
H. Buhrman; M. Christandl; C. Schaffner
2012
Complete Insecurity of Quantum Protocols for Classical Two-Party Computation
Type
Journal Article
Author
H. Buhrman; M. Christandl; C. Schaffner
Year
2012
Journal
Physical Review Letters
Abstract
A fundamental task in modern cryptography is the joint computation of a function which has two inputs, one from Alice and one from Bob, such that neither of the two can learn more about the other’s input than what is implied by the value of the function. In this Letter, we show that any quantum protocol for the computation of a classical deterministic function that outputs the result to both parties (two-sided computation) and that is secure against a cheating Bob can be completely broken by a cheating Alice. Whereas it is known that quantum protocols for this task cannot be completely secure, our result implies that security for one party implies complete insecurity for the other. Our findings stand in stark contrast to recent protocols for weak coin tossing and highlight the limits of cryptography within quantum mechanics. We remark that our conclusions remain valid, even if security is only required to be approximate and if the function that is computed for Bob is different from that of Alice.
In this work, we show that the asymptotic limit of the recoupling coefficients of the symmetric group is characterized by the existence of quantum states of three particles with given eigenvalues for their reduced density matrices. This parallels Wigner's observation that the semiclassical behavior of the 6j-symbols for SU(2)---fundamental to the quantum theory of angular momentum---is governed by the existence of Euclidean tetrahedra. We explain how to deduce solely from symmetry properties of the recoupling coefficients the strong subadditivity of the von Neumann entropy, first proved by Lieb and Ruskai, and discuss possible generalizations of our result.
Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state and, therefore, have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in terms of confidence regions—a concept originating from classical statistics. Confidence regions are subsets of the state space in which the true state lies with high probability, independently of any prior assumption on the distribution of the possible states. Our method for computing confidence regions can be applied to arbitrary measurements including fully coherent ones; it is practical and particularly well suited for tomography on systems consisting of a small number of qubits, which are currently in the focus of interest in experimental quantum information science.
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