Please come attend our Workshop on Quantum Marginals at the Isaac Newton Institute in Cambridge (October 14-18, 2013, part of the semester-long programme on Mathematical Challenges in Quantum Information Theory)! ☺
Stabilizer information inequalities from phase space distributions
D. Gross; M. Walter
2013
Stabilizer information inequalities from phase space distributions
Type
Online Database
Author
D. Gross; M. Walter
Year
2013
Abstract
The Shannon entropy of a collection of random variables is subject to a number of constraints, the best-known examples being monotonicity and strong subadditivity. It remains an open question to decide which of these"laws of information theory"are also respected by the von Neumann entropy of many-body quantum states. In this note, we consider a toy version of this difficult problem by analyzing the von Neumann entropy of stabilizer states. We find that the von Neumann entropy of stabilizer states satisfies all balanced information inequalities that hold in the classical case. Our argument relies on the fact any stabilizer states has a classical model, provided by the discrete Wigner function: The phase-space entropy of the Wigner function corresponds directly to the von Neumann entropy of the state, which allows us to reduce to the classical case. Our result has a natural counterpart for multi-mode Gaussian states, which sheds some light on the general properties of the construction.
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.
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.
When is a pure state of three qubits determined by its single-particle reduced density matrices?
A. Sawicki; M. Walter; M. Kus
2012
When is a pure state of three qubits determined by its single-particle reduced density matrices?
Type
Online Database
Author
A. Sawicki; M. Walter; M. Kus
Year
2012
Abstract
Using techniques from symplectic geometry, we determine when a pure state of three qubits is up to local unitaries uniquely determined by its reduced density matrices. We moreover show that this is always the case if one is given the additional promise that the quantum state is not convertible to the Greenberger-Horne-Zeilinger (GHZ) state by stochastic local operations and classical communication (SLOCC).
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.
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