arXiv subjects

Michael Walter

Publications and source records attributed to Michael Walter.

118 records · Page 7Linked to original sources

arXiv2014

Lower Bounds for Quantum Parameter Estimation

The laws of quantum mechanics place fundamental limits on the accuracy of measurements and therefore on the estimation of unknown parameters of a quantum system. In this work, we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success. Our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test. In group-covariant scenarios, we find that there is an ultimate bound for any estimation scheme which depends only on the representation-theoretic data of the probe system, and we evaluate its asymptotics in the limit of many systems, establishing a general "Heisenberg limit" for region estimation. We apply our results to several examples, in particular to phase estimation, where our bounds allow us to recover the well-known Heisenberg and shot-noise scaling.

quant-ph↗

arXiv2014

Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information

Entangled many-body states are an essential resource for quantum computing and interferometry. Determining the type of entanglement present in a system usually requires access to an exponential number of parameters. We show that in the case of pure multi-particle quantum states, features of the global entanglement can already be extracted from local information alone. This is achieved by associating with any given class of entanglement an entanglement polytope---a geometric object which characterizes the single-particle states compatible with that class. Our results, applicable to systems of arbitrary size and statistics, give rise to local witnesses for global pure-state entanglement, and can be generalized to states affected by low levels of noise.

quant-ph↗

arXiv2014

Multipartite Quantum States and their Marginals

Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions -- such as Pauli's exclusion principle or the monogamy of quantum entanglement -- which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic and rigorous study of the general relation between multipartite quantum states, i.e., states of quantum systems that are composed of several subsystems, and their marginals. In the first part, we focus on the one-body marginals of multipartite quantum states; in the second part, we study general quantum marginals from the perspective of entropy.

quant-ph↗

arXiv2014

Eigenvalue Distributions of Reduced Density Matrices

Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution's support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to 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 the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

quant-ph↗

arXiv2013

Stabilizer information inequalities from phase space distributions

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 article, 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 is built on the fact that stabilizer states have 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. We also discuss the relation of our results to recent work by Linden, Ruskai, and Winter.

quant-ph↗

arXiv2012

When is a pure state of three qubits determined by its single-particle reduced density matrices?

Using techniques from symplectic geometry, we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one-particle reduced density matrices exactly when their ordered spectra belong to the boundary of the, so called, Kirwan polytope. Otherwise, the states with given reduced density matrices are parameterized, up to local unitary equivalence, by two real variables. Given inevitable experimental imprecisions, this means that already for three qubits a pure quantum state can never be reconstructed from single-particle tomography. We moreover show that knowledge of the reduced density matrices is always sufficient 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), and discuss generalizations of our results to an arbitary number of qubits.

quant-ph↗

arXiv2012

Computing Multiplicities of Lie Group Representations

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.

cs.CC↗

arXiv2011

Au40: A Large Tetrahedral Magic Cluster

40 is a magic number for tetrahedral symmetry predicted in both nuclear physics and the electronic jellium model. We show that Au40 could be such a a magic cluster from density functional theory-based basin hopping for global minimization. The putative global minimum found for Au40 has a twisted pyramid structure, reminiscent of the famous tetrahedral Au20, and a sizable HOMO-LUMO gap of 0.69 eV, indicating its molecular nature. Analysis of the electronic states reveals that the gap is related to shell closings of the metallic electrons in a tetrahedrally distorted effective potential.

cond-mat.mes-hall↗

arXiv2010

Equivariant geometric K-homology for compact Lie group actions

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphism.

math.KT↗

arXiv2005

Optical absorption by magnesia-supported gold clusters and nanocatalysts: effects from the support, cluster and adsorbants

Polarization-resolved optical spectra of magnesia-supported gold clusters Au$_N$/MgO (N=1,2,4,8), bound at a surface color center $F_s$ of the MgO(100) face, are calculated from the time-dependent density functional theory. The optical lines for n=1,2 are dominated by transitions that involve strong hybridization between gold and $F_s$ states whereas for n=4,8 intracluster transitions dominate. The theoretical optical spectra are sensitive to cluster structure and adsorbants (here CO and O$_2$ molecules on Au$_8$/MgO) which suggests polarization-resolved optical spectroscopy as a powerful tool to investigate structures and functions of chemically active, supported clusters.

physics.atm-clus↗