### Abstracts - Contributed Talks/ Posters

###

*Chemical implications of variational second-order density
matrix theory: study of diatomic molecules along the potential energy
curve*

by

Helen van Aggelen

Ghent University

Coauthors: Brecht Verstichel, Paul Ayers, Patrick Bultinck, David
Cooper, Dimitri Van Neck

A semi-definite program was developed for variational optimization
of the second-order density matrix (DM2). It includes the usual
two-index N-representability conditions (the P-, Q- and G-condition
[1][2]), as well as conditions on spin. The variational DM2 method
is evaluated by calculating the dissociation process of a series
of 14-electron diatomic molecules, including N2, O22+, NO+, CN-
and CO.

This research focuses on the chemical properties of the optimized
DM2. Does this method give a consistent picture of a system's chemical
properties? The presented results bring serious chemical flaws to
the attention. Heteronuclear diatomics such as NO+ and CN- dissociate
into fractionally charged atoms, yielding dramatically incorrect
energies, dipole moments and atomic populations [3].

This problem is not solved, however, by adding the more stringent
conditions known as T1 and T2 [4]. A novel constraint is presented
which imposes the correct dissociation and enforces size-consistency
with little additional computational effort.

References:

1. J. Coleman, Rev.Mod.Phys. 35, 668 (1963)

2. C. Garrod and J. K. Percus, J.Math. Phys. 5 (12), 1756 (1964)

3. H. van Aggelen, P. Bultinck, B. Verstichel, D. Van Neck, and
P.W. Ayers, PCCP, DOI:10.1039/B907624G (2009)

4. R. M. Erdahl, Int. J. Quantum Chem. 13 (6), 697 (1978)

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*The Gaussian quantum marginal problem*

by

J. Eisert

University of Potsdam

Coauthors: T. Rudolph, T. Tyc, B. Sanders

If I have time to do it :), I will also bring the following poster/informal
talk:

The quantum marginal problem asks what local spectra are consistent
with a given spectrum of a joint state of a composite quantum system.
This setting, also referred to as the question of the compatibility
of local spectra, has several applications in quantum information
theory. Here, we introduce the analogue of this statement for Gaussian
states for any number of modes, and solve it in generality, for
pure and mixed states, both concerning necessary and sufficient
conditions. Formally, our result can be viewed as the symplectic
analogue of the Sing-Thompson Theorem (respectively Horn's Lemma),
characterizing the relationship between main diagonal elements and
singular values of a complex matrix: We find necessary and sufficient
conditions for vectors (d1, ..., dn) and (c1, ...,cn) to be the
symplectic eigenvalues and symplectic main diagonal elements of
a strictly positive real matrix, respectively.

More physically speaking, this result determines what local temperatures
or entropies are consistent with a pure or mixed Gaussian state
of several modes. We find that this result implies a solution to
the problem of sharing of entanglement in pure Gaussian states and
allows for estimating the global entropy of non-Gaussian states
based on local measurements. Implications to the actual preparation
of multi-mode continuous-variable entangled states are discussed.
We compare the findings with the marginal problem for qubits, the
solution of which for pure states has a strikingly similar and in
fact simple form.

Commun. Math. Phys. 280, 263 (2008)

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*Dissipative quantum theory: implications for quantum entanglement*

by

lohandjola l'okaso

university of kinshasa

Coauthors: mbiye kalumbu

Three inter-related topics are discussed here. (1) the Lindblad
dynamics of quantum dissipative systems; (2) quantum entanglement
in composite systems and its quantification based on the Tsallis
entropy; and (3) robustness of entanglement under dissipation. After
a brief review of the Lindblad theory of quantum dissipative systems
and the idea of quantum entanglement in composite quantum systems
illustrated by describing the three particle systems, the behavior
of entanglement under the influence of dissipative processes is
discussed. These issues are of importance in the discussion of quantum
nanometric systems of current research.

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*Symmetric extension of two-qubit states*

by

Geir Ove Myhr

University of Waterloo

Coauthors: Norbert Lütkenhaus (University of Waterloo)

We present analytical results for what is arguably the simplest
nontrivial quantum marginal problem: when does a two-qubit state
have a (1, 2)-symmetric extension? We can formulate this as an equivalent
marginal problem: given a bipartite state, when can one find a tripartite
state on ABC such that the marginal states on AB and AC is the given
state? The results can be summarized as follows:

Any state with a symmetric extension can be decomposed into a convex
sum of states with a pure symmetric extension (for all dimensions).

Any state with a pure symmetric extension has the same spectrum
as its marginal on B (for all dimensions).

Any state with the same spectrum as its marginal on B has a pure
symmetric extension (only true for two qubits).

We give a conjectured necessary and sufficient condition for symmetric
extendibility (only valid for two qubits).

The conjecture only depends on the purity and determinant of the
state and the purity of the marginal state on B.

We prove the conjecture in some special cases.

The conditions for symmetric extensions imply conditions for degradable
and antidegradable channels.

Phys. Rev. A 79, 062307 (2009); arXiv:0812.3667

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*Current status and open problems in the RDM method*

by

Maho Nakata

RIKEN

The RDM method; using 2-RDM as the basic variable and compute (minimize)
the ground state energy subjected to approximate (necessary) N-representability
condition is promising. Nevertheless, there are some serious problems
in conditions and solvers. We'd like to summarize what is the current
status and what are the problems.

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*Computational Complexity of interacting electrons and fundamental
limitations of Density Functional Theory*

by

Norbert Schuch

Max-Planck-Institute for Quantum Optics

Coauthors: Frank Verstraete

One of the central problems in quantum mechanics is to determine
the ground state properties of a system of electrons interacting
via the Coulomb potential. Since its introduction by Hohenberg,
Kohn, and Sham, Density Functional Theory (DFT) has become the most
widely used and successful method for simulating systems of interacting
electrons, making their original work one of the most cited in physics.
In this letter, we show that the field of computational complexity
imposes fundamental limitations on DFT, as an efficient description
of the associated universal functional would allow to solve any
problem in the class QMA (the quantum version of NP) and thus particularly
any problem in NP in polynomial time. This follows from the fact
that finding the ground state energy of the Hubbard model in an
external magnetic field is a hard problem even for a quantum computer,
while given the universal functional it can be computed efficiently
using DFT. Since a universal functional can be defined efficiently
using two-electron reduced states, this also shows that solving
the marginal problem for those states is QMA-hard.

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*Asymptotic decomposition of restricted representations: algorithms
and examples*

by

Robert Zeier

Technische Universitaet Muenchen

We compute convex polytopes related to the asymptotic decomposition
of restricted representations. We describe the approach of Berenstein
and Sjamaar. We present algorithms to solve important subproblems
for moderate dimensions. For example, we show how to compute the
relative Weyl set. We discuss computational challenges for higher
dimensions. We present examples in the case of two and three qubits,
i.e., for SU(4) and SU(8).

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