June 21, 2021

July 27-31, 2009
Workshop on Quantum Marginals and Density Matrices

Abstracts - Contributed Talks/ Posters


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


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)


The Gaussian quantum marginal problem
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)


Dissipative quantum theory: implications for quantum entanglement
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.


Symmetric extension of two-qubit states
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


Current status and open problems in the RDM method
Maho Nakata

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.


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


Asymptotic decomposition of restricted representations: algorithms and examples
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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