A number of important results in quantum information theory can be connected quite elegantly to the representation theory of the symmetric group through a quantum analogue of the classical information-theoretic "method of types" that arises naturally from the Schur-Weyl duality.We will give a brief introduction to this connection and briefly discuss some of the results that follow from it, such as quantum source compression rates, entanglement concentration rates, quantum entropy inequalities, and the admissisble spectra of partial density matrices from pure, multipartite entangled states.
Abstract The possible spectra of one-particle reduced density matrices that are compatible with a pure multipartite quantum system of finite dimension form a convex polytope. We introduce a new construction of inner- and outer-bounding polytopes that constrain the polytope for the entire quantum system. The outer bound is sharp. The inner polytope stems only from doubly excited states. We find all quantum systems, where the bounds coincide giving the entire polytope. We show, that those systems are: (i) any system of two particles (ii) L qubits, (iii) three fermions on N ⩽ 7 levels, (iv) any number of bosons on any number of levels and (v) fermionic Fock space on N ⩽ 5 levels. The methods we use come from symplectic geometry and representation theory of compact Lie groups. In particular, we study the images of proper momentum maps, where our method describes momentum images for all representations that are spherical.
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.
The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrödinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).
A bipartite state ${\ensuremath{\rho}}^{AB}$ has a
k-symmetric extension if there exists a (
k+1)-partite state ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ with marginals ${\ensuremath{\rho}}^{A{B}_{i}}={\ensuremath{\rho}}^{AB},\ensuremath{\forall}i$. The
k-symmetric extension is called bosonic if ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ is supported on the symmetric subspace of
B1B2...Bk. Understanding the structure of symmetric and bosonic extension has various applications in the theory of quantum entanglement, quantum key distribution, and the quantum marginal problem. In particular, bosonic extension gives a tighter bound for the quantum marginal problem based on separability. In general, it is known that a ${\ensuremath{\rho}}^{AB}$ admitting symmetric extension may not have bosonic extension. In this work, we show that, when the dimension of the subsystem
B is 2 (i.e., a qubit), ${\ensuremath{\rho}}^{AB}$ admits a
k-symmetric extension if and only if it has a
k-bosonic extension. Our result has an immediate application to the quantum marginal problem and indicates a special structure for qubit systems based on group representation theory.