In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension $2\otimes m\otimes n$ are SLOCC equivalent. In this note, we point out that a straightforward modification of the algorithm gives a simple enumeration of all SLOCC equivalence classes in the same space, with the class representatives expressed in the Kronecker canonical normal form of matrix pencils. Thus, two states are equivalent if and only if they have the same canonical form. As an example, we present representatives in canonical form for each of the 26 equivalence classes in $2\otimes 3\otimes n$ systems.In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension $2\otimes m\otimes n$ are SLOCC equivalent. In this note, we point out that a straightforward modification of the algorithm gives a simple enumeration of all SLOCC equivalence classes in the same space, with the class representatives expressed in the Kronecker canonical normal form of matrix pencils. Thus, two states are equivalent if and only if they have the same canonical form. As an example, we present representatives in canonical form for each of the 26 equivalence classes in $2\otimes 3\otimes n$ systems.

arXiv:0911.4058Nov 20, 20092 citations

Remarks on a quantum state extension problem

Reinhard F. Werner

Letters in Mathematical PhysicsMay 1, 199018 citations

Convexity of momentum map, Morse index, and quantum entanglement

Adam Sawicki, Michał Oszmaniec, Marek Kuś

We analyze from the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry, we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many "asymptotically" equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well-defined measure of entanglement) of the state Var [v] attains maximum. We provide an algorithm for finding critical sets of Var [v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using some developments in the momentum map geometry known as the Kirwan–Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.We analyze from the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry, we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many "asymptotically" equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well-defined measure of entanglement) of the state Var [v] attains maximum. We provide an algorithm for finding critical sets of Var [v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using some developments in the momentum map geometry known as the Kirwan–Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.

arXiv:1208.0556Reviews in Mathematical PhysicsMar 11, 201454 citations

A new class of quantum states: Examples and applications

Luigi Accardi

Springer eBooksApr 7, 20082 citations

Quasifree and Classical Quantum States

WORLD SCIENTIFIC eBooksApr 27, 20150 citations

In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension

2\otimes m\otimes n

are SLOCC equivalent. In this note, we point out that a straightforward modification of the algorithm gives a simple enumeration of all SLOCC equivalence classes in the same space, with the class representatives expressed in the Kronecker canonical normal form of matrix pencils. Thus, two states are equivalent if and only if they have the same canonical form. As an example, we present representatives in canonical form for each of the 26 equivalence classes in

2\otimes 3\otimes n

systems.