Locally Maximally Entangled States of Multipart Quantum Systems

Abstract

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to [BRV], which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions (d 1 , d 2 , . . . , d n ), and computes the dimension of the space S LM E /K of LME states up to local unitary transformations for all non-empty cases. Here we provide a pedagogical overview and physical interpretation of the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions (2, A, B) and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results have direct implications for the problem of characterizing SLOCC equivalence classes of quantum states, since points in S LM E /K correspond to natural families of SLOCC classes. Finally, we give the dimension of the stabilizer subgroup S ⊂ SL(d 1 , C) × · · · × SL(d n , C) for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial. sam.leutheusser@alumni.ubc.ca, jbryan@math.ubc.ca, reichst@math.ubc.ca, mav@phas.ubc.ca

Transformations among Pure Multipartite Entangled States via Local Operations are Almost Never Possible

“…Criteria for when a system with Hilbert space H n contains critical states were found in [33,34]. If Crit(H n ) is not empty the union of all orbits (in G) containing a critical state, i.e.…”

“…As mentioned before, Lemma 9 holds for arbitrary qudit-systems. Note further that criteria for when the stabilizer in G of a generic state is trivial were found in [34]. However, in order to obtain the strong implications in entanglement theory (see Sec.…”

A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems

“…Hence, if [Crit] = ∅, the criterion above solves the SLOCC-equivalence problem for generic states. The necessary and sufficient condition for the existence of a critical state has been presented in [32]. For arbitrary dimensional Hilbert spaces, a critical state 9 Note however, that some G-orbits in the null-cone can be distinguished by studying the gradient flow of the function − [38,[40][41][42]45].…”

“…However, its dimension does not coincide with the dimension of G since it admits continuous symmetries and its stabilizer has a 11 The reason for that is that the dimension of an open and dense set is the same as the dimension of the whole space. positive dimension [32] 12 . However, we have that dim G[GHZ] = dim P(H), as the GHZ-state is the only critical state for three qubits.…”

Maximal lexicographic spectra and ranks for states with fixed uniform margins

“…When n ∤ m it is interesting to give the construction of states with ranks from ⌈ m n ⌉ to mn. Recently, in [2] the authors discussed the construction of locally maximally entangled state of multipart quantum systems. By their results, we can decide whether there exist states with spectra ( 1 k , 1 k , ..., 1 k ) in C( 1 n I n , 1 m I m ) where 1 ≤ k ≤ mn.…”