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arXiv:1602.05278 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Length filtration of the separable states

Lin Chen, Dragomir Ž. Djoković

Published Nov 1, 2016• 2 citations
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Abstract

We investigate the separable states ρ of an arbitrary multi-partite quantum system with Hilbert space H of dimension d . The length L ( ρ ) of ρ is defined as the smallest number of pure product states having ρ as their mixture. The length filtration of the set of separable states, S , is the increasing chain ∅ ⊊ S 1 ′ ⊆ S 2 ′ ⊆ ⋯ , where S i ′ = { ρ ∈ S : L ( ρ ) ≤ i } . We define the maximum length, L max = max ρ ∈ S L ( ρ ) , critical length, L crit , and yet another special length, L c , which was defined by a simple formula in one of our previous papers. The critical length indicates the first term in the length filtration whose dimension is equal to Dim S . We show that in general d ≤ L c ≤ L crit ≤ L max ≤ d 2 . We conjecture that the equality L crit = L c holds for all finite-dimensional multi-partite quantum systems. Our main result is that L crit = L c for the bipartite systems having a single qubit as one of the parties. This is accomplished by computing the rank of the Jacobian matrix of a suitable map having S as its range.

DOI
10.1098/rspa.2016.0350

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We investigate the separable states ρ of an arbitrary multi-partite quantum system with Hilbert space H of dimension d . The length L ( ρ ) of ρ is defined as the smallest number of pure product states having ρ as their mixture. The length filtration of the set of separable states, S , is the increasing chain ∅ ⊊ S 1 ′ ⊆ S 2 ′ ⊆ ⋯ , where S i ′ = { ρ ∈ S : L ( ρ ) ≤ i } . We define the maximum length, L max = max ρ ∈ S L ( ρ ) , critical length, L crit , and yet another special length, L c , which was defined by a simple formula in one of our previous papers. The critical length indicates the first term in the length filtration whose dimension is equal to Dim S . We show that in general d ≤ L c ≤ L crit ≤ L max ≤ d 2 . We conjecture that the equality L crit = L c holds for all finite-dimensional multi-partite quantum systems. Our main result is that L crit = L c for the bipartite systems having a single qubit as one of the parties. This is accomplished by computing the rank of the Jacobian matrix of a suitable map having S as its range.
The optimal (pure state) ensemble length of a separable state, A, is the minimum number of (pure) product states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest, the lower bounds are sharp. A consequence of our results is that for all two particle systems, except possibly those with a qubit and those with a nine dimensional Hilbert space, and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state’s rank. In bipartite systems with each particle having the same Hilbert space, with probablity 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system with p qubits with probability 1 it is greater than 2 2p /(1 + 2p), which is almost the maximal rank squared. 1
The optimal (pure state) ensemble length of a separable state A is the minimum number of (pure) states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest the lower bounds are sharp. A consequence of our results is that for all two-particle systems, except possibly those with one qubit or those with a nine-dimensional Hilbert space, and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state’s rank. In bipartite systems with probabilty 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system of p qubits with probability 1 it is greater than 22p/(1+2p), i.e., almost the square of the rank.
Abstract We show that around any m-partite product state ρ p r o d = ρ 1 ⊗ … ⊗ ρ m of full rank (that is d e t ( ρ p r o d ) ≠ 0 ) , there exists a finite-sized closed ball of separable states centered around ρ p r o d whose radius is β := 2 1 − m / 2 λ m i n ( ρ p r o d ) . Here, λ m i n ( ρ p r o d ) is the smallest eigenvalue of ρ p r o d . We are assuming that the total Hilbert space is finite dimensional and we use the notion of distance induced by the Frobenius norm. Applying a scaling relation, we also give a new and simple sufficient criterion for multipartite separability based on trace: T r [ ρ ρ p r o d ] 2 / T r
We prove that every bipartite state ρρρ of a qubit coupled to an nnn-level system whose operator Schmidt rank equals three can be written as a mixture of rank(ρ)\text{rank}(ρ)rank(ρ) pure product states, and as a mixture of at most n+1n+1n+1 mixed product states in general. The second result is tight in the sense that there exist states that cannot be decomposed with a smaller number of terms. Our proof gives constructive methods to obtain the decompositions. It involves eigendecompositions of suitable unitary dilations of an operator explicitly determined by the state. Our framework recovers previously known results for the length of decompositions into pure product states and is also able to deal with decompositions into mixed states. For pure product states, our decomposition is different from existing ones.
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.
Motivated by the separability problem in quantum systems 2⊗4, 3⊗3 and 2⊗2⊗2, we study the maximal (proper) faces of the convex body, S 1 , of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space H = H 1 ⊗ H 2 ⊗ ⋯ ⊗ H n . To any subspace V ⊆ H , we associate a face F V of S 1 consisting of all states ρ ∈ S 1 whose range is contained in V . We prove that F V is a maximal face if and only if V is a hyperplane. If V =| ψ 〉 ⊥ , where | ψ 〉 is a product vector, we prove that Dim F V = d 2 − 1 − ∏ ( 2 d i − 1 ) , where d i = Dim H i and d = ∏ d i . We classify the maximal faces of S 1 in the cases 2⊗2 and 2⊗3. In particular, we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for 2⊗2, and 20 and 24 for 2⊗3. The boundary, ∂ S 1 , of S 1 is the union of all maximal faces. When d >6, it is easy to show that there exist full states on ∂ S 1