Two families of bipartite mixed quantum states are studied for which it is proved that the number of members in the optimal-decomposition ensemble—the ensemble realizing the entanglement of formation—is greater than the rank of the mixed state. We find examples for which the number of states in this optimal ensemble can be larger than the rank by an arbitrarily large factor. In one case the proof relies on the fact that the partial transpose of the mixed state has zero eigenvalues; in the other case the result arises from the properties of product bases that are completable only by embedding in a larger Hilbert space.Two families of bipartite mixed quantum states are studied for which it is proved that the number of members in the optimal-decomposition ensemble—the ensemble realizing the entanglement of formation—is greater than the rank of the mixed state. We find examples for which the number of states in this optimal ensemble can be larger than the rank by an arbitrarily large factor. In one case the proof relies on the fact that the partial transpose of the mixed state has zero eigenvalues; in the other case the result arises from the properties of product bases that are completable only by embedding in a larger Hilbert space.

arXiv:quant-ph/9904005Journal of Modern OpticsFeb 1, 200024 citations

Optimal Decompositions of Barely Separable States

David Peter Divincenzo, Barbara M. Terhal, Thapliyal, A V

Two families of bipartite mixed quantum states are studied for which it is proved that the number of members in the optimal-decomposition ensemble --- the ensemble realizing the entanglement of formation --- is greater than the rank of the mixed state. We find examples for which the number of states in this optimal ensemble can be larger than the rank by an arbitrarily large factor. In one case the proof relies on the fact that the partial transpose of the mixed state has zero eigenvalues; in the other case the result arises from the properties of product bases that are completable only by embedding in a larger Hilbert space.Two families of bipartite mixed quantum states are studied for which it is proved that the number of members in the optimal-decomposition ensemble --- the ensemble realizing the entanglement of formation --- is greater than the rank of the mixed state. We find examples for which the number of states in this optimal ensemble can be larger than the rank by an arbitrarily large factor. In one case the proof relies on the fact that the partial transpose of the mixed state has zero eigenvalues; in the other case the result arises from the properties of product bases that are completable only by embedding in a larger Hilbert space.

Apr 2, 199919 citations

Separable decompositions of 2xn states with operator Schmidt rank three

Perrine Vantalon, Nicolas Macris

We prove that every bipartite state $ρ$ of a qubit coupled to an $n$ -level system whose operator Schmidt rank equals three can be written as a mixture of $\text{rank}(ρ)$ pure product states, and as a mixture of at most $n+1$ 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 prove that every bipartite state $ρ$ of a qubit coupled to an $n$-level system whose operator Schmidt rank equals three can be written as a mixture of $\text{rank}(ρ)$ pure product states, and as a mixture of at most $n+1$ 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.

quant-pharXiv:2609.39669Oct 1, 20260 citations

Optimal Ensemble Length of Mixed Separable States

Robert B. Lockhart

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. 1The 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

Aug 17, 199914 citations

Optimal ensemble length of mixed separable states

Robert Lockhart

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.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.

arXiv:quant-ph/9908050Journal of Mathematical PhysicsOct 1, 200018 citations

Entanglement of Formation of an Arbitrary State of Two Qubits

William K. Wootters

The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state $\ensuremath{\rho}$ is the minimum average entanglement of an ensemble of pure states that represents \ensuremath{\rho}. An earlier paper conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula for special states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state $\ensuremath{\rho}$ is the minimum average entanglement of an ensemble of pure states that represents \ensuremath{\rho}. An earlier paper conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula for special states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.

arXiv:quant-ph/9709029Physical Review LettersMar 9, 19988,283 citations

We prove that every bipartite state

ρ

of a qubit coupled to an

n

-level system whose operator Schmidt rank equals three can be written as a mixture of

\text{rank}(ρ)

pure product states, and as a mixture of at most

n+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.