A Necessary and Sufficient Criterion for the Separability of Quantum State

Quantum entanglement has been regarded as one of the key physical resources in quantum information sciences. However, the determination of whether a mixed state is entangled or not is generally a hard issue, even for the bipartite system. In this work we propose an operational necessary and sufficient criterion for the separability of an arbitrary bipartite mixed state, by virtue of the multiplicative Horn’s problem. The work follows the work initiated by Horodecki et al. and uses the Bloch vector representation introduced to the separability problem by J. De Vicente. In our criterion, a complete and finite set of inequalities to determine the separability of compound system is obtained, which may be viewed as trade-off relations between the quantumness of subsystems. We apply the obtained result to explicit examples, e.g. the separable decomposition of arbitrary dimension Werner state and isotropic state.

Entanglement is a ubiquitous feature of quantum system and key element in quantum information processing, whereas yet far from fully understood 1 . A fundamental problem in the study of entanglement is the determination of the separability of quantum states. For pure state, the entangled states are those that cannot be expressed as the product of the subsystems, e.g. we say a bipartite pure state of A and B is entangled if it cannot be expressed in the product form like For the mixed state of a compound system, we say it is entangled if it cannot be written as a convex combination of product states. For example, a bipartite mixed state is separable (i.e. classically correlated 2 ) whenever it can be expressed as ( , ) are local density matrices of particles A and B; p i > 0 and p 1 The entanglement (non-separability) criterion for pure state is clear, by virtue of Schmidt or high order singular value decomposition for any-number-partite system 3 . However, none of the existing criteria for the separability of finite dimensional mixed states are satisfactory by far. They are generally either sufficient and necessary, but not practically usable; or easy to use, but only necessary (or only sufficient) 4 .
Over the past decades, one remarkable progress towards the operational characterization of a separable mixed state, the positive partial transposition (PPT) criterion 5 , was achieved by Peres twenty years ago. This separability criterion is only necessary and sufficient for 2 × 2 and 2 × 3 systems, rather than for arbitrary higher dimensional systems 6 . Though couple of necessary and sufficient criteria were developed afterwards [6][7][8] , they are generally difficult to handle in practice, or only applicable to low-rank density matrices 9 . With the dimension growing, the separability problem of a compound system tends to be NP-hard, even in the bipartite situation 10 . Recent investigations mostly focus on the necessary or sufficient conditions of witnessing entanglement or separability. The computable cross-norm or realignment (CCNR) criterion 11,12 and local uncertainty relations (LURs) 13 are proposed to detect entanglement. By virtue of the Bloch representation, separability criterion had been successfully formulated in matrix norms, which was found to be related to the CCNR criterion 14 . The optimization of entanglement witness observables may stand as a separability criterion 15 . For recent development, readers may refer to refs [16][17][18] and more comprehensive reviews 19,20 . It should be noted that even restricted to necessary or sufficient criterion, the corresponding inequalities tend to be an ever growing family. Therefore, to find a complete and finite set of inequalities to determine the separability of mixed states is theoretically important and practically necessary.
In this work, we present an applicable criterion for the separability of bipartite mixed state through exploring the multiplicative Horn's problem 21 . By expressing the quantum state in Bloch representation, the problem of factorizing a mixed state into sum of product states is transformed to the task of decomposing a matrix into the product of two other matrices. We find that the solution to the multiplicative Horn's problem yields a complete and finite set of inequalities, a new criterion, which in practice provides a systematic procedure for the decomposition of separable mixed states. Relations between this new criterion and other related ones are investigated through concrete examples, including the separable decomposition of arbitrary dimensional Werner and isotropic states. Results manifest that the criterion raised in this work is to our knowledge the most applicable one at present in determining the separability of entangled quantum states.

Results
The Bloch representation of quantum state. A quantum state in N-dimensional Hilbert space may be expressed as 22 , and comparing with Eq. (4), the necessary and sufficient condition of separability then turns to where the subscripts in r i → , s j → label different Bloch vectors rather the components of them, and the correlation matrix  has the matrix elements of  µν . The reduced density matrices of A and B can be derived from Eq. (4) by partial trace Here the local ranks are rank(ρ A ) = n and rank(ρ B ) = m, where n and m may be non-full local ranks of n < N and m < M. We then have the following observation (see 25  According to Observation 1, we need only consider the separability of mixed states with full local ranks. The full local rank state could be transformed to a normal form with maximally mixed subsystems 26 Note that in the literature there are studies about the normal form in the separability problem 27,28 . Hereafter, the quantum states ρ AB are implied to be in their normal form, and we have

The state ρ AB is separable if and only if there exist a number L such that
all the singular values are arranged in descending order.

Proof:
The if part is quite streightfoward For the only if part, suppose the singular value decompositions of M rp and M sp are Before proceeding to the Eq. (10), two prerequisite lemmas are necessary. Let I, J, K be certain subsets of natural numbers {1, …, n} with the same cardinality r, i.e., I = {i 1 , i 2 , …, i r }, J = {j 1 , j 2 , …, j r }, and K = {k 1 , k 2 , …, k r }, where the elements are arranged in increasing order so that i i i This lemma appears as Theorem 2 of ref. 29 , where the practical methods on how to generate T r n were also discussed, i.e., via the Horn's inductive procedure or Littlewood-Richardson coefficients.
for all (I, J, K) in T r n and all r < n. This is known as the multiplicative Horn's problem, see theorem 16 of ref. 29 for details. Historically, the multiplicative Horm's problem first appeared as the Thompson's conjecture 30 , and later was found can be solved for invertible matrices 31 . It was found to be true for real matrices 32 , and even extendable to the case of non-invertible matrices recently 21 (see Supplemental Material for a brief review of the proof).
The decomposition of Eq. (10) can be realized through the following theorem: There exists a real orthogonal matrix Q such that D α QD β has the singular values of D τ , if and only if the following is satisfied In the following we demonstrate our method in bipartite quantum system as an example. For more systematic and detailed applications, readers may refer to ref. 25 . It should be noted that theorems 1 and 2 are also suitable to the bi-separability of arbitrary multipartite states, and hence the method presented here is also applicable to the multi-separability problem, due to the reason that the Bloch representation generally turns the sum decomposition problem into a product decomposition one.
Applications. In Bloch representation of quantum state, we have the following two observations:

Observation 3. If r
→ is a Bloch vector of a density matrix, then the r → ′, whose components r r ′ = −  , then r Pr = →′ → is also a Bloch vector for arbitrary rotation P ∈ SO(N 2 − 1).
Here, the Observation 3 is established due to the fact that the transposition of a positive semidefinite Hermitian matrix keeps on being positive semidefinite, while the Observation 4 is just a corollary of Eq. (11) in ref. 14 (or see ref. 33 ). In the following, we demonstrate how the criterion works through concrete examples. 14

Example I: The relationship between Vicente's criterion
Taking Eq. (22) into Corollary 1, Theorem 1 of ref. 14 is arrived. On the other hand, from Observation 4 we have the following: , the quantum state  is separable.
with rank(T) = l ≥ 1, when working in the bases of → u i and v i → , we may construct the following matrix equation Here, Q ∈ SO(l + 1) with elements in the last row being = + Q p l j j ( 1) ; p j ≥ 0, and ∑ = = + p 1  We may set the probability distribution p j to be p Q j i l i ij Then replacing the p j in Eqs (27,28), we have According to Observation 4, the Corollary 2 is then established. Q.E.D. Corollary 2 agrees with the Proposition 3 of ref. 14

Example II: The relation with PPT 5 scheme
The partial transposition of a bipartite density matrix corresponds to the sign flips of columns or rows (not both) of  , whose indices are that of skew symmetric generators, i.e., λ λ = − µ µ T . The Observation 3 implies that the PPT criterion is necessary for separability. Conversely, the positivity of partially transposed density matrix SCientifiC REPORts | (2018) 8:1442 | DOI:10.1038/s41598-018-19709-z generally does not imply separability, that is, PPT is not sufficient. However, for qubit-qubit system, calculation indicates that the PPT of density operators gives τ ≤ ∑ ≤ 0 1 i i by means of the technique introduced in ref. 35 (see Supplemental Material). As 1 2 2(2 1) ≤ − , according to the Corollary 2, PPT also tells separability. Therefore it is a necessary and sufficient condition for qubit-qubit system, which agrees with the conclusionn proved by other method 6 .

Example III: For generalized Werner state and isotropic state
The relation between the Werner state and the isotropic state. The generalized Werner state and isotropic state in the Bloch vector representation read 14 : where S 1 represents the symmetric generators of λ λ = µ µ T , and S 2 denotes the skew symmetric generators of The partial transposition operation correlates the Werner state with isotropic states. According to Observation 3, we may readily find that the parameters in Eqs (30 and 31) satisfies Equation (32) tells us that, when considering the separability, only one of the two states need to be taken into account. Before proceeding to the separable decomposition, we first present serval straightforward but interesting results from Eq. (32): (1) The positivity condition p

Separable decomposition of the Werner state. Considering the Werner state with
Since the singular values of  are all equal, we may set   1 cos 1 (37) As being true for qubit and numerically verified for qutrit systems, we are tempted to make the following conjecture:   2 2 Inputting (33) and (34) Combining of Eqs (39) and (40) leads to for ρ W has the decomposition of two N 2 -simplexes in the ,  shall be written as , but not both), by the procedure of Eqs (38) to (40), we have  , both → r i and − → r i are Bloch vectors of density matrices.

Discussion
We have presented an applicable and operational necessary and sufficient criterion for the separability of bipartite mixed state. The criterion is exhibited in a finite set of inequalities relating the correlation matrix to the Bloch vectors of the quantum states of subsystems, which is shown to be complete by exploring the multiplicative Horn's problem. These inequalities may be treated as trade-off relations between the quantumness of the constituent parts, balanced by the correlation matrix. A state is separable if the decomposition can be performed within the convex hulls of the Bloch vectors of subsystems. As an illustration, separable decompositions for generalized Werner state and isotropic state are achieved in according to the new scheme. The proposed criterion sets up a geometric boundary in between the separability and entanglement for compound system, and provides a new perspective on the nonlocal nature of entanglement.