Is Absolute Separability Determined by the Partial Transpose?

Is Absolute Separability Determined by the Partial Transpose?

Srinivasan Arunachalam,111Centrum Wiskunde & Informatica (CWI) Amsterdam, The Netherlands  Nathaniel Johnston,222Institute for Quantum Computing and Department of Combinatorics & Optimization, University of Waterloo  and Vincent Russo333Institute for Quantum Computing and David R. Cheriton School of Computer Science, University of Waterloo
January 22, 2015
Abstract

The absolute separability problem asks for a characterization of the quantum states with the property that is separable for all unitary matrices . We investigate whether or not it is the case that is absolutely separable if and only if has positive partial transpose for all unitary matrices . In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer–Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.

1 Introduction

In quantum information theory, a quantum state (where denotes the space of complex matrices) is called separable [1] if there exist constants and states and such that and

Finding methods for determining whether a given quantum state is separable or entangled (i.e., not separable) is one of the most active areas of quantum information theory research [2, 3]. Although this problem is believed to be difficult in general [4, 5], many partial results are known. For example, the positive-partial-transpose (PPT) criterion states that if is separable then , where indicates positive semidefiniteness, is the identity map, and is the transpose map [6]. However, the converse of the PPT criterion only holds when [7, 8], so additional tests for separability are required in higher dimensions.

The most natural generalization of the PPT criterion says that a state is separable if and only if is positive semidefinite for all positive maps [9]. Thus each fixed positive gives a necessary condition for separability.

The absolute separability problem [10] (sometimes called the separability from spectrum problem [11]) asks for a characterization of the states with the property that is separable for all unitary matrices , which is equivalent to asking which sets of real numbers are such that every state with eigenvalues is separable. This question was first answered in the case in [12], where it was shown that is absolutely separable if and only if its eigenvalues satisfy , however the problem remains open in general.

One motivation for this problem comes from the fact that it is sometimes easier to determine the eigenvalues of a quantum state than it is to determine the entire structure of that state [13, 14]. Thus, the absolute separability problem asks for the strongest separability test that can be devised given this restricted information. In another direction, the exact largest size of a ball of separable states centered at the maximally-mixed state is known [15], and it is not difficult to show that every state within this ball is absolutely separable. However, there are also absolutely separable states outside of this ball, and it would be nice to have a characterization of where they are. Alternatively, we can think of states that are not absolutely separable as those that can be used to generate entanglement when the operations at our disposal are global unitary channels [16].

One approach to characterizing the states that are absolutely separable would be to instead fix some necessary test for separability and determine the set of states with the property that satisfies that separability test for all unitary matrices . This approach was initiated in [17], where the set of states that are absolutely PPT (i.e., states such that is positive semidefinite for all unitary ) were completely characterized. Similarly, the very recent paper [18] investigated states with the property that satisfies the reduction criterion for all unitary matrices (because the reduction criterion is weaker than the partial transpose criterion, we do not explicitly consider it in the present paper). We continue this work by considering the same problem for several other separability criteria.

It was shown in [19] that the set of absolutely PPT states coincides with the set of absolutely separable states when and is arbitrary, despite the fact that the set of PPT states is strictly larger than the set of separable states when and . The question was then asked whether or not the set of absolutely PPT states and absolutely separable states coincide when . In the present paper, we demonstrate that several standard methods of entanglement detection are unable to answer this question.

More specifically, we introduce a general method (Lemma 2) that can be used to show that a given entanglement witness or positive map cannot detect entanglement in any absolutely PPT state. Using this method, we prove several results of the form “if is absolutely PPT, then it is also absolutely <other separability criterion>”. For example, we show that every absolutely PPT is also “absolutely realignable”—i.e., always satisfies the realignment criterion introduced in [20, 21], even though there are PPT states that violate the realignment criterion. This difference between the usual separability problem and the absolute separability problem is illustrated in Figure 1. We also prove that the absolute separability and absolute PPT properties coincide when restricted to the well-known family of isotropic states.

Figure 1: The figure on the left represents the relationship between the set of separable states , the set of PPT states , and the set of states that satisfy the realignment criterion . The figure on the right represents the relationship between the set of absolutely PPT states and the set of states that are “absolutely realignable” (see Theorem 1).

2 Preliminaries

The proofs of our results rely heavily on semidefinite programming. Given Hermitian matrices and and a Hermiticity-preserving linear map (i.e., a map such that for all ), the semidefinite program associated with the triple is the following pair of optimization problems:

where is the dual map of defined by for all and . Semidefinite programs can be efficiently solved [22], and furthermore weak duality always holds, which tells us that for all feasible points and . In particular, this means that we can get upper bounds on the optimal value of the primal problem by simply finding a single feasible point for the dual problem (and similarly, feasible points of the primal problem give lower bounds on the optimal value of the dual problem). For a more thorough introduction to semidefinite programming, see [23, 24].

Given a linear map , we recall that its Choi matrix is the operator

where is the standard maximally-entangled pure state. It is well-known that is completely positive (i.e., satisfies whenever ) if and only if is positive semidefinite [25].

Our proofs will also be heavily reliant on the notion of entanglement witnesses, which are Hermitian operators with the property that for all separable , but for some (necessarily entangled) . Here we say that detects the entanglement in , and we note that every entangled is detected by some entanglement witness . Finally, we will also make frequent use of the family of Schatten -norms, defined for by

where we define , , and (and in these special cases, these norms are often called the trace norm, Frobenius norm, and operator norm, respectively).

The remainder of this article is organized as follows. In Section 3, we briefly review the characterization of states that are absolutely PPT that was originally derived in [17]. We then formally present the question in which we are interested in Section 4, and briefly discuss the implications of an answer to this question. The next sections are dedicated to showing that several well-known separability criteria are unable to detect entanglement in any absolutely PPT state, and are thus unable to answer the our question. In Section 6, we show that for specific classes of states that absolute separability and absolute PPT coincide. Finally, in Section 7, we conclude and list a number of open problems and directions for future research.

3 Absolute Positive Partial Transpose

We now briefly recall some of the key points of the characterization of absolutely PPT states given in [17]. Indeed, the main result of that paper shows that, for each , there exists a finite family of linear matrix inequalities (LMIs) with the property that is absolutely PPT if and only if its eigenvalues satisfy each of the LMIs.

In the case, the LMI that determines absolute PPT is

which is easily seen to be equivalent to the previously-discussed inequalities when and when .

In the case, there are two LMIs that determine absolute PPT:

(1)

That is, is absolutely PPT if and only if its eigenvalues satisfy both of the positive semidefiniteness conditions (1).

In general, once we have fixed we use to denote the matrices of eigenvalues whose positive semidefiniteness determine absolute PPT, and these matrices always look quite similar to the matrices (1) from the case. For example, each is of size , the diagonal entry of each is times one of the ’s, and each off-diagonal entry is the difference of two of the ’s. Furthermore, the top-left sub-matrix of is always of the form

(2)

so positive semidefiniteness of (2) is a necessary (but not sufficient when ) condition for to be absolutely PPT.

We note that the number of ’s that must be checked to be positive semidefinite grows exponentially in (for example, when the number of ’s is [26]), and their exact construction is slightly complicated. However, it is not important for our purposes to be familiar with their exact construction—the properties of these matrices that we presented above are all we need.

We now present, without proof, a lemma that is well-known in matrix analysis (see, for example, [27, Problem III.6.14]).

Lemma 1.

Let be Hermitian matrices with eigenvalues and , respectively. Then

We can make use of Lemma 1 to see that semidefinite programming can be used to determine whether or not a given entanglement witness is capable of detecting entanglement in an absolutely PPT state. In particular, if we have an entanglement witness with eigenvalues then can detect the entanglement in some absolutely PPT state if and only if the optimal value of the following semidefinite program is strictly less than zero:

(3)

Indeed, the constraints in the SDP (3) are simply enforcing the fact that are the eigenvalues of some absolutely PPT state. It then follows from Lemma 1 that the SDP (3) computes

4 The Absolute PPT Question

We now present the question that is at the heart of this work. Recall that the answer to this question was already shown to be “yes” in the case in [19].

Question 1.

Is it true that a quantum state is absolutely separable if and only if it is absolutely PPT?

The rest of the paper is devoted to investigating Question 1. In particular, we show that many of the standard techniques from entanglement theory cannot be used to help answer this question. We first need the following proposition.

Proposition 1.

Suppose that there exists a state that is absolutely PPT but not absolutely separable. Then has full rank.

Proof.

Suppose that is absolutely PPT with eigenvalues (notice that we set the smallest eigenvalue equal to , so that does not have full rank). Our goal is to show that is absolutely separable.

We recall from Section 3 that the matrix (2) must be positive semidefinite. However, by using the fact that , we then see that , which implies that (up to a positive scalar multiple), for some pure state . We now use [15, Theorem 1], which says that every operator of the form with is separable (and even absolutely separable). Since , it follows that is absolutely separable, as desired. ∎

We note that the proof of Proposition 1 shows that the only rank-deficient absolutely PPT states are (up to normalization) the orthogonal projections of rank , and these states are even in the Gurvits–Barnum ball of separability.

Proposition 1 immediately implies that the range criterion [28] for detecting entanglement, which states that the range of a separable state is spanned by product pure states, cannot possibly detect entanglement in any absolutely PPT state. To see this, simply note that the range of a full-rank state is the entire Hilbert space, which is always spanned by product states (such as the standard basis). Furthermore, Proposition 1 also shows that most of the “usual” ways of creating PPT entangled states cannot possibly create absolutely PPT entangled states, since many such methods result in states that are not of full rank (e.g., chessboard states [29], states constructed by unextendible product bases [30], the -parameter family of states constructed by the Horodeckis [31], and so on). Relatively few families of bound entangled states with full rank are known [32, 33], and we have not been able to find any that are absolutely PPT (see Section 6.3, for example).

5 The Absolute Separablity “Collapse”

In this section, we present the main results of the paper, which show that the set of absolutely PPT states is “closer” to the set of absolutely separable states than the set of PPT states is to the set of separable states in the following sense: there are (many) separability criteria that are capable of detecting entanglement in PPT states, but become weaker than the PPT criterion in the “absolute” regime (see Figure 1, for example). We already saw this for the range criterion in the previous section. We now prove that the same result holds for the realignment criterion [20, 21], the Choi map [25] and its generalizations [34], and the Breuer–Hall map [35, 36]. That is, each of these separability criteria are incapable of detecting any entanglement in absolutely PPT states.

Before dealing with any specific separability criteria, we first need the following very important lemma, which we will make repeated use of. This lemma lets us determine that an entanglement witness cannot detect entanglement in absolutely PPT states, based only on very limited information about the eigenvalues of the witness (specifically, its largest eigenvalue and the sum of its negative eigenvalues).

Lemma 2.

Let be a Hermitian operator with . Let be the maximum eigenvalue of and define to be the sum of its negative eigenvalues:

Furthermore, define a function by:

If and then for all absolutely PPT states .

Before proving the lemma, we note that we have numerically found that the function described by Lemma 2 is optimal at least in the case. That is, given any choice of and such that , we can numerically find a Hermitian operator and an absolutely PPT state such that , has a single negative eigenvalue equal to , the maximum eigenvalue of is , and .

The function is plotted in Figure 2, where we have highlighted some important special cases. For example, , , , and .

Sum of negative eigenvalues ()

Upper bound on

Figure 2: A plot of the upper bound given by Lemma 2. For example, point is , which tells us that if the sum of the negative eigenvalues () of equals then cannot detect entanglement in absolutely PPT states if the largest eigenvalue () of is . Point corresponds to and , corresponds to and , corresponds to and , corresponds to and , and corresponds to and (in which case the result is trivial).
Proof of Lemma 2.

We prove the result by showing that the semidefinite program (3) has optimal value whenever and . First, we replace the complicated set of LMI constraints for all in this SDP with the single constraint that the matrix (2) is positive semidefinite. Since this new SDP is a minimization problem subject to a weaker set of constraints, its optimal value is no larger than the optimal value of the SDP (3).

Second, we will make some simplifying assumptions about the eigenvalues . To this end, for now we fix some and satisfying the constraints of the SDP (3), and consider the following SDP, where we optimize over :

(4)

It is straightforward to see that the optimal solution of the SDP (4) occurs when (i.e., rather than having multiple negative ’s, we just have one of them as negative as possible). Similarly, increasing (subject to the constraints of the SDP (4)), or increasing while fixing will also decrease the value of the objective function, and similarly for increasing while fixing and . For example, when and , it suffices to consider the case when ( are determined by making as large as possible subject to , then as large as possible while subject to , and so on until ). In general, we set and (and for all ).

Since the optimal solution of the SDP (4) satisfies the conditions described in the previous paragraph regardless of , we can assume without loss of generality in the SDP (3) that satisfy those same conditions. That is, it suffices to show that the optimal value of the following SDP is , where we recall that and are fixed constants in this SDP, and we optimize over :

(5)

The dual problem can be constructed using standard techniques of semidefinite programming as found in [24].

(6)

It thus suffices to find a feasible point of the above dual problem with . We note that code that implements the above SDP in MATLAB via the CVX package [37] can be downloaded from [38]. We now split into three cases, depending on which branch of we are working with.

Case a): . In this case, we have , , and . It is then straightforward to verify the following defines a feasible point of the dual problem of the semidefinite program (5):

The only condition in the dual problem that is not obviously satisfied is the fact that . However, this follows from the fact that for this particular choice of and . Since this dual feasible point has , it follows that the semidefinite program (5) has optimal value , as desired.

Case b): . This case follows immediately from choosing in case a) and noting that we can choose the function described by the lemma to be non-decreasing.

Case c): . In this case, we have , , and . It is then straightforward to verify the following defines a feasible point of the dual problem of the semidefinite program (5):

Similar to case a), we have for this particular choice of and , so the above point indeed satisfies all of the constraints of the dual problem. Since , it follows that the semidefinite program (5) has optimal value , which completes the proof. ∎

5.1 The Realignment Criterion

The realignment criterion [20, 21] for entanglement states that all separable states satisfy , where is the linear “realignment” map defined on elementary tensors by . Thus if then we know that is entangled, and we say that the realignment criterion detected the entanglement in . This criterion is particularly useful, as it is one of the simplest tests that can detect entanglement in PPT states. The main result of this section shows that the realignment criterion cannot detect entanglement in any absolutely PPT states.

To phrase our result in another way, we can consider the sets of absolutely PPT states and “absolutely realignable” states:

Our result states that , so the realignment criterion becomes a weaker entanglement test than the PPT criterion in the “absolute” setting:

Theorem 1.

If is absolutely PPT then . That is, the realignment criterion cannot detect entanglement in any absolutely PPT state.

It will be helpful to note that for any state , one can write in terms of its operator-Schmidt decomposition, which is defined as

where for all and the sets of operators and form orthonormal bases of and in the Hilbert–Schmidt inner product. There is a well-known correspondence between the realignment criterion and the operator-Schmidt decomposition of any state . Specifically, it is the case that (a proof of this fact can be found in [39]), so any state with is entangled. In this case, it is straightforward to see that the operator

(7)

is an entanglement witness that detects the entanglement in , since orthonormality of and implies that . Thus, to show that for all absolutely PPT states, it suffices to show that operators of the form (7) have whenever is absolutely PPT. In order to prove this result, we first need the following two auxiliary lemmas.

Lemma 3.

Let and be orthonormal sets in the Hilbert–Schmidt inner product. Then for all .

Proof.

Define vectors by and . By the Cauchy–Schwarz inequality, it follows that

To prove our result, we show that , with a similar argument holding for . We first rewrite as . Note that the identity operator can be rewritten as

(8)

Computing the norms of the quantities in Equation (8), we obtain

where the first equality follows from the definition of the Frobenius norm, and by noting that the set of operators are orthonormal. Since , it directly follows that , as desired. ∎

Lemma 4.

Let be an entanglement witness of the form (7), scaled so that . Then

Proof.

Consider the Hermitian operator and let be its normalization: . For brevity, define . Then

where the first inequality above comes from simply noting that and the second inequality comes from Lemma 3 and noting that is an increasing function of (for ). ∎

We are now in a position to prove Theorem 1.

Proof of Theorem 1.

Our goal is to make use of Lemma 2, so we want to find bounds on and the maximum eigenvalue of , where is an entanglement witness of the form (7), scaled so that . Bounds on both of these quantities follow straightforwardly from Lemma 4. More specifically, it is always the case that for matrices, so Lemma 4 immediately implies that

Similarly, always, so we get the following bounds on and :

Furthermore, when (which is the only case we need to consider, since it is already known that absolute separability and absolute PPT coincide when ) we then have the looser bounds and . Since , it follows from Lemma 2 that if is absolutely PPT then , so , as desired. ∎

5.2 The Choi Map

The Choi map [40] is a positive map on that is defined as follows:

This map is one of the most well-known positive maps because it was one of the first maps found with the property that there are states such that , but . In other words, the Choi map was one of the first known examples of a positive map that can detect entanglement in PPT states. Furthermore, it is extremal in the set of positive maps [41].

Our main result of this section is that the Choi map cannot detect entanglement in absolutely PPT states. Equivalently, we show that the set of “absolutely Choi map” states:

is a superset of the absolutely PPT states: . Thus the Choi map is a weaker entanglement test than the PPT criterion in the “absolute” setting:

Theorem 2.

If is absolutely PPT then .

As with the previous section, our first goal here is to rephrase the condition in terms of entanglement witnesses, so that we can make use of Lemma 2. To this end, simply note that if then there exists a pure state such that . Thus , so

(9)

is a witness that detects entanglement in . It thus suffices to show that witnesses of the form (9) cannot detect entanglement in absolutely PPT states . In order to prove this claim, we present the following lemma, which bounds the eigenvalues of .

Lemma 5.

Let be a unit vector. Then the eigenvalues of are contained within the interval .

Before proving this result, we note that both of its bounds on the eigenvalues are tight. A minimal eigenvalue of is obtained when (the standard maximally-entangled state), and a maximal eigenvalue of is obtained when .

Proof.

Let be the eigenvalues of . Our first goal is to show that . To this end, first notice that is trace-preserving, so we have

(10)

Also notice that

(11)

where is the diamond norm [42] defined by

By subtracting Inequality (11) from Equation (10), we see that

(12)

The diamond norm can be computed via semidefinite programming [43], and in particular is equal to the optimal value of the following problem [44]:

(13)

where we optimize over and denote as the partial trace with respect to the second subsystem. It is straightforward to verify that the following are feasible values of and , written with respect to the standard basis of :

Since , it follows that (it is not difficult to show that it actually equals , but we only need the upper bound). By plugging this upper bound into Inequality (12), we see that , as desired.

Our next goal is to show that . To show this, we first note that

By making use of [45, Theorem 4], we see that the in this quantity can be omitted, giving

(14)

It was shown in [46] that the quantity on the right in Inequality (14) equals

(15)

Optimizing over separable states in general is expected to be difficult, but we can compute upper bounds of the quantity (15) via the semidefinite programming methods of [46, 47]. In particular, the following SDP takes the supremum over the set of PPT states, rather than the set of separable states, and thus computes an upper bound of the quantity (15) (and hence of ):

(16)

where we optimize over density matrices in the primal problem and over Hermitian in the dual problem. It is straightforward to use semidefinite programming solvers to numerically verify that the optimal value of this semidefinite program is , from which it follows that . To obtain a completely analytic proof of this fact, it suffices to find a single positive semidefinite such that . One such matrix is as follows: