Detecting Consistency of Overlapping Quantum Marginals by Separability
The quantum marginal problem asks whether a set of given density matrices are consistent, i.e., whether they can be the reduced density matrices of a global quantum state. Not many non-trivial analytic necessary (or sufficient) conditions are known for the problem in general. We propose a method to detect consistency of overlapping quantum marginals by considering the separability of some derived states. Our method works well for the -symmetric extension problem in general, and for the general overlapping marginal problems in some cases. Our work is, in some sense, the converse to the well-known -symmetric extension criterion for separability.
The quantum marginal problem, also known as the consistency problem, asks for the conditions under which there exists an -particle density matrix whose reduced density matrices (quantum marginals) on the subsets of particles equal to the given density matrices for all Klyachko (2006). The related problem in fermionic (bosonic) systems is the so-called -representability problem. It asks whether a -fermionic (bosonic) density matrix is the reduced density matrix of some -fermion (boson) state . The -representability problem inherits a long history in quantum chemistry Coleman (1963); Erdahl (1972).
The quantum marginal problem and the -representability problem are in general very difficult. They were shown to be the complete problems of the complexity class QMA, even for the relatively simple case where the given marginals are two-particle states Liu (2006); Liu et al. (2007); Wei et al. (2010). In other words, even with the help of a quantum computer, it is very unlikely that the quantum marginal problems can be solved efficiently in the worst case. In this sense, the best hope to have simple analytic conditions for the quantum marginal problem is to find either necessary or sufficient conditions in certain special cases.
When the given marginals are states of non-overlapping subsets of particles, and one is interested in a global pure state consistent with the given marginals, both the quantum marginal problem and the -representability problem were solved Klyachko (2004, 2006); Altunbulak and Klyachko (2008); Schilling et al. (2013); Walter et al. (2013); Sawicki et al. (2014). However, not much is known for the general problem with overlapping subsystems. For the tripartite case of particles , , , the strong subadditivity of von Neumann entropy enforces non-trivial necessary conditions for the consistency of and such as Carlen et al. (2013). In a similar spirit, certain quantitative monogamy of entanglement type of results (see e.g. Coffman et al. (2000)) also put non-trivial necessary conditions. Necessary and sufficient conditions are generally not known, except in very few special situations Smith (1965); Chen et al. (2014); Carlen et al. (2013) when is small.
In this work, we propose a simple but powerful analytic necessary condition for arguably the simplest overlapping quantum marginal problem, known as the -symmetric extension problem. That is, we will consider quantum marginal problems of particles , for a given density matrix , and require that there is a global quantum state whose marginals on equal to the given for . The classical analog of this particular case is trivial and there is a consistent global probability distribution as long as the marginals agree on . In the quantum case, however, the problem remains unsolved even for .
We prove the separability of certain derived state as a necessary condition for the -symmetric extension problem. A quantum state is separable if it can be written as the convex combination for a probability distribution and states and . It is now well-known that the -symmetric extension of provides a hierarchy of separability criteria for , which converges exactly to the set of separable states when goes to infinity Doherty et al. (2002). This result is essentially given by the quantum de Finetti’s theorem Stormer (1969); Hudson and Moody (1976); Doherty et al. (2002); Renner (2007); Christandl et al. (2007); Harrow (2013). Our method can, in some sense, be thought of as a converse to the -symmetric extension criterion of separability. We will use separability instead as a criterion to test -symmetric extendability of a bipartite state. This, however, does not cause any circular reasoning problem—we can instead use other known separability criteria, such as the positive partial transpose condition Peres (1996); Horodecki et al. (1996), to give necessary conditions for the -symmetric extension problems.
In particular, our method computes a linear combination of the given density matrix and its reduced density matrix . The separability of is then shown to be a necessary condition of the corresponding -symmetric extension problem for .
Interestingly, the condition can also be applied to the more general setting of overlapping quantum marginal problems where the given marginals on are different. We reduce them to the -symmetric extension problems of . This averaging method may give trivial conditions in adversarial situations. But it will nevertheless provide non-trivial conditions better than many known results when the given density matrices , though different, are related in some way.
Necessary conditions for the -symmetric extension problems.— Let , be two Hilbert spaces of dimension and , respectively. For a Hilbert space , let be the set of density matrices on . For a bipartite state , we consider the following overlapping quantum marginal problem: whether there exists a state whose marginals on equal to for all . The problem is also called the -symmetric extension problem of Doherty et al. (2002, 2004, 2005); Navascués et al. (2009); Brandão and Christandl (2012) and the global state is called a -symmetric extension of . If such a global state exists, one can choose it to be invariant under permutations of Doherty et al. (2002); Myhr (2011).
If the state is separable, then it is also obviously -symmetric extendable for any . Interestingly, the converse of the statement is also true. That is, if is -symmetric extendable for all , then must be separable Doherty et al. (2004). This provides a complete hierarchy of separability criteria. The -symmetric extension problem can be formulated as a semidefinite programming (SDP), providing a numerical procedure to detect entanglement in a mixed state (see e.g. Vandenberghe and Boyd (1996)).
In this paper, we want to know for a given , whether is -symmetric extendable. One can of course use the semidefinite programming to solve the problem, but the size of the SDP formulation will grow exponentially with , rendering the approach impractical even numerically for large . We will instead use the separability of some derived state to detect the -extendability of . The important thing is that the dimension of the state is independent of .
For convenience, we will also consider a variant of the -symmetric extension problem called the -bosonic extension problem. For Hilbert spaces of dimension , let be the symmetric subspace of . A state has a -bosonic extension if it has a -symmetric extension whose support on is in the symmetric subspace .
Our main observation is the following theorem. In the theorem, and are two Hilbert spaces of dimension and respectively.
If a bipartite state has a -symmetric extension, then the bipartite state
If a bipartite state has a -bosonic extension, then the bipartite state
Let be Hilbert spaces of dimension and let be a state supported on the symmetric subspace . Consider the following superoperator :
where is the Haar measure over the pure states of and is the permutation operator defined by
We claim that
for all state where is the -particle marginal of . The claim follows from the Chiribella’s theorem Chiribella (2011); we give a proof here for its importance to our work. By the fact that any state supported on the symmetric subspace can be written as the linear combination of states of the form (see the Appendix of Chiribella (2011)), it suffices to prove the claim in Eq. (4) for . For all ,
There are permutations such that and permutations and the claim follows from Eq. (3).
We now prove Theorem 1.
Proof of Theorem 1.
Examples of Bell-diagonal states.— First consider the simple case of , and are qubit systems (). Since for any two-qubit state, the existence of a -symmetric extension implies that of a -bosonic extension (see Proposition 21 of Myhr (2011)), we can use the stronger condition of Eq. (2) also for the symmetric extension problem. For simplicity, we will investigate our condition for -symmetric extension for the class of Bell-diagonal states. A state is Bell-diagonal if it is of the form
where , and
are the four Bell states.
A simple computation tells that our condition that being separable is equivalent to for all . This is a close approximation of the exact condition of -symmetric extendability given in Myhr and Lütkenhaus (2009); Chen et al. (2014); Cerf (2000); Niu and Griffiths (1998); Cubitt et al. (2008):
The regions of given by these two conditions are plotted in Fig. 1. The volume of the exact set is approximately and the volume of the polytope given by our condition is , which is only about larger.
For comparison purposes, we have also plotted the conditions given by the strong subadditivity (SSA). For Bell-diagonal states, the SSA condition simplifies to . We find that our condition and the SSA condition are incomparable—the non-extendability can sometimes be detected by our condition but not the SSA condition, and vice versa. See Fig. 2 for details.
Examples of Werner states.— In our next example, we analyze our conditions for the -symmetric extension problem of the Werner states Werner (1989a, 1990). A two-qudit Werner state is a state invariant under the operator for all unitary and has the following form
where is the parameter, and are the states proportional to the projection of the symmetric subspace and anti-symmetric subspace respectively. The Werner state is separable if and only if . The state is separable when . Therefore, by Theorem 1, is not -symmetric extendable if . We note that our bound, though not optimal, is a close approximation of the necessary and sufficient condition proved in Johnson and Viola (2013) for the -symmetric extendability of Werner states. This also proves that the -symmetric extension and -bosonic extension problems are generally different. In particular, it also implies that the in the linear combination in Eq. (1) is essential for the -symmetric extension problem.
Applications to the overlapping marginal problems.— We now extend our method to the more general situation with different marginals on . That is, one asks whether there exists a state whose marginals on is the given density matrices for all . This consistency problem for bipartite marginals is of vital importance in many-body physics and quantum chemistry, where the Hamiltonians of the system in general involve only two-body interactions Coleman (1963); Erdahl (1972); Zeng et al. (2015).
In order to use the necessary condition derived in the previous section, we observe the following fact.
If the marginals with are consistent, then the bipartite state
has -symmetric extension.
If with are consistent, then there exists a state , such that its reduced density matrix on the system is for all . Now consider the state
where is the symmetric group of elements. Then is a -symmetric extension of . ∎
This then allows us to use Theorem 1 and Lemma 2 to detect consistency of bipartite marginals. Consider the example of a three-qubit system with , and for , both of which are two-qubit Werner states. For two-qubit states, -symmetric extendability implies -bosonic extendability. Hence, we can use the condition of Eq. (2), which implies that and are consistent only if . This in fact gives a quantitative entanglement monogamy inequality Werner (1989b); Coffman et al. (2000); Terhal (2003); Bae and Acin (2006); Osborne and Verstraete (2006) for Werner states.
We compare our condition to that given by the Coffman-Kundu-Wootters (CKW) entanglement monogamy inequality Coffman et al. (2000),
where , are the concurrences Hill and Wootters (1997); Wootters (1998) between and respectively, while is the concurrence between subsystems and for Werner states. As shown in Fig. 2(a), our condition (the pentagon defined by and ) is always better than the condition given by the CKW inequality (the union of the yellow and green regions).
We have also computed the SSA condition for this particular case and plotted the regions of the SSA condition and our condition in Fig. 2(b). Again, the SSA condition (the union of the yellow and green regions) is incomparable with ours.
Generalizations.— Our method extends to the following more general settings. Let be a given density matrix. The -bosonic extension problem of asks whether there is a global state whose marginal on is . Following a similar argument as in the proof of Lemma 2 and using the Chiribella’s theorem Chiribella (2011); Harrow (2013), one obtains a necessary condition generalizing Lemma 2. Namely,
is an -party separable state. Here,
is a distribution satisfying , and is the superoperator given by
where and is the projection onto the symmetric subspace .
At the moment, however, we do not know how to generalize the formula in Theorem 1 to this multi-party setting as the procedure of tracing out does not commute with the projection in general. We leave it as an open problem for future work.
Summary and discussion.— We have proposed a method to detect consistency of overlapping quantum marginals. The key idea is to construct some other density matrix from the linear combinations of the local density matrices and test the separability of the derived density matrix. Our idea is closely related to the finite quantum de Finetti’s theorem Diaconis and Freedman (1980); Doherty et al. (2002); Renner (2007); Christandl et al. (2007), which states that the -particle marginal of a symmetric -particle state cannot be too far from an -particle separate state, with a distance bounded by for fixed and . Therefore, if an -particle state is too far from a separable state, then it cannot be the marginal of a symmetric -particle state. However, to directly check the distance to the nearest separable state is not easy. Moreover, the bound given in the known versions of finite quantum de Finetti’s theorem are in general not tight, so when is small those bound may not be useful.
For comparison, our method gives simple necessarily conditions, which are evidently good even for small. Our method can also lead to improved bound in the finite de Finetti’s theorem. For instance, as a direct consequence of Theorem 1, we can obtain that for any -symmetric extendible state , its distance to separable states is upper bounded by
which slightly improves that of Christandl et al. (2007).
is always separable. Notice that Eq. (12) implies that , so we have This gives an interesting sufficient condition of separability for : if then is separable. We may also compare this with the known necessary condition of separability for Horodecki and Horodecki (1999); Cerf et al. (1999): if is separable, then
We thank John Watrous for helpful discussions. ZJ acknowledges support from NSERC and ARO. NY and BZ are supported by NSERC.
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