Abstract One of the goals of science is to understand the relation between a whole and its parts, as exemplified by the problem of certifying the entanglement of a system from the knowledge of its reduced states. Here, we focus on a different but related question: can a collection of marginal information reveal new marginal information? We answer this affirmatively and show that (non-) entangled marginal states may exhibit (meta)transitivity of entanglement, i.e., implying that a different target marginal must be entangled. By showing that the global n -qubit state compatible with certain two-qubit marginals in a tree form is unique, we prove that transitivity exists for a system involving an arbitrarily large number of qubits. We also completely characterize—in the sense of providing both the necessary and sufficient conditions—when (meta)transitivity can occur in a tripartite scenario when the two-qudit marginals given are either the Werner states or the isotropic states. Our numerical results suggest that in the tripartite scenario, entanglement transitivity is generic among the marginals derived from pure states.
Abstract One of the goals of science is to understand the relation between a whole and its parts, as exemplified by the problem of certifying the entanglement of a system from the knowledge of its reduced states. Here, we focus on a different but related question: can a collection of marginal information reveal new marginal information? We answer this affirmatively and show that (non-) entangled marginal states may exhibit (meta)transitivity of entanglement, i.e., implying that a different target marginal must be entangled. By showing that the global n -qubit state compatible with certain two-qubit marginals in a tree form is unique, we prove that transitivity exists for a system involving an arbitrarily large number of qubits. We also completely characterize—in the sense of providing both the necessary and sufficient conditions—when (meta)transitivity can occur in a tripartite scenario when the two-qudit marginals given are either the Werner states or the isotropic states. Our numerical results suggest that in the tripartite scenario, entanglement transitivity is generic among the marginals derived from pure states.
Abstract Generic bipartite pure states of sufficiently large dimensions are overwhelmingly likely to be highly entangled. However, unlike the bipartite setting, the entanglement of generic multipartite pure states, and specifically their multipartite marginals , is far less understood. Here, we show that sufficiently large marginals of generic multipartite pure states, accounting for approximately half or more of the subsystems, are entangled across all bipartitions. These pure states are thus robust to losses in entanglement distribution. Moreover, even without assuming that the global state is pure, a small number of overlapping entangled marginals of generic pure states—as we show in this work—must induce entanglement in other marginals when some mild dimension constraints are satisfied, i.e. entanglement transitivity is a generic feature of various many-body pure states. Numerically, we further observe that the genericity of (1) entangled marginals, (2) unique global compatibility, and (3) entanglement transitivity may also hold beyond the analytically established dimension constraints. We also discuss potential applications of these features of generic pure states in quantum information processing.
Abstract Despite the monogamous nature of nonlocal correlations, in a Bell test involving three parties A, B, and C, the nonlocality in two bipartite subsystems (e.g., AB and BC) may force the remaining bipartite subsystem (e.g., AC) to exhibit nonlocality. Although this intriguing effect of nonlocality transitivity has been identified in the non -quantum non-signaling world since 2011, whether such transitivity could manifest within quantum theory has remained unresolved. Here, we answer this open problem affirmatively—both analytically and numerically—at the level of quantum states, thereby showing that a quantum-realizable notion of nonlocality transitivity exists. In our analytic construction, we prove and use the fact that copies of the W -state marginals uniquely determine the global compatible state, thus establishing another instance when the parts determine the whole. Moreover, we present a simple method to construct quantum states and correlations that are nonlocal in all their non-unipartite marginals. We also discuss the implications of our results in (semi-) device-independent cryptographic and certification tasks.
Imagine three parties, Alice, Bob, and Charlie, who share a state of three qubits such that all two-party reduced states A-B, A-C, and B-C are separable. Suppose that they have information only about these marginals but not about the global state. According to recent results, there exists an example for a set of three separable two-party reduced states that is only compatible with an entangled global state. In this paper, we show a stronger result by exhibiting separable two-party reduced states A-B, A-C, and B-C, such that any global state compatible with these marginals is nonlocal. Hence, we obtain that nonlocality of multipartite states can be certified from information only about separable marginals.
Understanding what can be inferred about a multiparticle quantum system given only the knowledge of its subparts is a highly nontrivial task. Clearly, if a global system does not contain an information resource of some kind, neither do its subparts. For the case of entanglement as an information resource, it is known that the converse of this last statement is not true: Some nonentangled reduced states are only compatible with global states which are entangled. We extend this result to correlations and provide local marginal correlations that are only compatible with global genuinely tripartite nonlocal correlations. Quantum nonlocality can thus be deduced from the mere observation of local marginal correlations.
In many-body quantum physics, the global state encodes all correlations, but the information most naturally available is often local. This makes it important to understand when a small collection of reduced density matrices already fixes the whole state. Such uniqueness questions are especially important for tomography and for the study of entanglement, because agreement with local data from a pure state does not by itself exclude a different mixed global state. We prove a generic uniqueness theorem for finite-dimensional tripartite systems. Previous results were restricted either to uniqueness against pure competitors or to uniqueness against all competitors under stronger dimension assumptions or balanced subsystem dimensions. Our result removes both restrictions.
We introduce the resource marginal problems , which concern the possibility of having a resource-free target subsystem compatible with a g i v e n collection of marginal density matrices. By identifying an appropriate choice of resource R and target subsystem T, our problems reduce, respectively, to the well-known marginal problems for quantum states and the problem of determining if a given quantum system is a resource. More generally, we say that a set of marginal states is resource-free incompatible with a target subsystem T if all global states compatible with this set must result in a resourceful state in T of type R. We show that this incompatibility i n d u c e s a resource theory that can be quantified by a monotone and obtain necessary and sufficient conditions for this monotone to be computable as a conic program with finite optimum. We further show, via the corresponding witnesses, that (1) resource-free incompatibility is equivalent to an operational advantage in some channel-discrimination tasks, and (2) some specific cases of such tasks fully characterize the convertibility between marginal density matrices exhibiting resource-free incompatibility. Through our framework, one sees a clear connection between any marginal problem – which implicitly involves some notion of incompatibility – for quantum states and a resource theory for quantum states. We also establish a close connection between the physical relevance of resource marginal problems and the ground state properties of certain many-body Hamiltonians. In terms of application, the universality of our framework leads, for example, to a further quantitative understanding of the incompatibility associated with the recently-proposed entanglement marginal problems and entanglement transitivity problems.