Symmetric extendibility for a class of qudit states

We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global N -partite quantum state. Intuitively, we say that the multipartite state "joins" the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible N -partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in d dimensions, and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is m-n sharability of a Werner state across N subsystems, which may be seen as a variant of the N -representability problem to the case where subsystems are partitioned into two groupings of m and n parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we determine necessary and sufficient conditions for three-party joinability and 1-n sharability for arbitrary d. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed.

Section: Introductionmentioning

confidence: 99%

“…3. Pictorial summary of sharability properties of qubitWerner and isotropic states, according to Eqs (26). and(27).…”

mentioning

confidence: 99%

Compatible quantum correlations: Extension problems for Werner and isotropic states

Johnson

Viola

2013

“…Unfortunately, for any higher dimensions a full characterization involving only spectra is highly unlikely [22]. There have been some efforts made for special cases but no general results found [41][42][43]. Nevertheless, our physical picture based on the convexity of B and the symmetry of the system may shed light on the understanding of symmetric extendibility for higher dimensional systems.…”

Section: Theoremmentioning

confidence: 96%

Symmetric extension of two-qubit states

Chen

Kribs

et al. 2014

A bipartite state ρAB is symmetric extendible if there exists a tripartite state ρ ABB ′ whose AB and AB ′ marginal states are both identical to ρAB. Symmetric extendibility of bipartite states is of vital importance in quantum information because of its central role in separability tests, one-way distillation of EPR pairs, one-way distillation of secure keys, quantum marginal problems, and anti-degradable quantum channels. We establish a simple analytic characterization for symmetric extendibility of any two-qubit quantum state ρAB; specifically, tr(ρ 2 B ) ≥ tr(ρ 2 AB ) − 4 √ det ρAB. Given the intimate relationship between the symmetric extension problem and the quantum marginal problem, our result also provides the first analytic necessary and sufficient condition for the quantum marginal problem with overlapping marginals.PACS numbers: 03.65. Ud, 03.67.Dd, 03.67.Mn The notion of symmetric extendibility for a bipartite quantum state ρ AB was introduced in [1] as a test for entanglement. A bipartite density operator ρ AB is symmetric extendible if there exists a tripartite state ρ ABB ′ such that tr B ′ (ρ ABB ′ ) = tr B (ρ ABB ′ ). A state ρ AB without symmetric extension is evidently entangled, and to decide such an extendibility for ρ AB can be formulated in terms of semi-definite programming (SDP) [2]. This then leads to effective numerical tests and bounds [3][4][5][6] that allow for entanglement detection for some well-known positive-partial-transpose (PPT) states [7][8][9][10][11].States with symmetric extension also have a clear operational meaning for quantum information processing [12]. One simple idea is that if a bipartite state ρ AB is symmetric extendible, then one cannot distill any entanglement from ρ AB by protocols only involving local operations and oneway classical communication (from A to B) [13], because of entanglement monogamy [14]. Furthermore, using the ChoiJamiolkowski isomorphism, symmetric extendibility of bipartite states also provides a test for anti-degradable quantum channels [15], and one-way quantum capacity of quantum channels [13].A similar idea applies to the protocols for quantum key distribution (QKD), which aim to establish a shared secret key between two parties (for a review, see [16]). The corresponding QKD protocols can be viewed as having two phases: in a first phase, the two parties establish joint classical correlations by performing measurements on an untrusted bipartite quantum state, while in a second phase a secret key is being distilled from these correlations by a public discussion protocol (via authenticated classical channels) which typically involves classical error correction and privacy amplification [17][18][19][20]. If the underlying bipartite state ρ AB is symmetric extendible, then no secret key can be distilled by a process involving only one-way communication. Therefore, the foremost task of the public discussion protocol is to break this symmetric extendibility by some bi-directional post-selection process. Failure to find such a protocol me...

“…where a ≥ b and x := a + (m − 1)b ≤ 1. It is known [27,Theorem 3] that such states are symmetrically extendable for all m ≥ 2 if and only if…”

Section: A Simplex Codesmentioning

confidence: 99%

Numerical evidence for bound secrecy from two-way postprocessing in quantum key distribution

Khatri

Lütkenhaus

2017

Bound secret information is classical information that contains secrecy but from which secrecy cannot be extracted. The existence of bound secrecy has been conjectured but is currently unproven, and in this work we provide analytical and numerical evidence for its existence. Specifically, we consider two-way postprocessing protocols in prepare-and-measure quantum key distribution based on the well-known six-state signal states. In terms of the quantum bit-error rate Q of the classical data, such protocols currently exist for Q < 5− √ 5 10