Non-full-rank bound entangled states satisfying the range criterion (6 pages)

Saronath Halder, Ritabrata Sengupta

Y.-X. Chen, L. Chen

Pratapaditya Bej, Saronath Halder

Somshubhro Bandyopadhyay, Sibasish Ghosh A, Vwani Roychowdhury

A systematic method for generating bound entangled states in any bipartite system, with ranks ranging from five to full rank, is presented. These states are constructed by mixing separable states with unextendable product basis--generated bound entangled states that are positive under partial transposition (PPT). A subset of this class of PPT bound entangled states, having less than full rank, is shown to satisfy the range criterion [P. Horodecki, Phys. Lett. A 232, 333 (1997)].

Xuan Du Trinh, Nicholas Pardave, Angie Huang +2 more

Quantum repeaters extend entanglement by swapping noisy elementary links. We prove that full-rank noise forbids this at long range: for any entangled full-rank two-qubit link, there is a finite depth beyond which no end-to-end entanglement can be established regardless of the measurement outcomes on intermediate qubits, even under any adaptive postselected strategy. This limit is set by one spectral parameter of the link, giving a no-go criterion for repeater routing. In contrast, we construct link state families of rank three and rank two that admit postselected measurement outcome branches of exponentially small probability but with strictly positive concurrence at every finite depth. Swapping experiments on a superconducting processor show that links of equal initial concurrence but different rank behave differently under postselected swapping. In the language of many-body physics, the chain is a matrix-product density operator, and full-rank bonds forbid long-range localizable entanglement, while rank-deficient bonds can sustain it.

Saronath Halder, Ritabrata Sengupta

It is known that bound entangled states are mixed entangled states from which no pure entanglement can be distilled by local operations and classical communication. This holds true even if large number of identical copies of the state are shared between spatially separated parties. In this work we consider bipartite {\it noisy} bound entangled states with positive partial transpose, that is, such a state can be written as a convex combination of a so-called edge state and a separable state ({\it noise}). Starting from a particular class of unextendible product bases (UPBs) we first show a systematic method to construct low-rank noisy bound entangled states which satisfy the range criterion. Next, we consider different classes of uncompletable product bases (UCPBs) to construct noisy bound entangled states. A subset of these bound entangled states has the property that they satisfy the range criterion. However, UCPBs are important to produce noisy bound entangled states and thus, the problem of constructing UCPBs from different classes of UPBs is also considered in this work. In each case explicit examples are constructed. In fact, present bound entangled states help us to learn about the robustness of edge states. We also discuss about the detection these states.