Quantum Shannon theory: capacities and entanglement distillation

5 problems

• Noise threshold for positive quantum capacity of the depolarizing channel

  Open

  The simplest noisy quantum channel replaces a qubit by a random error with probability p. The noise level above which it can no longer carry any quantum information at all is not known.

• Do bound entangled states with negative partial transpose exist?

  Open

  Some entangled states cannot be purified into perfect entangled pairs, no matter how many copies are used. All known examples pass a simple test called positive partial transpose, and it is open whether states that fail the test can also be impossible to purify.

• PPT-squared conjecture: is composing two PPT channels entanglement breaking

  Partially resolved

  Some noisy channels are so weak at carrying entanglement that their partial transpose stays positive. The conjecture says that applying two such channels in a row always destroys all entanglement.

• Is the quantum capacity of a channel computable?

  Open

  The quantum capacity of a channel is defined by a limit over ever more channel uses, and some channels show positive capacity only after arbitrarily many uses. Whether any algorithm can compute the capacity to a given accuracy, or even decide whether it is zero, is unknown.

• Two-copy distillability of the two-ququart Werner state at $\alpha = -1/2$

  Open

  A sharper test case: one specific entangled state of two four-level systems. The task is to show that even two copies of it cannot be processed into an entangled pair.