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The quantum coupling of two given quantum states denotes the set of bipartite states whose marginal states are these given two states. In this paper, we provide tight inequalities to describe the structure of quantum coupling. These inequalities directly imply that the trace distance between two quantum states cannot be determined by the quantum analog of the earth mover’s distance, thus ruling out the equality version of the quantum Kantorovich–Rubinstein theorem for trace distance even in the finite-dimensional case. In addition, we provide an inequality that can be regarded as a quantum generalization of the Kantorovich–Rubinstein theorem. Then, we generalize our inequalities and apply them to the three tripartite quantum marginal problems. Numerical tests with a three-qubit system show that our criteria are much stronger than the known criteria: the strong subadditivity of entropy and the monogamy of entanglement.

We show that all $n$-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, admit Hardy-type proofs of non-locality without inequalities or probabilities. More precisely, we show that for all such states, there are local, one-qubit observables such that the resulting probability tables are logically contextual in the sense of Abramsky and Brandenburger, this being the general form of the Hardy-type property. Moreover, our proof is constructive: given a state, we show how to produce the witnessing local observables. In fact, we give an algorithm to do this. Although the algorithm is reasonably straightforward, its proof of correctness is non-trivial. A further striking feature is that we show that $n+2$ local observables suffice to witness the logical contextuality of any $n$-qubit state: two each for two for the parties, and one each for the remaining $n-2$ parties., Comment: 23 pages. Submitted for publication

Quantum walks are very useful tools in designing quantum algorithms. Amplitude amplification is a key technique to increase the success probability of a quantum-walk-based algorithm, and it is quadratically faster than classical probabilistic amplification. However, amplitude amplification only applies to quantum walks with one-shot hitting time, where no measurements except a final one are performed, and not to quantum walks with concurrent hitting time, where measurements happen or absorbing boundaries exist at each step. In this paper, we propose a procedure to modify quantum walks with concurrent hitting time by removing measurements from them. This procedure enables us to use amplitude amplification to design algorithms based on the modified quantum walks which are faster than those based on the original walks with a concurrent hitting time and more robust than those based on the corresponding walks with a one-shot hitting time. © 2013 American Physical Society.