We consider the problem of implementing two-party interactive quantum

communication over noisy channels, a necessary endeavor if we wish to

fully reap quantum advantages for communication.

For an arbitrary protocol with n messages, designed for

noiseless qudit channels, our main result is a simulation method that fails with probability less than

$2^{-\Theta(n\epsilon)}$ and uses a qudit channel $n(1 + \Theta

(\sqrt{\epsilon}))$ times, of which an $\epsilon$ fraction can be

corrupted adversarially.

The simulation is thus capacity achieving to leading order, and

we conjecture that it is optimal up to a constant factor in

the $\sqrt{\epsilon}$ term.

Furthermore, the simulation is in a model that does not require

pre-shared resources such as randomness or entanglement between the

communicating parties.

Surprisingly, this outperforms the best-known overhead of $1 +

O(\sqrt{\epsilon \log \log 1/\epsilon})$ in the corresponding

\emph{classical} model, which is also conjectured to be optimal

[Haeupler, FOCS'14].

Our work also improves over the best previously known quantum result

where the overhead is a non-explicit large constant [Brassard \emph{et

al.}, FOCS'14] for low $\epsilon$.

1 aLeung, Debbie1 aNayak, Ashwin1 aShayeghi, Ala1 aTouchette, Dave1 aYao, Penghui1 aYu, Nengkun uhttp://acm-stoc.org/stoc2018/STOC-2018-Accepted.html01598nas a2200241 4500008004100000245005800041210005800099260001500157300001100172490000800183520093100191100001301122700001401135700001801149700001601167700001801183700001801201700001301219700001601232700001401248700002201262856007201284 2017 eng d00aQuantum state tomography via reduced density matrices0 aQuantum state tomography via reduced density matrices c2017/01/09 a0204010 v1183 aQuantum state tomography via local measurements is an efficient tool for characterizing quantum states. However it requires that the original global state be uniquely determined (UD) by its local reduced density matrices (RDMs). In this work we demonstrate for the first time a class of states that are UD by their RDMs under the assumption that the global state is pure, but fail to be UD in the absence of that assumption. This discovery allows us to classify quantum states according to their UD properties, with the requirement that each class be treated distinctly in the practice of simplifying quantum state tomography. Additionally we experimentally test the feasibility and stability of performing quantum state tomography via the measurement of local RDMs for each class. These theoretical and experimental results advance the project of performing efficient and accurate quantum state tomography in practice.

1 aXin, Tao1 aLu, Dawei1 aKlassen, Joel1 aYu, Nengkun1 aJi, Zhengfeng1 aChen, Jianxin1 aMa, Xian1 aLong, Guilu1 aZeng, Bei1 aLaflamme, Raymond uhttp://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.02040101164nas a2200169 4500008004100000245007500041210006900116260001500185300001100200490000700211520067400218100001800892700001800910700001600928700001400944856003600958 2016 eng d00aDetecting Consistency of Overlapping Quantum Marginals by Separability0 aDetecting Consistency of Overlapping Quantum Marginals by Separa c2016/03/03 a0321050 v933 a The quantum marginal problem asks whether a set of given density matrices are consistent, i.e., whether they can be the reduced density matrices of a global quantum state. Not many non-trivial analytic necessary (or sufficient) conditions are known for the problem in general. We propose a method to detect consistency of overlapping quantum marginals by considering the separability of some derived states. Our method works well for the $k$-symmetric extension problem in general, and for the general overlapping marginal problems in some cases. Our work is, in some sense, the converse to the well-known $k$-symmetric extension criterion for separability. 1 aChen, Jianxin1 aJi, Zhengfeng1 aYu, Nengkun1 aZeng, Bei uhttp://arxiv.org/abs/1509.0659102567nas a2200145 4500008004100000245007800041210006900119260001500188520211000203100001802313700002002331700001702351700001602368856003702384 2016 eng d00aExponential Separation of Quantum Communication and Classical Information0 aExponential Separation of Quantum Communication and Classical In c2016/11/283 aWe exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that these two complexity measures are incomparable. As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold. The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha.

As another application of the techniques that we develop, we give a simple proof for an optimal trade-off between Alice's and Bob's communication while computing the related Greater-Than function on n bits: say Bob communicates at most b bits, then Alice must send n/exp(O(b)) bits to Bob. This holds even when allowing pre-shared entanglement. We also present a classical protocol achieving this bound.