Complexity and capacity bounds for quantum channels

We generalise some well-known graph parameters to operator systems by considering their underlying quantum channels. In particular, we introduce the quantum complexity as the dimension of the smallest co-domain Hilbert space a quantum channel requires to realise a given operator system as its non-commutative confusability graph. We describe quantum complexity as a generalised minimum semidefinite rank and, in the case of a graph operator system, as a quantum intersection number. The quantum complexity and a closely related quantum version of orthogonal rank turn out to be upper bounds for the Shannon zero-error capacity of a quantum channel, and we construct examples for which these bounds beat the best previously known general upper bound for the capacity of quantum channels, given by the quantum Lov\'asz theta number

Maximum privacy without coherence, zero-error

We study the possible difference between the quantum and the private capacities of a quantum channel in the zero-error setting. For a family of channels introduced by Leung et al. [Phys. Rev. Lett. 113, 030512 (2014)], we demonstrate an extreme difference: the zero-error quantum capacity is zero, whereas the zero-error private capacity is maximum given the quantum output dimension

Separation between quantum Lovász number and entanglement-assisted zero-error classical capacity

© 1963-2012 IEEE. Quantum Lovász number is a quantum generalization of the Lovász number in graph theory. It is the best known efficiently computable upper bound of the entanglement-assisted zero-error classical capacity of a quantum channel. However, it remains an intriguing open problem whether quantum entanglement can always enhance the zero-error capacity to achieve the quantum Lovász number. In this paper, by constructing a particular class of qutrit-to-qutrit channels, we show that there exists a strict gap between the entanglement-assisted zero-error capacity and the quantum Lovász number. Interestingly, for this class of quantum channels, the quantum generalization of fractional packing number is strictly larger than the zero-error capacity assisted with feedback or no-signaling correlations, which differs from the case of classical channels

Quantum proof systems and entanglement theory

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009.Includes bibliographical references (p. 99-106).Quantum complexity theory is important from the point of view of not only theory of computation but also quantum information theory. In particular, quantum multi-prover interactive proof systems are defined based on complexity theory notions, while their characterization can be formulated using LOCC operations. On the other hand, the main resource in quantum information theory is entanglement, which can be considered as a monotonic decreasing quantity under LOCC maps. Indeed, any result in quantum proof systems can be translated to entanglement theory, and vice versa. In this thesis I mostly focus on quantum Merlin-Arthur games as a proof system in quantum complexity theory. I present a new complete problem for the complexity class QMA. I also show that computing both the Holevo capacity and the minimum output entropy of quantum channels are NP-hard. Then I move to the multiple-Merlin-Arthur games and show that assuming some additivity conjecture for entanglement of formation, we can amplify the gap in QMA(2) protocols. Based on the same assumption, I show that the QMA(k)-hierarchy collapses to QMA(2). I also prove that QMAlog(2), which is defined the same as QMA(2) except that the size of witnesses is logarithmic, with the gap n-(3+e) contains NP. Finally, motivated by the previous results, I show that the positive partial transpose test gives no bound on the trace distance of a given bipartite state from the set of separable states.by Salman Abolfathe Beikidezfuli.Ph.D

A Survey on Quantum Channel Capacities

Quantum information processing exploits the quantum nature of information. It offers fundamentally new solutions in the field of computer science and extends the possibilities to a level that cannot be imagined in classical communication systems. For quantum communication channels, many new capacity definitions were developed in comparison to classical counterparts. A quantum channel can be used to realize classical information transmission or to deliver quantum information, such as quantum entanglement. Here we review the properties of the quantum communication channel, the various capacity measures and the fundamental differences between the classical and quantum channels.Comment: 58 pages, Journal-ref: IEEE Communications Surveys and Tutorials (2018) (updated & improved version of arXiv:1208.1270