643 research outputs found
A Resource Framework for Quantum Shannon Theory
Quantum Shannon theory is loosely defined as a collection of coding theorems,
such as classical and quantum source compression, noisy channel coding
theorems, entanglement distillation, etc., which characterize asymptotic
properties of quantum and classical channels and states. In this paper we
advocate a unified approach to an important class of problems in quantum
Shannon theory, consisting of those that are bipartite, unidirectional and
memoryless.
We formalize two principles that have long been tacitly understood. First, we
describe how the Church of the larger Hilbert space allows us to move flexibly
between states, channels, ensembles and their purifications. Second, we
introduce finite and asymptotic (quantum) information processing resources as
the basic objects of quantum Shannon theory and recast the protocols used in
direct coding theorems as inequalities between resources. We develop the rules
of a resource calculus which allows us to manipulate and combine resource
inequalities. This framework simplifies many coding theorem proofs and provides
structural insights into the logical dependencies among coding theorems.
We review the above-mentioned basic coding results and show how a subset of
them can be unified into a family of related resource inequalities. Finally, we
use this family to find optimal trade-off curves for all protocols involving
one noisy quantum resource and two noiseless ones.Comment: 60 page
A family of quantum protocols
We introduce two dual, purely quantum protocols: for entanglement
distillation assisted by quantum communication (``mother'' protocol) and for
entanglement assisted quantum communication (``father'' protocol). We show how
a large class of ``children'' protocols (including many previously known ones)
can be derived from the two by direct application of teleportation or
super-dense coding. Furthermore, the parent may be recovered from most of the
children protocols by making them ``coherent''. We also summarize the various
resource trade-offs these protocols give rise to.Comment: 5 pages, 1 figur
How robust is a quantum gate in the presence of noise?
We define several quantitative measures of the robustness of a quantum gate
against noise. Exact analytic expressions for the robustness against
depolarizing noise are obtained for all unitary quantum gates, and it is found
that the controlled-not is the most robust two-qubit quantum gate, in the sense
that it is the quantum gate which can tolerate the most depolarizing noise and
still generate entanglement. Our results enable us to place several analytic
upper bounds on the value of the threshold for quantum computation, with the
best bound in the most pessimistic error model being 0.5.Comment: 14 page
On the capacities of bipartite Hamiltonians and unitary gates
We consider interactions as bidirectional channels. We investigate the
capacities for interaction Hamiltonians and nonlocal unitary gates to generate
entanglement and transmit classical information. We give analytic expressions
for the entanglement generating capacity and entanglement-assisted one-way
classical communication capacity of interactions, and show that these
quantities are additive, so that the asymptotic capacities equal the
corresponding 1-shot capacities. We give general bounds on other capacities,
discuss some examples, and conclude with some open questions.Comment: V3: extensively rewritten. V4: a mistaken reference to a conjecture
by Kraus and Cirac [quant-ph/0011050] removed and a mistake in the order of
authors in Ref. [53] correcte
Random and free observables saturate the Tsirelson bound for CHSH inequality
Maximal violation of the CHSH-Bell inequality is usually said to be a feature
of anticommuting observables. In this work we show that even random observables
exhibit near-maximal violations of the CHSH-Bell inequality. To do this, we use
the tools of free probability theory to analyze the commutators of large random
matrices. Along the way, we introduce the notion of "free observables" which
can be thought of as infinite-dimensional operators that reproduce the
statistics of random matrices as their dimension tends towards infinity. We
also study the fine-grained uncertainty of a sequence of free or random
observables, and use this to construct a steering inequality with a large
violation
Property testing of unitary operators
In this paper, we systematically study property testing of unitary operators.
We first introduce a distance measure that reflects the average difference
between unitary operators. Then we show that, with respect to this distance
measure, the orthogonal group, quantum juntas (i.e. unitary operators that only
nontrivially act on a few qubits of the system) and Clifford group can be all
efficiently tested. In fact, their testing algorithms have query complexities
independent of the system's size and have only one-sided error. Then we give an
algorithm that tests any finite subset of the unitary group, and demonstrate an
application of this algorithm to the permutation group. This algorithm also has
one-sided error and polynomial query complexity, but it is unknown whether it
can be efficiently implemented in general
Two-way quantum communication channels
We consider communication between two parties using a bipartite quantum
operation, which constitutes the most general quantum mechanical model of
two-party communication. We primarily focus on the simultaneous forward and
backward communication of classical messages. For the case in which the two
parties share unlimited prior entanglement, we give inner and outer bounds on
the achievable rate region that generalize classical results due to Shannon. In
particular, using a protocol of Bennett, Harrow, Leung, and Smolin, we give a
one-shot expression in terms of the Holevo information for the
entanglement-assisted one-way capacity of a two-way quantum channel. As
applications, we rederive two known additivity results for one-way channel
capacities: the entanglement-assisted capacity of a general one-way channel,
and the unassisted capacity of an entanglement-breaking one-way channel.Comment: 21 pages, 3 figure
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