19,745 research outputs found
Zero-error channel capacity and simulation assisted by non-local correlations
Shannon's theory of zero-error communication is re-examined in the broader
setting of using one classical channel to simulate another exactly, and in the
presence of various resources that are all classes of non-signalling
correlations: Shared randomness, shared entanglement and arbitrary
non-signalling correlations. Specifically, when the channel being simulated is
noiseless, this reduces to the zero-error capacity of the channel, assisted by
the various classes of non-signalling correlations. When the resource channel
is noiseless, it results in the "reverse" problem of simulating a noisy channel
exactly by a noiseless one, assisted by correlations. In both cases, 'one-shot'
separations between the power of the different assisting correlations are
exhibited. The most striking result of this kind is that entanglement can
assist in zero-error communication, in stark contrast to the standard setting
of communicaton with asymptotically vanishing error in which entanglement does
not help at all. In the asymptotic case, shared randomness is shown to be just
as powerful as arbitrary non-signalling correlations for noisy channel
simulation, which is not true for the asymptotic zero-error capacities. For
assistance by arbitrary non-signalling correlations, linear programming
formulas for capacity and simulation are derived, the former being equal (for
channels with non-zero unassisted capacity) to the feedback-assisted zero-error
capacity originally derived by Shannon to upper bound the unassisted zero-error
capacity. Finally, a kind of reversibility between non-signalling-assisted
capacity and simulation is observed, mirroring the famous "reverse Shannon
theorem".Comment: 18 pages, 1 figure. Small changes to text in v2. Removed an
unnecessarily strong requirement in the premise of Theorem 1
Improving zero-error classical communication with entanglement
Given one or more uses of a classical channel, only a certain number of
messages can be transmitted with zero probability of error. The study of this
number and its asymptotic behaviour constitutes the field of classical
zero-error information theory, the quantum generalisation of which has started
to develop recently. We show that, given a single use of certain classical
channels, entangled states of a system shared by the sender and receiver can be
used to increase the number of (classical) messages which can be sent with no
chance of error. In particular, we show how to construct such a channel based
on any proof of the Bell-Kochen-Specker theorem. This is a new example of the
use of quantum effects to improve the performance of a classical task. We
investigate the connection between this phenomenon and that of
``pseudo-telepathy'' games. The use of generalised non-signalling correlations
to assist in this task is also considered. In this case, a particularly elegant
theory results and, remarkably, it is sometimes possible to transmit
information with zero-error using a channel with no unassisted zero-error
capacity.Comment: 6 pages, 2 figures. Version 2 is the same as the journal version plus
figure 1 and the non-signalling box exampl
Quantum enigma machines and the locking capacity of a quantum channel
The locking effect is a phenomenon which is unique to quantum information
theory and represents one of the strongest separations between the classical
and quantum theories of information. The Fawzi-Hayden-Sen (FHS) locking
protocol harnesses this effect in a cryptographic context, whereby one party
can encode n bits into n qubits while using only a constant-size secret key.
The encoded message is then secure against any measurement that an eavesdropper
could perform in an attempt to recover the message, but the protocol does not
necessarily meet the composability requirements needed in quantum key
distribution applications. In any case, the locking effect represents an
extreme violation of Shannon's classical theorem, which states that
information-theoretic security holds in the classical case if and only if the
secret key is the same size as the message. Given this intriguing phenomenon,
it is of practical interest to study the effect in the presence of noise, which
can occur in the systems of both the legitimate receiver and the eavesdropper.
This paper formally defines the locking capacity of a quantum channel as the
maximum amount of locked information that can be reliably transmitted to a
legitimate receiver by exploiting many independent uses of a quantum channel
and an amount of secret key sublinear in the number of channel uses. We provide
general operational bounds on the locking capacity in terms of other well-known
capacities from quantum Shannon theory. We also study the important case of
bosonic channels, finding limitations on these channels' locking capacity when
coherent-state encodings are employed and particular locking protocols for
these channels that might be physically implementable.Comment: 37 page
Multi-party zero-error classical channel coding with entanglement
We study the effects of quantum entanglement on the performance of two
classical zero-error communication tasks among multiple parties. Both tasks are
generalizations of the two-party zero-error channel-coding problem, where a
sender and a receiver want to perfectly communicate messages through a one-way
classical noisy channel. If the two parties are allowed to share entanglement,
there are several positive results that show the existence of channels for
which they can communicate strictly more than what they could do with classical
resources. In the first task, one sender wants to communicate a common message
to multiple receivers. We show that if the number of receivers is greater than
a certain threshold then entanglement does not allow for an improvement in the
communication for any finite number of uses of the channel. On the other hand,
when the number of receivers is fixed, we exhibit a class of channels for which
entanglement gives an advantage. The second problem we consider features
multiple collaborating senders and one receiver. Classically, cooperation among
the senders might allow them to communicate on average more messages than the
sum of their individual possibilities. We show that whenever a channel allows
single-sender entanglement-assisted advantage, then the gain extends also to
the multi-sender case. Furthermore, we show that entanglement allows for a
peculiar amplification of information which cannot happen classically, for a
fixed number of uses of a channel with multiple senders.Comment: Some proofs have been modifie
Quantum Reverse Shannon Theorem
Dual to the usual noisy channel coding problem, where a noisy (classical or
quantum) channel is used to simulate a noiseless one, reverse Shannon theorems
concern the use of noiseless channels to simulate noisy ones, and more
generally the use of one noisy channel to simulate another. For channels of
nonzero capacity, this simulation is always possible, but for it to be
efficient, auxiliary resources of the proper kind and amount are generally
required. In the classical case, shared randomness between sender and receiver
is a sufficient auxiliary resource, regardless of the nature of the source, but
in the quantum case the requisite auxiliary resources for efficient simulation
depend on both the channel being simulated, and the source from which the
channel inputs are coming. For tensor power sources (the quantum generalization
of classical IID sources), entanglement in the form of standard ebits
(maximally entangled pairs of qubits) is sufficient, but for general sources,
which may be arbitrarily correlated or entangled across channel inputs,
additional resources, such as entanglement-embezzling states or backward
communication, are generally needed. Combining existing and new results, we
establish the amounts of communication and auxiliary resources needed in both
the classical and quantum cases, the tradeoffs among them, and the loss of
simulation efficiency when auxiliary resources are absent or insufficient. In
particular we find a new single-letter expression for the excess forward
communication cost of coherent feedback simulations of quantum channels (i.e.
simulations in which the sender retains what would escape into the environment
in an ordinary simulation), on non-tensor-power sources in the presence of
unlimited ebits but no other auxiliary resource. Our results on tensor power
sources establish a strong converse to the entanglement-assisted capacity
theorem.Comment: 35 pages, to appear in IEEE-IT. v2 has a fixed proof of the Clueless
Eve result, a new single-letter formula for the "spread deficit", better
error scaling, and an improved strong converse. v3 and v4 each make small
improvements to the presentation and add references. v5 fixes broken
reference
No-signalling-assisted zero-error capacity of quantum channels and an information theoretic interpretation of the lovász number
© 2015 IEEE. We study the one-shot zero-error classical capacity of a quantum channel assisted by quantum no-signalling correlations, and the reverse problem of exact simulation of a prescribed channel by a noiseless classical one. Quantum no-signalling correlations are viewed as two-input and twooutput completely positive and trace preserving maps with linear constraints enforcing that the device cannot signal. Both problems lead to simple semidefinite programmes (SDPs) that depend only on the Choi-Kraus (operator) space of the channel. In particular, we show that the zero-error classical simulation cost is precisely the conditional min-entropy of the Choi-Jamiozkowski matrix of the given channel. The zero-error classical capacity is given by a similar-looking but different SDP; the asymptotic zero-error classical capacity is the regularization of this SDP, and in general, we do not know of any simple form. Interestingly, however, for the class of classical-quantum channels, we show that the asymptotic capacity is given by a much simpler SDP, which coincides with a semidefinite generalization of the fractional packing number suggested earlier by Aram Harrow. This finally results in an operational interpretation of the celebrated Lovász θ function of a graph as the zero-error classical capacity of the graph assisted by quantum no-signalling correlations, the first information theoretic interpretation of the Lovász number
Entanglement can increase asymptotic rates of zero-error classical communication over classical channels
It is known that the number of different classical messages which can be
communicated with a single use of a classical channel with zero probability of
decoding error can sometimes be increased by using entanglement shared between
sender and receiver. It has been an open question to determine whether
entanglement can ever increase the zero-error communication rates achievable in
the limit of many channel uses. In this paper we show, by explicit examples,
that entanglement can indeed increase asymptotic zero-error capacity, even to
the extent that it is equal to the normal capacity of the channel.
Interestingly, our examples are based on the exceptional simple root systems E7
and E8.Comment: 14 pages, 2 figur
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
Converse bounds for private communication over quantum channels
This paper establishes several converse bounds on the private transmission
capabilities of a quantum channel. The main conceptual development builds
firmly on the notion of a private state, which is a powerful, uniquely quantum
method for simplifying the tripartite picture of privacy involving local
operations and public classical communication to a bipartite picture of quantum
privacy involving local operations and classical communication. This approach
has previously led to some of the strongest upper bounds on secret key rates,
including the squashed entanglement and the relative entropy of entanglement.
Here we use this approach along with a "privacy test" to establish a general
meta-converse bound for private communication, which has a number of
applications. The meta-converse allows for proving that any quantum channel's
relative entropy of entanglement is a strong converse rate for private
communication. For covariant channels, the meta-converse also leads to
second-order expansions of relative entropy of entanglement bounds for private
communication rates. For such channels, the bounds also apply to the private
communication setting in which the sender and receiver are assisted by
unlimited public classical communication, and as such, they are relevant for
establishing various converse bounds for quantum key distribution protocols
conducted over these channels. We find precise characterizations for several
channels of interest and apply the methods to establish several converse bounds
on the private transmission capabilities of all phase-insensitive bosonic
channels.Comment: v3: 53 pages, 3 figures, final version accepted for publication in
IEEE Transactions on Information Theor
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