636 research outputs found
Entanglement cost and quantum channel simulation
This paper proposes a revised definition for the entanglement cost of a
quantum channel . In particular, it is defined here to be the
smallest rate at which entanglement is required, in addition to free classical
communication, in order to simulate calls to , such that the
most general discriminator cannot distinguish the calls to
from the simulation. The most general discriminator is one who tests the
channels in a sequential manner, one after the other, and this discriminator is
known as a quantum tester [Chiribella et al., Phys. Rev. Lett., 101, 060401
(2008)] or one who is implementing a quantum co-strategy [Gutoski et al., Symp.
Th. Comp., 565 (2007)]. As such, the proposed revised definition of
entanglement cost of a quantum channel leads to a rate that cannot be smaller
than the previous notion of a channel's entanglement cost [Berta et al., IEEE
Trans. Inf. Theory, 59, 6779 (2013)], in which the discriminator is limited to
distinguishing parallel uses of the channel from the simulation. Under this
revised notion, I prove that the entanglement cost of certain
teleportation-simulable channels is equal to the entanglement cost of their
underlying resource states. Then I find single-letter formulas for the
entanglement cost of some fundamental channel models, including dephasing,
erasure, three-dimensional Werner--Holevo channels, epolarizing channels
(complements of depolarizing channels), as well as single-mode pure-loss and
pure-amplifier bosonic Gaussian channels. These examples demonstrate that the
resource theory of entanglement for quantum channels is not reversible.
Finally, I discuss how to generalize the basic notions to arbitrary resource
theories.Comment: 28 pages, 7 figure
Sequential decoding of a general classical-quantum channel
Since a quantum measurement generally disturbs the state of a quantum system,
one might think that it should not be possible for a sender and receiver to
communicate reliably when the receiver performs a large number of sequential
measurements to determine the message of the sender. We show here that this
intuition is not true, by demonstrating that a sequential decoding strategy
works well even in the most general "one-shot" regime, where we are given a
single instance of a channel and wish to determine the maximal number of bits
that can be communicated up to a small failure probability. This result follows
by generalizing a non-commutative union bound to apply for a sequence of
general measurements. We also demonstrate two ways in which a receiver can
recover a state close to the original state after it has been decoded by a
sequence of measurements that each succeed with high probability. The second of
these methods will be useful in realizing an efficient decoder for fully
quantum polar codes, should a method ever be found to realize an efficient
decoder for classical-quantum polar codes.Comment: 12 pages; accepted for publication in the Proceedings of the Royal
Society
Strong and uniform convergence in the teleportation simulation of bosonic Gaussian channels
In the literature on the continuous-variable bosonic teleportation protocol
due to [Braunstein and Kimble, Phys. Rev. Lett., 80(4):869, 1998], it is often
loosely stated that this protocol converges to a perfect teleportation of an
input state in the limit of ideal squeezing and ideal detection, but the exact
form of this convergence is typically not clarified. In this paper, I
explicitly clarify that the convergence is in the strong sense, and not the
uniform sense, and furthermore, that the convergence occurs for any input state
to the protocol, including the infinite-energy Basel states defined and
discussed here. I also prove, in contrast to the above result, that the
teleportation simulations of pure-loss, thermal, pure-amplifier, amplifier, and
additive-noise channels converge both strongly and uniformly to the original
channels, in the limit of ideal squeezing and detection for the simulations.
For these channels, I give explicit uniform bounds on the accuracy of their
teleportation simulations. I then extend these uniform convergence results to
particular multi-mode bosonic Gaussian channels. These convergence statements
have important implications for mathematical proofs that make use of the
teleportation simulation of bosonic Gaussian channels, some of which have to do
with bounding their non-asymptotic secret-key-agreement capacities. As a
byproduct of the discussion given here, I confirm the correctness of the proof
of such bounds from my joint work with Berta and Tomamichel from [Wilde,
Tomamichel, Berta, IEEE Trans. Inf. Theory 63(3):1792, March 2017].
Furthermore, I show that it is not necessary to invoke the energy-constrained
diamond distance in order to confirm the correctness of this proof.Comment: 19 pages, 3 figure
Recoverability in quantum information theory
The fact that the quantum relative entropy is non-increasing with respect to
quantum physical evolutions lies at the core of many optimality theorems in
quantum information theory and has applications in other areas of physics. In
this work, we establish improvements of this entropy inequality in the form of
physically meaningful remainder terms. One of the main results can be
summarized informally as follows: if the decrease in quantum relative entropy
between two quantum states after a quantum physical evolution is relatively
small, then it is possible to perform a recovery operation, such that one can
perfectly recover one state while approximately recovering the other. This can
be interpreted as quantifying how well one can reverse a quantum physical
evolution. Our proof method is elementary, relying on the method of complex
interpolation, basic linear algebra, and the recently introduced Renyi
generalization of a relative entropy difference. The theorem has a number of
applications in quantum information theory, which have to do with providing
physically meaningful improvements to many known entropy inequalities.Comment: v5: 26 pages, generalized lower bounds to apply when supp(rho) is
contained in supp(sigma
Recoverability for Holevo's just-as-good fidelity
Holevo's just-as-good fidelity is a similarity measure for quantum states
that has found several applications. One of its critical properties is that it
obeys a data processing inequality: the measure does not decrease under the
action of a quantum channel on the underlying states. In this paper, I prove a
refinement of this data processing inequality that includes an additional term
related to recoverability. That is, if the increase in the measure is small
after the action of a partial trace, then one of the states can be nearly
recovered by the Petz recovery channel, while the other state is perfectly
recovered by the same channel. The refinement is given in terms of the trace
distance of one of the states to its recovered version and also depends on the
minimum eigenvalue of the other state. As such, the refinement is universal, in
the sense that the recovery channel depends only on one of the states, and it
is explicit, given by the Petz recovery channel. The appendix contains a
generalization of the aforementioned result to arbitrary quantum channels.Comment: 6 pages, submission to ISIT 201
Quantum reading capacity: General definition and bounds
Quantum reading refers to the task of reading out classical information
stored in a read-only memory device. In any such protocol, the transmitter and
receiver are in the same physical location, and the goal of such a protocol is
to use these devices (modeled by independent quantum channels), coupled with a
quantum strategy, to read out as much information as possible from a memory
device, such as a CD or DVD. As a consequence of the physical setup of quantum
reading, the most natural and general definition for quantum reading capacity
should allow for an adaptive operation after each call to the channel, and this
is how we define quantum reading capacity in this paper. We also establish
several bounds on quantum reading capacity, and we introduce an
environment-parametrized memory cell with associated environment states,
delivering second-order and strong converse bounds for its quantum reading
capacity. We calculate the quantum reading capacities for some exemplary memory
cells, including a thermal memory cell, a qudit erasure memory cell, and a
qudit depolarizing memory cell. We finally provide an explicit example to
illustrate the advantage of using an adaptive strategy in the context of
zero-error quantum reading capacity.Comment: v3: 17 pages, 2 figures, final version published in IEEE Transactions
on Information Theor
-Logarithmic negativity
The logarithmic negativity of a bipartite quantum state is a widely employed
entanglement measure in quantum information theory, due to the fact that it is
easy to compute and serves as an upper bound on distillable entanglement. More
recently, the -entanglement of a bipartite state was shown to be the
first entanglement measure that is both easily computable and has a precise
information-theoretic meaning, being equal to the exact entanglement cost of a
bipartite quantum state when the free operations are those that completely
preserve the positivity of the partial transpose [Wang and Wilde, Phys. Rev.
Lett. 125(4):040502, July 2020]. In this paper, we provide a non-trivial link
between these two entanglement measures, by showing that they are the extremes
of an ordered family of -logarithmic negativity entanglement measures,
each of which is identified by a parameter . In this
family, the original logarithmic negativity is recovered as the smallest with
, and the -entanglement is recovered as the largest with
. We prove that the -logarithmic negativity satisfies
the following properties: entanglement monotone, normalization, faithfulness,
and subadditivity. We also prove that it is neither convex nor monogamous.
Finally, we define the -logarithmic negativity of a quantum channel as
a generalization of the notion for quantum states, and we show how to
generalize many of the concepts to arbitrary resource theories.Comment: v3: 15 pages, accepted for publication in Physical Review
Exact entanglement cost of quantum states and channels under PPT-preserving operations
This paper establishes single-letter formulas for the exact entanglement cost
of generating bipartite quantum states and simulating quantum channels under
free quantum operations that completely preserve positivity of the partial
transpose (PPT). First, we establish that the exact entanglement cost of any
bipartite quantum state under PPT-preserving operations is given by a
single-letter formula, here called the -entanglement of a quantum
state. This formula is calculable by a semidefinite program, thus allowing for
an efficiently computable solution for general quantum states. Notably, this is
the first time that an entanglement measure for general bipartite states has
been proven not only to possess a direct operational meaning but also to be
efficiently computable, thus solving a question that has remained open since
the inception of entanglement theory over two decades ago. Next, we introduce
and solve the exact entanglement cost for simulating quantum channels in both
the parallel and sequential settings, along with the assistance of free
PPT-preserving operations. The entanglement cost in both cases is given by the
same single-letter formula and is equal to the largest -entanglement
that can be shared by the sender and receiver of the channel. It is also
efficiently computable by a semidefinite program.Comment: 54 pages, 8 figures; comments are welcome
Characterizing the performance of continuous-variable Gaussian quantum gates
The required set of operations for universal continuous-variable quantum
computation can be divided into two primary categories: Gaussian and
non-Gaussian operations. Furthermore, any Gaussian operation can be decomposed
as a sequence of phase-space displacements and symplectic transformations.
Although Gaussian operations are ubiquitous in quantum optics, their
experimental realizations generally are approximations of the ideal Gaussian
unitaries. In this work, we study different performance criteria to analyze how
well these experimental approximations simulate the ideal Gaussian unitaries.
In particular, we find that none of these experimental approximations converge
uniformly to the ideal Gaussian unitaries. However, convergence occurs in the
strong sense, or if the discrimination strategy is energy bounded, then the
convergence is uniform in the Shirokov-Winter energy-constrained diamond norm
and we give explicit bounds in this latter case. We indicate how these
energy-constrained bounds can be used for experimental implementations of these
Gaussian unitaries in order to achieve any desired accuracy.Comment: v3: 26 pages, 10 figures, final version accepted for publication in
Physical Review Researc
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