29 research outputs found
Transmitting Quantum Information Reliably across Various Quantum Channels
Transmitting quantum information across quantum channels is an important task. However quantum information is delicate, and is easily corrupted. We address the task of protecting quantum information from an information theoretic perspective -- we encode some message qudits into a quantum code, send the encoded quantum information across the noisy quantum channel, then recover the message qudits by decoding. In this dissertation, we discuss the coding problem from several perspectives.}
The noisy quantum channel is one of the central aspects of the quantum coding problem, and hence quantifying the noisy quantum channel from the physical model is an important problem.
We work with an explicit physical model -- a pair of initially decoupled quantum harmonic oscillators interacting with a spring-like coupling, where the bath oscillator is initially in a thermal-like state. In particular, we treat the completely positive and trace preserving map on the system as a quantum channel, and study the truncation of the channel by truncating its Kraus set. We thereby derive the matrix elements of the Choi-Jamiolkowski operator of the corresponding truncated channel, which are truncated transition amplitudes. Finally, we give a computable approximation for these truncated transition amplitudes with explicit error bounds, and perform a case study of the oscillators in the off-resonant and weakly-coupled regime numerically.
In the context of truncated noisy channels, we revisit the notion of approximate error correction of finite dimension codes. We derive a computationally simple lower bound on the worst case entanglement fidelity of a quantum code, when the truncated recovery map of Leung et. al. is rescaled. As an application, we apply our bound to construct a family of multi-error correcting amplitude damping codes that are permutation-invariant. This demonstrates an explicit example where the specific structure of the noisy channel allows code design out of the stabilizer formalism via purely algebraic means.
We study lower bounds on the quantum capacity of adversarial channels, where we restrict the selection of quantum codes to the set of concatenated quantum codes.
The adversarial channel is a quantum channel where an adversary corrupts a fixed fraction of qudits sent across a quantum channel in the most malicious way possible. The best known rates of communicating over adversarial channels are given by the quantum Gilbert-Varshamov (GV) bound, that is known to be attainable with random quantum codes. We generalize the classical result of Thommesen to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region.
We next study upper bounds on the quantum capacity of some low dimension quantum channels.
The quantum capacity of a quantum channel is the maximum rate at which quantum information can be transmitted reliably across it, given arbitrarily many uses of it. While it is known that random quantum codes can be used to attain the quantum capacity, the quantum capacity of many classes of channels is undetermined, even for channels of low input and output dimension. For example, depolarizing channels are
important quantum channels, but do not have tight numerical bounds.
We obtain upper bounds on the quantum capacity of some unital and non-unital channels
-- two-qubit Pauli channels, two-qubit depolarizing channels, two-qubit locally symmetric channels,
shifted qubit depolarizing channels, and shifted two-qubit Pauli channels --
using the coherent information of some degradable channels. We use the notion
of twirling quantum channels, and Smith and Smolin's method of constructing
degradable extensions of quantum channels extensively. The degradable channels we
introduce, study and use are two-qubit amplitude damping channels. Exploiting the
notion of covariant quantum channels, we give sufficient conditions for the quantum
capacity of a degradable channel to be the optimal value of a concave program with
linear constraints, and show that our two-qubit degradable amplitude damping channels have this property
Amortized entanglement of a quantum channel and approximately teleportationsimulable channels
This paper defines the amortized entanglement of a quantum channel as the largest difference in entanglement between the output and the input of the channel, where entanglement is quantified by an arbitrary entanglement measure. We prove that the amortized entanglement of a channel obeys several desirable properties, and we also consider special cases such as the amortized relative entropy of entanglement and the amortized Rains relative entropy. These latter quantities are shown to be single-letter upper bounds on the secret-key-agreement and PPT-assisted quantum capacities of a quantum channel, respectively. Of especial interest is a uniform continuity bound for these latter two special cases of amortized entanglement, in which the deviation between the amortized entanglement of two channels is bounded from above by a simple function of the diamond norm of their difference and the output dimension of the channels. We then define approximately teleportation- and positive-partial-transpose-simulable (PPT-simulable) channels as those that are close in diamond norm to a channel which is either exactly teleportationor PPT-simulable, respectively. These results then lead to single-letter upper bounds on the secret-key-agreement and PPT-assisted quantum capacities of channels that are approximately teleportation- or PPT-simulable, respectively. Finally, we generalize many of the concepts in the paper to the setting of general resource theories, defining the amortized resourcefulness of a channel and the notion of ν-freely-simulable channels, connecting these concepts in an operational way as well
Approximate Degradable Quantum Channels
Degradable quantum channels are an important class of completely positive
trace-preserving maps. Among other properties, they offer a single-letter
formula for the quantum and the private classical capacity and are
characterized by the fact that a complementary channel can be obtained from the
channel by applying a degrading channel. In this work we introduce the concept
of approximate degradable channels, which satisfy this condition up to some
finite . That is, there exists a degrading channel which upon
composition with the channel is -close in the diamond norm to the
complementary channel. We show that for any fixed channel the smallest such
can be efficiently determined via a semidefinite program.
Moreover, these approximate degradable channels also approximately inherit all
other properties of degradable channels. As an application, we derive improved
upper bounds to the quantum and private classical capacity for certain channels
of interest in quantum communication.Comment: v3: minor changes, published version. v2: 21 pages, 2 figures,
improved bounds on the capacity for approximate degradable channels based on
[arXiv:1507.07775], an author adde
The superadditivity effects of quantum capacity decrease with the dimension for qudit depolarizing channels
Quantum channel capacity is a fundamental quantity in order to understand how
good can quantum information be transmitted or corrected when subjected to
noise. However, it is generally not known how to compute such quantities, since
the quantum channel coherent information is not additive for all channels,
implying that it must be maximized over an unbounded number of channel uses.
This leads to the phenomenon known as superadditivity, which refers to the fact
that the regularized coherent information of channel uses exceeds one-shot
coherent information. In this article, we study how the gain in quantum
capacity of qudit depolarizing channels relates to the dimension of the systems
considered. We make use of an argument based on the no-cloning bound in order
to proof that the possible superadditive effects decrease as a function of the
dimension for such family of channels. In addition, we prove that the capacity
of the qudit depolarizing channel coincides with the coherent information when
. We conclude that when high dimensional qudits
experiencing depolarizing noise are considered, the coherent information of the
channel is not only an achievable rate but essentially the maximum possible
rate for any quantum block code.Comment: 7 pages, 2 figure
Trade-offs on number and phase shift resilience in bosonic quantum codes
Minimizing the number of particles used by a quantum code is helpful, because
every particle incurs a cost. One quantum error correction solution is to
encode quantum information into one or more bosonic modes. We revisit
rotation-invariant bosonic codes, which are supported on Fock states that are
gapped by an integer apart, and the gap imparts number shift resilience
to these codes. Intuitively, since phase operators and number shift operators
do not commute, one expects a trade-off between resilience to number-shift and
rotation errors. Here, we obtain results pertaining to the non-existence of
approximate quantum error correcting -gapped single-mode bosonic codes with
respect to Gaussian dephasing errors. We show that by using arbitrarily many
modes, -gapped multi-mode codes can yield good approximate quantum error
correction codes for any finite magnitude of Gaussian dephasing errors.Comment: 8 pages, 3 figures, 2 column
Resonant Multilevel Amplitude Damping Channels
We introduce a new set of quantum channels: resonant multilevel amplitude
damping (ReMAD) channels. Among other instances, they can describe energy
dissipation effects in multilevel atomic systems induced by the interaction
with a zero-temperature bosonic environment. At variance with the already known
class of multilevel amplitude damping (MAD) channels, this new class of maps
allows the presence of an environment unable to discriminate transitions with
identical energy gaps. After characterizing the algebra of their composition
rules, by analyzing the qutrit case, we show that this new set of channels can
exhibit degradability and antidegradability in vast regions of the allowed
parameter space. There we compute their quantum capacity and private classical
capacity. We show that these capacities can be computed exactly also in regions
of the parameter space where the channels aren't degradable nor antidegradable
Causal limit on quantum communication
The capacity of a channel is known to be equivalent to the highest rate at
which it can generate entanglement. Analogous to entanglement, the notion of a
causality measure characterises the temporal aspect of quantum correlations.
Despite holding an equally fundamental role in physics, temporal quantum
correlations have yet to find their operational significance in quantum
communication. Here we uncover a connection between quantum causality and
channel capacity. We show the amount of temporal correlations between two ends
of the noisy quantum channel, as quantified by a causality measure, implies a
general upper bound on its channel capacity. The expression of this new bound
is simpler to evaluate than most previously known bounds. We demonstrate the
utility of this bound by applying it to a class of shifted depolarizing
channels, which results in improvement over previously calculated bounds for
this class of channels.Comment: 9 pages, 3 figure
Quantum entanglement
All our former experience with application of quantum theory seems to say:
{\it what is predicted by quantum formalism must occur in laboratory}. But the
essence of quantum formalism - entanglement, recognized by Einstein, Podolsky,
Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a
new resource as real as energy.
This holistic property of compound quantum systems, which involves
nonclassical correlations between subsystems, is a potential for many quantum
processes, including ``canonical'' ones: quantum cryptography, quantum
teleportation and dense coding. However, it appeared that this new resource is
very complex and difficult to detect. Being usually fragile to environment, it
is robust against conceptual and mathematical tools, the task of which is to
decipher its rich structure.
This article reviews basic aspects of entanglement including its
characterization, detection, distillation and quantifying. In particular, the
authors discuss various manifestations of entanglement via Bell inequalities,
entropic inequalities, entanglement witnesses, quantum cryptography and point
out some interrelations. They also discuss a basic role of entanglement in
quantum communication within distant labs paradigm and stress some
peculiarities such as irreversibility of entanglement manipulations including
its extremal form - bound entanglement phenomenon. A basic role of entanglement
witnesses in detection of entanglement is emphasized.Comment: 110 pages, 3 figures, ReVTex4, Improved (slightly extended)
presentation, updated references, minor changes, submitted to Rev. Mod. Phys
Quantum capacity analysis of multi-level amplitude damping channels
The set of Multi-level Amplitude Damping (MAD) quantum channels is introduced
as a generalization of the standard qubit Amplitude Damping Channel to quantum
systems of finite dimension . In the special case of , by exploiting
degradability, data-processing inequalities, and channel isomorphism, we
compute the associated quantum and private classical capacities for a rather
wide class of maps, extending the set of solvable models known so far. We
proceed then to the evaluation of the entanglement assisted, quantum and
classical, capacities