795 research outputs found
Compressibility of Mixed-State Signals
We present a formula that determines the optimal number of qubits per message
that allows asymptotically faithful compression of the quantum information
carried by an ensemble of mixed states. The set of mixed states determines a
decomposition of the Hilbert space into the redundant part and the irreducible
part. After removing the redundancy, the optimal compression rate is shown to
be given by the von Neumann entropy of the reduced ensemble.Comment: 7 pages, no figur
Lower bound for the quantum capacity of a discrete memoryless quantum channel
We generalize the random coding argument of stabilizer codes and derive a
lower bound on the quantum capacity of an arbitrary discrete memoryless quantum
channel. For the depolarizing channel, our lower bound coincides with that
obtained by Bennett et al. We also slightly improve the quantum
Gilbert-Varshamov bound for general stabilizer codes, and establish an analogue
of the quantum Gilbert-Varshamov bound for linear stabilizer codes. Our proof
is restricted to the binary quantum channels, but its extension of to l-adic
channels is straightforward.Comment: 16 pages, REVTeX4. To appear in J. Math. Phys. A critical error in
fidelity calculation was corrected by using Hamada's result
(quant-ph/0112103). In the third version, we simplified formula and
derivation of the lower bound by proving p(Gamma)+q(Gamma)=1. In the second
version, we added an analogue of the quantum Gilbert-Varshamov bound for
linear stabilizer code
Generalization of entanglement to convex operational theories: Entanglement relative to a subspace of observables
We define what it means for a state in a convex cone of states on a space of
observables to be generalized-entangled relative to a subspace of the
observables, in a general ordered linear spaces framework for operational
theories. This extends the notion of ordinary entanglement in quantum
information theory to a much more general framework. Some important special
cases are described, in which the distinguished observables are subspaces of
the observables of a quantum system, leading to results like the identification
of generalized unentangled states with Lie-group-theoretic coherent states when
the special observables form an irreducibly represented Lie algebra. Some open
problems, including that of generalizing the semigroup of local operations with
classical communication to the convex cones setting, are discussed.Comment: 19 pages, to appear in proceedings of Quantum Structures VII, Int. J.
Theor. Phy
Verschraenkung versus Stosszahlansatz: Disappearance of the Thermodynamic Arrow in a High-Correlation Environment
The crucial role of ambient correlations in determining thermodynamic
behavior is established. A class of entangled states of two macroscopic systems
is constructed such that each component is in a state of thermal equilibrium at
a given temperature, and when the two are allowed to interact heat can flow
from the colder to the hotter system. A dilute gas model exhibiting this
behavior is presented. This reversal of the thermodynamic arrow is a
consequence of the entanglement between the two systems, a condition that is
opposite to molecular chaos and shown to be unlikely in a low-entropy
environment. By contrast, the second law is established by proving Clausius'
inequality in a low-entropy environment. These general results strongly support
the expectation, first expressed by Boltzmann and subsequently elaborated by
others, that the second law is an emergent phenomenon that requires a
low-entropy cosmological environment, one that can effectively function as an
ideal information sink.Comment: 4 pages, REVTeX
Trading quantum for classical resources in quantum data compression
We study the visible compression of a source E of pure quantum signal states,
or, more formally, the minimal resources per signal required to represent
arbitrarily long strings of signals with arbitrarily high fidelity, when the
compressor is given the identity of the input state sequence as classical
information. According to the quantum source coding theorem, the optimal
quantum rate is the von Neumann entropy S(E) qubits per signal.
We develop a refinement of this theorem in order to analyze the situation in
which the states are coded into classical and quantum bits that are quantified
separately. This leads to a trade--off curve Q(R), where Q(R) qubits per signal
is the optimal quantum rate for a given classical rate of R bits per signal.
Our main result is an explicit characterization of this trade--off function
by a simple formula in terms of only single signal, perfect fidelity encodings
of the source. We give a thorough discussion of many further mathematical
properties of our formula, including an analysis of its behavior for group
covariant sources and a generalization to sources with continuously
parameterized states. We also show that our result leads to a number of
corollaries characterizing the trade--off between information gain and state
disturbance for quantum sources. In addition, we indicate how our techniques
also provide a solution to the so--called remote state preparation problem.
Finally, we develop a probability--free version of our main result which may be
interpreted as an answer to the question: ``How many classical bits does a
qubit cost?'' This theorem provides a type of dual to Holevo's theorem, insofar
as the latter characterizes the cost of coding classical bits into qubits.Comment: 51 pages, 7 figure
Necessary Condition for the Quantum Adiabatic Approximation
A gapped quantum system that is adiabatically perturbed remains approximately
in its eigenstate after the evolution. We prove that, for constant gap, general
quantum processes that approximately prepare the final eigenstate require a
minimum time proportional to the ratio of the length of the eigenstate path to
the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We
also give a necessary condition for the adiabatic approximation that depends on
local properties of the path, which is appropriate when the gap varies.Comment: 5 pages, 1 figur
Three-dimensionality of space and the quantum bit: an information-theoretic approach
It is sometimes pointed out as a curiosity that the state space of quantum
two-level systems, i.e. the qubit, and actual physical space are both
three-dimensional and Euclidean. In this paper, we suggest an
information-theoretic analysis of this relationship, by proving a particular
mathematical result: suppose that physics takes place in d spatial dimensions,
and that some events happen probabilistically (not assuming quantum theory in
any way). Furthermore, suppose there are systems that carry "minimal amounts of
direction information", interacting via some continuous reversible time
evolution. We prove that this uniquely determines spatial dimension d=3 and
quantum theory on two qubits (including entanglement and unitary time
evolution), and that it allows observers to infer local spatial geometry from
probability measurements.Comment: 13 + 22 pages, 9 figures. v4: some clarifications, in particular in
Section V / Appendix C (added Example 39
Quantum communication using a bounded-size quantum reference frame
Typical quantum communication schemes are such that to achieve perfect
decoding the receiver must share a reference frame with the sender. Indeed, if
the receiver only possesses a bounded-size quantum token of the sender's
reference frame, then the decoding is imperfect, and we can describe this
effect as a noisy quantum channel. We seek here to characterize the performance
of such schemes, or equivalently, to determine the effective decoherence
induced by having a bounded-size reference frame. We assume that the token is
prepared in a special state that has particularly nice group-theoretic
properties and that is near-optimal for transmitting information about the
sender's frame. We present a decoding operation, which can be proven to be
near-optimal in this case, and we demonstrate that there are two distinct ways
of implementing it (corresponding to two distinct Kraus decompositions). In
one, the receiver measures the orientation of the reference frame token and
reorients the system appropriately. In the other, the receiver extracts the
encoded information from the virtual subsystems that describe the relational
degrees of freedom of the system and token. Finally, we provide explicit
characterizations of these decoding schemes when the system is a single qubit
and for three standard kinds of reference frame: a phase reference, a Cartesian
frame (representing an orthogonal triad of spatial directions), and a reference
direction (representing a single spatial direction).Comment: 17 pages, 1 figure, comments welcome; v2 published versio
A Lower Bound for Quantum Phase Estimation
We obtain a query lower bound for quantum algorithms solving the phase
estimation problem. Our analysis generalizes existing lower bound approaches to
the case where the oracle Q is given by controlled powers Q^p of Q, as it is
for example in Shor's order finding algorithm. In this setting we will prove a
log (1/epsilon) lower bound for the number of applications of Q^p1, Q^p2, ...
This bound is tight due to a matching upper bound. We obtain the lower bound
using a new technique based on frequency analysis.Comment: 7 pages, 1 figur
Discrete phase space based on finite fields
The original Wigner function provides a way of representing in phase space
the quantum states of systems with continuous degrees of freedom. Wigner
functions have also been developed for discrete quantum systems, one popular
version being defined on a 2N x 2N discrete phase space for a system with N
orthogonal states. Here we investigate an alternative class of discrete Wigner
functions, in which the field of real numbers that labels the axes of
continuous phase space is replaced by a finite field having N elements. There
exists such a field if and only if N is a power of a prime; so our formulation
can be applied directly only to systems for which the state-space dimension
takes such a value. Though this condition may seem limiting, we note that any
quantum computer based on qubits meets the condition and can thus be
accommodated within our scheme. The geometry of our N x N phase space also
leads naturally to a method of constructing a complete set of N+1 mutually
unbiased bases for the state space.Comment: 60 pages; minor corrections and additional references in v2 and v3;
improved historical introduction in v4; references to quantum error
correction in v5; v6 corrects the value quoted for the number of similarity
classes for N=
- …