247 research outputs found
Local asymptotic normality for qubit states
We consider n identically prepared qubits and study the asymptotic properties
of the joint state \rho^{\otimes n}. We show that for all individual states
\rho situated in a local neighborhood of size 1/\sqrt{n} of a fixed state
\rho^0, the joint state converges to a displaced thermal equilibrium state of a
quantum harmonic oscillator. The precise meaning of the convergence is that
there exist physical transformations T_{n} (trace preserving quantum channels)
which map the qubits states asymptotically close to their corresponding
oscillator state, uniformly over all states in the local neighborhood.
A few consequences of the main result are derived. We show that the optimal
joint measurement in the Bayesian set-up is also optimal within the pointwise
approach. Moreover, this measurement converges to the heterodyne measurement
which is the optimal joint measurement of position and momentum for the quantum
oscillator. A problem of local state discrimination is solved using local
asymptotic normality.Comment: 16 pages, 3 figures, published versio
Two quantum analogues of Fisher information from a large deviation viewpoint of quantum estimation
We discuss two quantum analogues of Fisher information, symmetric logarithmic
derivative (SLD) Fisher information and Kubo-Mori-Bogoljubov (KMB) Fisher
information from a large deviation viewpoint of quantum estimation and prove
that the former gives the true bound and the latter gives the bound of
consistent superefficient estimators. In another comparison, it is shown that
the difference between them is characterized by the change of the order of
limits.Comment: LaTeX with iopart.cls, iopart12.clo, iopams.st
Statistical properties of random density matrices
Statistical properties of ensembles of random density matrices are
investigated. We compute traces and von Neumann entropies averaged over
ensembles of random density matrices distributed according to the Bures
measure. The eigenvalues of the random density matrices are analyzed: we derive
the eigenvalue distribution for the Bures ensemble which is shown to be broader
then the quarter--circle distribution characteristic of the Hilbert--Schmidt
ensemble. For measures induced by partial tracing over the environment we
compute exactly the two-point eigenvalue correlation function.Comment: 8 revtex pages with one eps file included, ver. 2 - minor misprints
correcte
Hilbert--Schmidt volume of the set of mixed quantum states
We compute the volume of the convex N^2-1 dimensional set M_N of density
matrices of size N with respect to the Hilbert-Schmidt measure. The hyper--area
of the boundary of this set is also found and its ratio to the volume provides
an information about the complex structure of M_N. Similar investigations are
also performed for the smaller set of all real density matrices. As an
intermediate step we analyze volumes of the unitary and orthogonal groups and
of the flag manifolds.Comment: 13 revtex pages, ver 3: minor improvement
On the volume of the set of mixed entangled states II
The problem of of how many entangled or, respectively, separable states there
are in the set of all quantum states is investigated. We study to what extent
the choice of a measure in the space of density matrices describing
N--dimensional quantum systems affects the results obtained. We demonstrate
that the link between the purity of the mixed states and the probability of
entanglement is not sensitive to the measure chosen. Since the criterion of
partial transposition is not sufficient to distinguish all separable states for
N > 6, we develop an efficient algorithm to calculate numerically the
entanglement of formation of a given mixed quantum state, which allows us to
compute the volume of separable states for N=8 and to estimate the volume of
the bound entangled states in this case.Comment: 14 pages in Latex, Revtex + epsf; 7 figures in .ps included (one new
figure in the revised version, several minor changes
Local asymptotic normality for finite dimensional quantum systems
We extend our previous results on local asymptotic normality (LAN) for
qubits, to quantum systems of arbitrary finite dimension . LAN means that
the quantum statistical model consisting of identically prepared
-dimensional systems with joint state converges as
to a statistical model consisting of classical and quantum
Gaussian variables with fixed and known covariance matrix, and unknown means
related to the parameters of the density matrix . Remarkably, the limit
model splits into a product of a classical Gaussian with mean equal to the
diagonal parameters, and independent harmonic oscillators prepared in thermal
equilibrium states displaced by an amount proportional to the off-diagonal
elements.
As in the qubits case, LAN is the main ingredient in devising a general two
step adaptive procedure for the optimal estimation of completely unknown
-dimensional quantum states. This measurement strategy shall be described in
a forthcoming paper.Comment: 64 page
Bures volume of the set of mixed quantum states
We compute the volume of the N^2-1 dimensional set M_N of density matrices of
size N with respect to the Bures measure and show that it is equal to that of a
N^2-1 dimensional hyper-halfsphere of radius 1/2. For N=2 we obtain the volume
of the Uhlmann 3-D hemisphere, embedded in R^4. We find also the area of the
boundary of the set M_N and obtain analogous results for the smaller set of all
real density matrices. An explicit formula for the Bures-Hall normalization
constants is derived for an arbitrary N.Comment: 15 revtex pages, 2 figures in .eps; ver. 3, Eq. (4.19) correcte
Unknown Quantum States: The Quantum de Finetti Representation
We present an elementary proof of the quantum de Finetti representation
theorem, a quantum analogue of de Finetti's classical theorem on exchangeable
probability assignments. This contrasts with the original proof of Hudson and
Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced
mathematics and does not share the same potential for generalization. The
classical de Finetti theorem provides an operational definition of the concept
of an unknown probability in Bayesian probability theory, where probabilities
are taken to be degrees of belief instead of objective states of nature. The
quantum de Finetti theorem, in a closely analogous fashion, deals with
exchangeable density-operator assignments and provides an operational
definition of the concept of an ``unknown quantum state'' in quantum-state
tomography. This result is especially important for information-based
interpretations of quantum mechanics, where quantum states, like probabilities,
are taken to be states of knowledge rather than states of nature. We further
demonstrate that the theorem fails for real Hilbert spaces and discuss the
significance of this point.Comment: 30 pages, 2 figure
Quantum Iterated Function Systems
Iterated functions system (IFS) is defined by specifying a set of functions
in a classical phase space, which act randomly on an initial point. In an
analogous way, we define a quantum iterated functions system (QIFS), where
functions act randomly with prescribed probabilities in the Hilbert space. In a
more general setting a QIFS consists of completely positive maps acting in the
space of density operators. We present exemplary classical IFSs, the invariant
measure of which exhibits fractal structure, and study properties of the
corresponding QIFSs and their invariant states.Comment: 12 pages, 1 figure include
A priori probability that a qubit-qutrit pair is separable
We extend to arbitrarily coupled pairs of qubits (two-state quantum systems)
and qutrits (three-state quantum systems) our earlier study (quant-ph/0207181),
which was concerned with the simplest instance of entangled quantum systems,
pairs of qubits. As in that analysis -- again on the basis of numerical
(quasi-Monte Carlo) integration results, but now in a still higher-dimensional
space (35-d vs. 15-d) -- we examine a conjecture that the Bures/SD (statistical
distinguishability) probability that arbitrarily paired qubits and qutrits are
separable (unentangled) has a simple exact value, u/(v Pi^3)= >.00124706, where
u = 2^20 3^3 5 7 and v = 19 23 29 31 37 41 43 (the product of consecutive
primes). This is considerably less than the conjectured value of the Bures/SD
probability, 8/(11 Pi^2) = 0736881, in the qubit-qubit case. Both of these
conjectures, in turn, rely upon ones to the effect that the SD volumes of
separable states assume certain remarkable forms, involving "primorial"
numbers. We also estimate the SD area of the boundary of separable qubit-qutrit
states, and provide preliminary calculations of the Bures/SD probability of
separability in the general qubit-qubit-qubit and qutrit-qutrit cases.Comment: 9 pages, 3 figures, 2 tables, LaTeX, we utilize recent exact
computations of Sommers and Zyczkowski (quant-ph/0304041) of "the Bures
volume of mixed quantum states" to refine our conjecture
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