27 research outputs found
Multipartite quantum correlations: symplectic and algebraic geometry approach
We review a geometric approach to classification and examination of quantum
correlations in composite systems. Since quantum information tasks are usually
achieved by manipulating spin and alike systems or, in general, systems with a
finite number of energy levels, classification problems are usually treated in
frames of linear algebra. We proposed to shift the attention to a geometric
description. Treating consistently quantum states as points of a projective
space rather than as vectors in a Hilbert space we were able to apply powerful
methods of differential, symplectic and algebraic geometry to attack the
problem of equivalence of states with respect to the strength of correlations,
or, in other words, to classify them from this point of view. Such
classifications are interpreted as identification of states with `the same
correlations properties' i.e. ones that can be used for the same information
purposes, or, from yet another point of view, states that can be mutually
transformed one to another by specific, experimentally accessible operations.
It is clear that the latter characterization answers the fundamental question
`what can be transformed into what \textit{via} available means?'. Exactly such
an interpretations, i.e, in terms of mutual transformability can be clearly
formulated in terms of actions of specific groups on the space of states and is
the starting point for the proposed methods.Comment: 29 pages, 9 figures, 2 tables, final form submitted to the journa
Convexity of momentum map, Morse index, and quantum entanglement
We analyze form the topological perspective the space of all SLOCC
(Stochastic Local Operations with Classical Communication) classes of pure
states for composite quantum systems. We do it for both distinguishable and
indistinguishable particles. In general, the topology of this space is rather
complicated as it is a non-Hausdorff space. Using geometric invariant theory
(GIT) and momentum map geometry we propose a way to divide the space of all
SLOCC classes into mathematically and physically meaningful families. Each
family consists of possibly many `asymptotically' equivalent SLOCC classes.
Moreover, each contains exactly one distinguished SLOCC class on which the
total variance (a well defined measure of entanglement) of the state Var[v]
attains maximum. We provide an algorithm for finding critical sets of Var[v],
which makes use of the convexity of the momentum map and allows classification
of such defined families of SLOCC classes. The number of families is in general
infinite. We introduce an additional refinement into finitely many groups of
families using the recent developments in the momentum map geometry known as
Ness stratification. We also discuss how to define it equivalently using the
convexity of the momentum map applied to SLOCC classes. Moreover, we note that
the Morse index at the critical set of the total variance of state has an
interpretation of number of non-SLOCC directions in which entanglement
increases and calculate it for several exemplary systems. Finally, we introduce
the SLOCC-invariant measure of entanglement as a square root of the total
variance of state at the critical point and explain its geometric meaning.Comment: 37 pages, 2 figures, changes in the manuscript structur
When is a pure state of three qubits determined by its single-particle reduced density matrices?
Using techniques from symplectic geometry, we prove that a pure state of
three qubits is up to local unitaries uniquely determined by its one-particle
reduced density matrices exactly when their ordered spectra belong to the
boundary of the, so called, Kirwan polytope. Otherwise, the states with given
reduced density matrices are parameterized, up to local unitary equivalence, by
two real variables. Given inevitable experimental imprecisions, this means that
already for three qubits a pure quantum state can never be reconstructed from
single-particle tomography. We moreover show that knowledge of the reduced
density matrices is always sufficient if one is given the additional promise
that the quantum state is not convertible to the Greenberger--Horne--Zeilinger
(GHZ) state by stochastic local operations and classical communication (SLOCC),
and discuss generalizations of our results to an arbitary number of qubits.Comment: 19 page
Photonic quantum information processing: a review
Photonic quantum technologies represent a promising platform for several
applications, ranging from long-distance communications to the simulation of
complex phenomena. Indeed, the advantages offered by single photons do make
them the candidate of choice for carrying quantum information in a broad
variety of areas with a versatile approach. Furthermore, recent technological
advances are now enabling first concrete applications of photonic quantum
information processing. The goal of this manuscript is to provide the reader
with a comprehensive review of the state of the art in this active field, with
a due balance between theoretical, experimental and technological results. When
more convenient, we will present significant achievements in tables or in
schematic figures, in order to convey a global perspective of the several
horizons that fall under the name of photonic quantum information.Comment: 36 pages, 6 figures, 634 references. Updated version with minor
changes and extended bibliograph
Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography
Fermion Sampling: a robust quantum computational advantage scheme using fermionic linear optics and magic input states
Random Bosonic States for Robust Quantum Metrology
We study how useful random states are for quantum metrology, i.e., whether they surpass the classical limits imposed on precision in the canonical phase sensing scenario. First, we prove that random pure states drawn from the Hilbert space of distinguishable particles typically do not lead to superclassical scaling of precision even when allowing for local unitary optimization. Conversely, we show that random pure states from the symmetric subspace typically achieve the optimal Heisenberg scaling without the need for local unitary optimization. Surprisingly, the Heisenberg scaling is observed for random isospectral states of arbitrarily low purity and preserved under loss of a fixed number of particles. Moreover, we prove that for pure states, a standard photon-counting interferometric measurement suffices to typically achieve resolution following the Heisenberg scaling for all values of the phase at the same time. Finally, we demonstrate that metrologically useful states can be prepared with short random optical circuits generated from three types of beam splitters and a single nonlinear (Kerr-like) transformation