91 research outputs found
On the tensor convolution and the quantum separability problem
We consider the problem of separability: decide whether a Hermitian operator
on a finite dimensional Hilbert tensor product is separable or entangled. We
show that the tensor convolution defined for certain mappings on an almost
arbitrary locally compact abelian group, give rise to formulation of an
equivalent problem to the separability one.Comment: 13 pages, two sections adde
Detection of Multiparticle Entanglement: Quantifying the Search for Symmetric Extensions
We provide quantitative bounds on the characterisation of multiparticle
separable states by states that have locally symmetric extensions. The bounds
are derived from two-particle bounds and relate to recent studies on quantum
versions of de Finetti's theorem. We discuss algorithmic applications of our
results, in particular a quasipolynomial-time algorithm to decide whether a
multiparticle quantum state is separable or entangled (for constant number of
particles and constant error in the LOCC or Frobenius norm). Our results
provide a theoretical justification for the use of the Search for Symmetric
Extensions as a practical test for multiparticle entanglement.Comment: 5 pages, 1 figur
Quantifying nonclassicality: global impact of local unitary evolutions
We show that only those composite quantum systems possessing nonvanishing
quantum correlations have the property that any nontrivial local unitary
evolution changes their global state. We derive the exact relation between the
global state change induced by local unitary evolutions and the amount of
quantum correlations. We prove that the minimal change coincides with the
geometric measure of discord (defined via the Hilbert- Schmidt norm), thus
providing the latter with an operational interpretation in terms of the
capability of a local unitary dynamics to modify a global state. We establish
that two-qubit Werner states are maximally quantum correlated, and are thus the
ones that maximize this type of global quantum effect. Finally, we show that
similar results hold when replacing the Hilbert-Schmidt norm with the trace
norm.Comment: 5 pages, 1 figure. To appear in Physical Review
Geometric discord and Measurement-induced nonlocality for well known bound entangled states
We employ geometric discord and measurement induced nonlocality to quantify
non classical correlations of some well-known bipartite bound entangled states,
namely the two families of Horodecki's (, and
dimensional) bound entangled states and that of Bennett etal's in
dimension. In most of the cases our results are analytic and both
the measures attain relatively small value. The amount of quantumness in the
bound entangled state of Benatti etal and the state
having the same matrix representation (in computational basis) is same.
Coincidently, the Werner and isotropic states also exhibit the
same property, when seen as dimensional states.Comment: V2: Title changed, one more state added; 11 pages (single column), 2
figures, accepted in Quantum Information Processin
Approximating the Set of Separable States Using the Positive Partial Transpose Test
The positive partial transpose test is one of the main criteria for detecting
entanglement, and the set of states with positive partial transpose is
considered as an approximation of the set of separable states. However, we do
not know to what extent this criterion, as well as the approximation, are
efficient. In this paper, we show that the positive partial transpose test
gives no bound on the distance of a density matrix from separable states. More
precisely, we prove that, as the dimension of the space tends to infinity, the
maximum trace distance of a positive partial transpose state from separable
states tends to 1. Using similar techniques, we show that the same result holds
for other well-known separability criteria such as reduction criterion,
majorization criterion and symmetric extension criterion. We also bring an
evidence that the sets of positive partial transpose states and separable
states have totally different shapes.Comment: 12 pages, published versio
Measurement-induced nonlocality based on the relative entropy
We quantify the measurement-induced nonlocality [Luo and Fu, Phys. Rev. Lett.
106, 120401 (2011)] from the perspective of the relative entropy. This
quantification leads to an operational interpretation for the
measurementinduced nonlocality, namely, it is the maximal entropy increase
after the locally invariant measurements. The relative entropy of nonlocality
is upper bounded by the entropy of the measured subsystem. We establish a
relationship between the relative entropy of nonlocality and the geometric
nonlocality based on the Hilbert- Schmidt norm, and show that it is equal to
the maximal distillable entanglement. Several trade-off relations are obtained
for tripartite pure states. We also give explicit expressions for the relative
entropy of nonlocality for Bell-diagonal states.Comment: 5 pages, 1 figures, version accepted Phys. Rev. A, PHYSICAL REVIEW A
85, 042325 (2012
Limitations to sharing entanglement
We discuss limitations to sharing entanglement known as monogamy of
entanglement. Our pedagogical approach commences with simple examples of
limited entanglement sharing for pure three-qubit states and progresses to the
more general case of mixed-state monogamy relations with multiple qudits.Comment: Pedagogical review article on monogamy and polygamy of entanglemen
On quantum mean-field models and their quantum annealing
This paper deals with fully-connected mean-field models of quantum spins with
p-body ferromagnetic interactions and a transverse field. For p=2 this
corresponds to the quantum Curie-Weiss model (a special case of the
Lipkin-Meshkov-Glick model) which exhibits a second-order phase transition,
while for p>2 the transition is first order. We provide a refined analytical
description both of the static and of the dynamic properties of these models.
In particular we obtain analytically the exponential rate of decay of the gap
at the first-order transition. We also study the slow annealing from the pure
transverse field to the pure ferromagnet (and vice versa) and discuss the
effect of the first-order transition and of the spinodal limit of metastability
on the residual excitation energy, both for finite and exponentially divergent
annealing times. In the quantum computation perspective this quantity would
assess the efficiency of the quantum adiabatic procedure as an approximation
algorithm.Comment: 44 pages, 23 figure
Faithful Squashed Entanglement
Squashed entanglement is a measure for the entanglement of bipartite quantum
states. In this paper we present a lower bound for squashed entanglement in
terms of a distance to the set of separable states. This implies that squashed
entanglement is faithful, that is, strictly positive if and only if the state
is entangled. We derive the bound on squashed entanglement from a bound on
quantum conditional mutual information, which is used to define squashed
entanglement and corresponds to the amount by which strong subadditivity of von
Neumann entropy fails to be saturated. Our result therefore sheds light on the
structure of states that almost satisfy strong subadditivity with equality. The
proof is based on two recent results from quantum information theory: the
operational interpretation of the quantum mutual information as the optimal
rate for state redistribution and the interpretation of the regularised
relative entropy of entanglement as an error exponent in hypothesis testing.
The distance to the set of separable states is measured by the one-way LOCC
norm, an operationally-motivated norm giving the optimal probability of
distinguishing two bipartite quantum states, each shared by two parties, using
any protocol formed by local quantum operations and one-directional classical
communication between the parties. A similar result for the Frobenius or
Euclidean norm follows immediately. The result has two applications in
complexity theory. The first is a quasipolynomial-time algorithm solving the
weak membership problem for the set of separable states in one-way LOCC or
Euclidean norm. The second concerns quantum Merlin-Arthur games. Here we show
that multiple provers are not more powerful than a single prover when the
verifier is restricted to one-way LOCC operations thereby providing a new
characterisation of the complexity class QMA.Comment: 24 pages, 1 figure, 1 table. Due to an error in the published
version, claims have been weakened from the LOCC norm to the one-way LOCC
nor
Improved hardness results for the guided local Hamiltonian problem
Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with 6-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to 1/2 with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry
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