469 research outputs found
The random link approximation for the Euclidean traveling salesman problem
The traveling salesman problem (TSP) consists of finding the length of the
shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where
the cities are distributed randomly and independently in a d-dimensional unit
hypercube. Working with periodic boundary conditions and inspired by a
remarkable universality in the kth nearest neighbor distribution, we find for
the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with
beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive
analytical predictions for these quantities using the random link
approximation, where the lengths between cities are taken as independent random
variables. From the ``cavity'' equations developed by Krauth, Mezard and
Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3,
numerical results show that the random link approximation is a good one, with a
discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we
argue that the approximation is exact up to O(1/d^2) and give a conjecture for
beta_E(d), in terms of a power series in 1/d, specifying both leading and
subleading coefficients.Comment: 29 pages, 6 figures; formatting and typos correcte
Phase-Conjugated Inputs Quantum Cloning Machines
A quantum cloning machine is introduced that yields identical optimal
clones from replicas of a coherent state and replicas of its phase
conjugate. It also optimally produces phase-conjugated clones at no
cost. For well chosen input asymmetries , this machine is shown to
provide better cloning fidelities than the standard cloner. The
special cases of the optimal balanced cloner () and the optimal
measurement () are investigated.Comment: 4 pages (RevTex), 2 figure
The optimal cloning of quantum coherent states is non-Gaussian
We consider the optimal cloning of quantum coherent states with single-clone
and joint fidelity as figures of merit. Both optimal fidelities are attained
for phase space translation covariant cloners. Remarkably, the joint fidelity
is maximized by a Gaussian cloner, whereas the single-clone fidelity can be
enhanced by non-Gaussian operations: a symmetric non-Gaussian 1-to-2 cloner can
achieve a single-clone fidelity of approximately 0.6826, perceivably higher
than the optimal fidelity of 2/3 in a Gaussian setting. This optimal cloner can
be realized by means of an optical parametric amplifier supplemented with a
particular source of non-Gaussian bimodal states. Finally, we show that the
single-clone fidelity of the optimal 1-to-infinity cloner, corresponding to a
measure-and-prepare scheme, cannot exceed 1/2. This value is achieved by a
Gaussian scheme and cannot be surpassed even with supplemental bound entangled
states.Comment: 4 pages, 2 figures, revtex; changed title, extended list of authors,
included optical implementation of optimal clone
Quantum Distribution of Gaussian Keys with Squeezed States
A continuous key distribution scheme is proposed that relies on a pair of
canonically conjugate quantum variables. It allows two remote parties to share
a secret Gaussian key by encoding it into one of the two quadrature components
of a single-mode electromagnetic field. The resulting quantum cryptographic
information vs disturbance tradeoff is investigated for an individual attack
based on the optimal continuous cloning machine. It is shown that the
information gained by the eavesdropper then simply equals the information lost
by the receiver.Comment: 5 pages, RevTe
Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator
We re-derive the Schr\"{o}dinger-Robertson uncertainty principle for the
position and momentum of a quantum particle. Our derivation does not directly
employ commutation relations, but works by reduction to an eigenvalue problem
related to the harmonic oscillator, which can then be further exploited to find
a larger class of constrained uncertainty relations. We derive an uncertainty
relation under the constraint of a fixed degree of Gaussianity and prove that,
remarkably, it is saturated by all eigenstates of the harmonic oscillator. This
goes beyond the common knowledge that the (Gaussian) ground state of the
harmonic oscillator saturates the uncertainty relation.Comment: 9 pages, 3 figure
Majorization relations and entanglement generation in a beam splitter
We prove that a beam splitter, one of the most common optical components, fulfills several classes of majorization relations, which govern the amount of quantum entanglement that it can generate. First, we show that the state resulting from k photons impinging on a beam splitter majorizes the corresponding state with any larger photon number k > k, implying that the entanglement monotonically grows with k. Then we examine parametric infinitesimal majorization relations as a function of the beam-splitter transmittance and find that there exists a parameter region where majorization is again fulfilled, implying a monotonic increase of entanglement by moving towards a balanced beam splitter. We also identify regions with a majorization default, where the output states become incomparable. In this latter situation, we find examples where catalysis may nevertheless be used in order to recover majorization. The catalyst states can be as simple as a path-entangled single-photon state or a two-mode squeezed vacuum state
Schmidt balls around the identity
Robustness measures as introduced by Vidal and Tarrach [PRA, 59, 141-155]
quantify the extent to which entangled states remain entangled under mixing.
Analogously, we introduce here the Schmidt robustness and the random Schmidt
robustness. The latter notion is closely related to the construction of Schmidt
balls around the identity. We analyse the situation for pure states and provide
non-trivial upper and lower bounds. Upper bounds to the random Schmidt-2
robustness allow us to construct a particularly simple distillability
criterion. We present two conjectures, the first one is related to the radius
of inner balls around the identity in the convex set of Schmidt number
n-states. We also conjecture a class of optimal Schmidt witnesses for pure
states.Comment: 7 pages, 1 figur
Scheme for the implementation of a universal quantum cloning machine via cavity-assisted atomic collisions in cavity QED
We propose a scheme to implement the universal quantum cloning
machine of Buzek et.al [Phys. Rev.A 54, 1844(1996)] in the context of cavity
QED. The scheme requires cavity-assisted collision processes between atoms,
which cross through nonresonant cavity fields in the vacuum states. The cavity
fields are only virtually excited to face the decoherence problem. That's why
the requirements on the cavity quality factor can be loosened.Comment: to appear in PR
Scaling Separability Criterion: Application To Gaussian States
We introduce examples of three- and four-mode entangled Gaussian mixed states
that are not detected by the scaling and Peres-Horodecki separability criteria.
The presented modification of the scaling criterion resolves this problem. Also
it is shown that the new criterion reproduces the main features of the scaling
pictures for different cases of entangled states, while the previous versions
lead to completely different outcomes. This property of the presented scheme is
evidence of its higher generality.Comment: 7 pages, 4 figure
Polarization state of a biphoton: quantum ternary logic
Polarization state of biphoton light generated via collinear
frequency-degenerate spontaneous parametric down-conversion is considered. A
biphoton is described by a three-component polarization vector, its arbitrary
transformations relating to the SU(3) group. A subset of such transformations,
available with retardation plates, is realized experimentally. In particular,
two independent orthogonally polarized beams of type-I biphotons are
transformed into a beam of type-II biphotons. Polarized biphotons are suggested
as ternary analogs of two-state quantum systems (qubits)
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