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S. Massar

Publications and source records attributed to S. Massar.

At least 55 records · Page 3Linked to original sources
arXiv1999

State estimation for large ensembles

We consider the problem of estimating the state of a large but finite number $N$ of identical quantum systems. In the limit of large $N$ the problem simplifies. In particular the only relevant measure of the quality of the estimation is the mean quadratic error matrix. Here we present a bound on the mean quadratic error which is a new quantum version of the Cramér-Rao inequality. This new bound expresses in a succinct way how in the quantum case one can trade information about one parameter for information about another parameter. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states--a striking example of non-locality without entanglement for mixed but not for pure states. Cramér-Rao bounds are generally derived under the assumption that the estimator is unbiased. We also prove that under additional regularity conditions our bound also holds for biased estimators. Finally we prove that when the unknown states belong to a 2 dimensional Hilbert space our quantum Cramér-Rao bound can always be attained and we provide an explicit measurement strategy that attains our bound. This therefore provides a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. For qubits in the same mixed state, this also provides an optimal estimation strategy if one only considers separable measurements.

quant-ph↗
arXiv1999

How much information can be obtained by a quantum measurement?

How much information about an unknown quantum state can be obtained by a measurement? We propose a model independent answer: the information obtained is equal to the minimum entropy of the outputs of the measurement, where the minimum is taken over all measurements which measure the same ``property'' of the state. This minimization is necessary because the measurement outcomes can be redundant, and this redundancy must be eliminated. We show that this minimum entropy is less or equal than the von Neumann entropy of the unknown states. That is a measurement can extract at most one meaningful bit from every qubit carried by the unknown states.

quant-ph↗
arXiv1999

Classical Teleportation of a Quantum Bit

Classical teleportation is defined as a scenario where the sender is given the classical description of an arbitrary quantum state while the receiver simulates any measurement on it. This scenario is shown to be achievable by transmitting only a few classical bits if the sender and receiver initially share local hidden variables. Specifically, a communication of 2.19 bits is sufficient on average for the classical teleportation of a qubit, when restricted to von Neumann measurements. The generalization to positive-operator-valued measurements is also discussed.

quant-ph↗
arXiv1999

Unitary and non-unitary evolution in quantum cosmology

We analyse when and why unitarity violations might occur in quantum cosmology restricted to minisuperspace. To this end we discuss in detail backscattering transitions between expanding and contracting solutions of the Wheeler-DeWitt equation. We first show that upon neglecting only backscattering, one obtains an intermediate regime in which matter evolves unitarily but which does not correspond to any Schrödinger equation in a given geometry since gravitational backreaction effects are taken into account at the quantum level. We then show that backscattering amplitudes are exponentially smaller than matter transition amplitudes. Both results follow from an adiabatic treatment valid for macroscopic universes. To understand how backscattering and the intermediate regime should be interpreted, we review the problem of electronic transitions induced by nuclear motion since it is mathematically very similar. In this problem, transition amplitudes are obtained from the conserved current. The same applies to quantum cosmology and shows that the unique consistent interpretation is based on the current when backscattering is neglected. We then review why, in a relativistic context, backscattering is interpreted as pair production whereas it is not in the non relativistic case. In each example the correct interpretation is obtained by coupling the system to an external quantum device. From the absence of such external systems in cosmology, we conclude that backscattering does not have a unique consistent interpretation in quantum cosmology.

gr-qc↗
arXiv1998

Optimal Quantum Clocks

A quantum clock must satisfy two basic constraints. The first is a bound on the time resolution of the clock given by the difference between its maximum and minimum energy eigenvalues. The second follows from Holevo's bound on how much classical information can be encoded in a quantum system. We show that asymptotically, as the dimension of the Hilbert space of the clock tends to infinity, both constraints can be satisfied simultaneously. The experimental realization of such an optimal quantum clock using trapped ions is discussed.

quant-ph↗
arXiv1998

Purifying noisy entanglement requires collective measurements

Known entanglement purification protocols for mixed states use collective measurements on several copies of the state in order to increase the entanglement of some of them. We address the question of whether it is possible to purify the entanglement of a state by processing each copy separately. While this is possible for pure states, we show that this is impossible, in general, for mixed states. The importance of this result both conceptually and for experimental realization of purification is discussed. We also give explicit invariants of an entangled state of two qubits under local actions and classical communication.

quant-ph↗
arXiv1998

On the Gravitational Back Reaction to Hawking Radiation

We show that a surface term should be added to the Einstein-Hilbert action in order to properly describe quantum transitions occurring around a black hole. The introduction of this boundary term has been advocated by Teitelboim and collaborators and it has been used in the computation of the black hole entropy. Here, we use it to compute the gravitational corrections to the transition amplitudes giving rise to Hawking radiation. This surface term implies that the probability to emit a particle is given by $e^{- ΔA/4}$ where $ΔA$ is the change in the area of the black hole horizon induced by the emission. Its inclusion at the level of the amplitudes therefore relates quantum black hole radiation to the first law of black hole dynamics. In both cases indeed, the term expressing the change in area directly results from the same boundary term introduced for the same reason: to obtain a well defined action principle.

gr-qc↗
arXiv1997

Optimal Quantum Cloning Machines

We present Quantum Cloning Machines (QCM) that transform N identical qubits into $M>N$ identical copies and we prove that the fidelity (quality) of these copies is optimal. The connection between cloning and measurement is discussed in detail. When the number of clones M tends towards infinity, the fidelity of each clone tends towards the optimal fidelity that can be obtained by a measurement on the input qubits. More generally, the QCM are universal devices to translate quantum information into classical information.

quant-ph↗
arXiv1997

Interacting Charged Particles in an Electric Field and the Unruh Effect

We compute the transition amplitudes between charged particles of mass $M$ and $m$ accelerated by a constant electric field and interacting by the exchange of quanta of a third field. We work in second quantization in order to take into account both recoil effects induced by transitions and the vacuum instability of the charged fields. In spite of both effects, when the exchanged particle is neutral, the equilibrium ratio of the populations is simply $\exp(π(M^2 - m^2)/eE)$. Thus, in the limit $(M-m)/M \to 0$, one recovers Unruh's result characterized by the temperature $a/2π$ where $a$ is the acceleration. When the exchanged particle is charged, its vacuum instability prevents a simple description of the equilibrium state. However, in the limit wherein the charge of the exchanged particle tends to zero, the equilibrium distribution is once more Boltzmanian, but characterized not only by a temperature but also by the electric potential felt by the exchanged particle. This work therefore confirms that thermodynamics in the presence of horizons does not rely on a semi-classical treatment. The relationship with thermodymanics of charged black holes is stressed.

hep-th↗
arXiv1997

Particle creation and non-adiabatic transitions in quantum cosmology

The aim of this paper is to compute transitions amplitudes in quantum cosmology, and in particular pair creation amplitudes and radiative transitions. To this end, we apply a double adiabatic development to the solutions of the Wheeler-DeWitt equation restricted to mini-superspace wherein gravity is described by the scale factor $a$. The first development consists in working with instantaneous eigenstates, in $a$, of the matter Hamiltonian. The second development is applied to the gravitational part of the wave function and generalizes the usual WKB approximation. We then obtain an exact equation which replaces the Wheeler-DeWitt equation and determines the evolution, i.e. the dependence in $a$, of the coefficients of this double expansion. When working in the gravitational adiabatic approximation, the simplified equation delivers the unitary evolution of transition amplitudes occurring among instantaneous eigenstates. Upon abandoning this approximation, one finds that there is an additional coupling among matter states living in expanding and contracting universes. Moreover one has to face also the Klein paradox, i.e. the generation of backward waves from an initially forward wave. The interpretation and the consequences of these unusual features are only sketched in the present paper. Finally, the examples of pair creation and radiative transitions are analyzed in detail to establish when and how the above mentioned unitary evolution coincides with the Schr\" odinger evolution.

gr-qc↗
arXiv1997

Gravitational Instanton for Black Hole Radiation

Hawking radiation is derived from the existence of a euclidean instanton which lives in the euclidean black hole geometry. Upon taking into account the gravitational field of the instanton itself, its action is exactly equal to one quarter the change in the horizon area. This result also applies to the Schwinger process, the Unruh process, and particle creation in deSitter space. The implications for horizon thermodynamics are discussed.

gr-qc↗
arXiv1996

The Schwinger Mechanism, the Unruh Effect and the Production of Accelerated Black Holes

We compute the corrections to the transition amplitudes of an accelerated Unruh ``box'' that arise when the accelerated box is replaced by a ``two level ion'' immersed in a constant electric field and treated in second quantization. There are two kinds of corrections, those due to recoil effects induced by the momentum transfers and those due to pair creation. Taken together, these corrections show that there is a direct relationship between pair creation amplitudes described by the Heisenberg-Euler-Schwinger mechanism and the Unruh effect, i.e. the thermalisation of accelerated systems at temperature $a/ 2 π$ where $a$ is the acceleration. In particular, there is a thermodynamical consistency between both effects whose origin is that the euclidean action governing pair creation rates acts as an entropy in delivering the Unruh temperature. Upon considering pair creation of charged black holes in an electric field, these relationships explain why black holes are created from vacuum in thermal equilibrium, i.e. with their Hawking temperature equal to their Unruh temperature.

hep-th↗
arXiv1996

Black Hole Horizon Fluctuations

It is generally admitted that gravitational interactions become large at an invariant distance of order $1$ from the black hole horizon. We show that due to the ``atmosphere'' of high angular particles near the horizon strong gravitational interactions already occur at an invariant distance of the order of $\sqrt[3]{M}$. The implications of these results for the origin of black hole radiation, the meaning of black hole entropy and the information puzzle are discussed.

hep-th↗
arXiv1996

Applications of the Complex Geometric "Phase" for Meta-stable Systems

Garrison and Wright showed that upon undergoing cyclic quantum evolution a meta-stable state acquires both a geometric phase and a geometric decay probability. This is described by a complex geometric ``phase'' associated with the cyclic evolution of two states and is closely related to the two state formalism developed by Aharonov et al.. Applications of the complex geometric phase to the Born--Oppenheimer approximation and the Aharonov--Bohm effect are considered. A simple experiment based on the optical properties of absorbing birefringent crystals is proposed.

quant-ph↗
arXiv1996

Adiabatic Measurements on Metastable Systems

In several situations, most notably when describing metastable states, a system can evolve according to an effective non hermitian Hamiltonian. To each eigenvalue of a non hermitian Hamiltonian is associated an eigenstate $\vertϕ\rangle$ which evolves forward in time and an eigenstate $\langleψ\vert$ which evolves backward in time. Quantum measurements on such systems are analyzed in detail with particular emphasis on adiabatic measurements in which the measuring device is coupled weakly to the system. It is shown that in this case the outcome of the measurement of an observable $A$ is the weak value $\langleψ\vert A\vertϕ\rangle / \langleψ\vertϕ\rangle $ associated to the two-state vector $\langleψ\vert$ $\vertϕ\rangle$ corresponding to one of the eigenvalues of the non hermitian Hamiltonian. The possibility of performing such measurements in a laboratory is discussed.

quant-ph↗
arXiv1995

Hawking Radiation Without Transplanckian Frequencies

In a recent work, Unruh showed that Hawking radiation is unaffected by a truncation of free field theory at the Planck scale. His analysis was performed numerically and based on a hydrodynamical model. In this work, by analytical methods, the mathematical and physical origin of Unruh's result is revealed. An alternative truncation scheme which may be more appropriate for black hole physics is proposed and analyzed. In both schemes the thermal Hawking radiation remains unaffected even though transplanckian energies no longer appear. The universality of this result is explained by working in momentum space. In that representation, in the presence of a horizon, the d'Alembertian equation becomes a singular first order equation. In addition, the boundary conditions corresponding to vacuum before the black hole formed are that the in--modes contain positive momenta only. Both properties remain valid when the spectrum is truncated and they suffice to obtain Hawking radiation.

hep-th↗
arXiv1995

From Vacuum Fluctuations to Radiation: Accelerated Detectors and Black Holes.(2)

The energy and particle fluxes emitted by an accelerated two level atom are analysed in detail. It is shown both perturbatively and non perturbatively that the total number of emitted photons is equal to the number of transitions characterizing thermal equilibrium thereby confirming that each internal transition is accompanied by the emission of a Minkowski quantum. The mean fluxes are then decomposed according to the final state of the atom and the notion of conditional flux is introduced. This notion is generalized so as to study the energy content of the vacuum fluctuations that induce the transitions of the accelerated atom. The physical relevance of these conditional fluxes is displayed and contact is made with the formalism of Aharonov et al. The same decomposition is then applied to isolate, in the context of black hole radiation, the energy content of the particular vacuum fluctuations which are converted into on mass shell quanta. It is shown that initially these fluctuations are located around the light like geodesic that shall generate the horizon and have exponentially large energy densities. Upon exiting from the star they break up into two pieces. The external one is red shifted and becomes an on mass shell quantum, the other, its ''partner", ends up in the singularity. We avail ourselves of this analysis to study back reaction effects to the production of a single quantum.

gr-qc↗