Fisher information in quantum statistics

Braunstein and Caves (1994) proposed to use Helstrom's {\em quantum information} number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing else than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on $n$ identical copies of the given quantum system. We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two stage adaptive measurement procedure. When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of ``distinguishability of close-by states'' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.Comment: This version replaces the previous versions of February 1999 (titled 'An Example of Non-Attainability of Expected Quantum Information') and that of November 1999. Proofs and results are much improved. To appear in J. Phys.

Transforming squeezed light into a large amplitude coherent state superposition

A quantum superposition of two coherent states of light with small amplitude can be obtained by subtracting a photon from a squeezed vacuum state. In experiments this preparation can be made conditioned on the detection of a photon in the field from a squeezed light source. We propose and analyze an extended measurement strategy which allows generation of high fidelity coherent state superpositions with larger amplitude.Comment: 6 pages, 4 figures, v2: published versio

Entanglement and optimal strings of qubits for memory channels

We investigate the problem of enhancement of mutual information by encoding classical data into entangled input states of arbitrary length and show that while there is a threshold memory or correlation parameter beyond which entangled states outperform the separable states, resulting in a higher mutual information, this memory threshold increases toward unity as the length of the string increases. These observations imply that encoding classical data into entangled states may not enhance the classical capacity of quantum channels.Comment: 14 pages, 8 figures, latex, accepted for publication in Physical Review

Error Analysis For Encoding A Qubit In An Oscillator

In the paper titled "Encoding A Qubit In An Oscillator" Gottesman, Kitaev, and Preskill [quant-ph/0008040] described a method to encode a qubit in the continuous Hilbert space of an oscillator's position and momentum variables. This encoding provides a natural error correction scheme that can correct errors due to small shifts of the position or momentum wave functions (i.e., use of the displacement operator). We present bounds on the size of correctable shift errors when both qubit and ancilla states may contain errors. We then use these bounds to constrain the quality of input qubit and ancilla states.Comment: 5 pages, 8 figures, submitted to Physical Review

Advanced LIGO's ability to detect apparent violations of the cosmic censorship conjecture and the no-hair theorem through compact binary coalescence detections

We study the ability of the advanced Laser Interferometer Gravitational-wave Observatory (aLIGO) to detect apparent violations of the cosmic censorship conjecture and the no-hair theorem. The cosmic censorship conjecture, which is believed to be true in the theory of general relativity, limits the spin-to-mass-squared ratio of a Kerr black hole. The no-hair theorem, which is also believed to be true in the theory of general relativity, suggests a particular value for the tidal Love number of a non-rotating black hole. Using the Fisher matrix formalism, we examine the measurability of the spin and tidal deformability of compact binary systems involving at least one putative black hole. Using parameter measurement errors and correlations obtained from the Fisher matrix, we determine the smallest detectable violation of bounds implied by the cosmic censorship conjecture and the no-hair theorem. We examine the effect of excluding unphysical areas of parameter space when determining the smallest detectable apparent violations, and we examine the effect of different post-Newtonian corrections to the amplitude of the compact binary coalescence gravitational waveform. In addition, we perform a brief study of how the recently calculated 3.0 pN and 3.5 pN spin-orbit corrections to the phase affect spin and mass parameter measurability. We find that physical priors on the symmetric mass ratio and higher harmonics in the gravitational waveform could significantly affect the ability of aLIGO to investigate cosmic censorship and the no-hair theorem for certain systems.Comment: 21 pages, 7 figures, 6 table

Three-body Thomas-Ehrman shifts of analog states of 17^{17}Ne and 17^{17}N

The lowest-lying states of the Borromean nucleus $^{17}$Ne ($^{15}$O+$p$ + $p$) and its mirror nucleus $^{17}$N ($^{15}$N+$n$ + $n$) are compared by using the hyperspheric adiabatic expansion. Three-body resonances are computed by use of the complex scaling method. The measured size of $^{15}$O and the low-lying resonances of $^{16}$F ($^{15}$O+$p$) are first used as constraints to determine both central and spin-dependent two-body interactions. The interaction obtained reproduces relatively accurately both experimental three-body spectra. The Thomas-Ehrman shifts, involving excitation energy differences, are computed and found to be less than 3% of the total Coulomb energy shift for all states.Comment: 9 pages, 3 postscript figures, revtex style. To be published in Phys. Rev.

The Wishart short rate model

We consider a short rate model, driven by a stochastic process on the cone of positive semidefinite matrices. We derive sufficient conditions ensuring that the model replicates normal, inverse or humped yield curves

Entanglement in a first order quantum phase transition

The phase diagram of spins 1/2 embedded in a magnetic field mutually interacting antiferromagnetically is determined. Contrary to the ferromagnetic case where a second order quantum phase transition occurs, a first order transition is obtained at zero field. The spectrum is computed for a large number of spins and allows one to study the ground state entanglement properties which displays a jump of its concurrence at the critical point.Comment: 4 pages, 3 EPS figure