Correlations of observables in chaotic states of macroscopic quantum systems

We study correlations of observables in energy eigenstates of chaotic systems of a large size $N$. We show that the bipartite entanglement of two subsystems is quite strong, whereas macroscopic entanglement of the total system is absent. It is also found that correlations, either quantum or classical, among less than $N/2$ points are quite small. These results imply that chaotic states are stable. Invariance of these properties under local operations is also shown.Comment: 5 pages, 2 figure

On local-hidden-variable no-go theorems

The strongest attack against quantum mechanics came in 1935 in the form of a paper by Einstein, Podolsky and Rosen. It was argued that the theory of quantum mechanics could not be called a complete theory of Nature, for every element of reality is not represented in the formalism as such. The authors then put forth a proposition: we must search for a theory where, upon knowing everything about the system, including possible hidden variables, one could make precise predictions concerning elements of reality. This project was ultimatly doomed in 1964 with the work of Bell Bell, who showed that the most general local hidden variable theory could not reproduce correlations that arise in quantum mechanics. There exist mainly three forms of no-go theorems for local hidden variable theories. Although almost every physicist knows the consequences of these no-go theorems, not every physicist is aware of the distinctions between the three or even their exact definitions. Thus we will discuss here the three principal forms of no-go theorems for local hidden variable theories of Nature. We will define Bell inequalities, Bell inequalities without inequalities and pseudo-telepathy. A discussion of the similarities and differences will follow.Comment: 7 pages, no figure, replaced "Bell inequalities" with "Bell theorems" and updated the reference

Quantum dynamics and breakdown of classical realism in nonlinear oscillators

The dynamics of a quantum nonlinear oscillator is studied in terms of its quasi-flow, a dynamical mapping of the classical phase plane that represents the time-evolution of the quantum observables. Explicit expressions are derived for the deformation of the classical flow by the quantum nonlinearity in the semiclassical limit. The breakdown of the classical trajectories under the quantum nonlinear dynamics is quantified by the mismatch of the quasi-flow carried by different observables. It is shown that the failure of classical realism can give rise to a dynamical violation of Bell's inequalities.Comment: RevTeX 4 pages, no figure

Test of Bell's Inequality using the Spin Filter Effect in Ferromagnetic Semiconductor Micro-structures

A theoretical proposal for testing Bell's inequality in mesoscopic systems is presented. We show that the entanglement of two electron spins can be detected in the spin filter effect in the mesoscopic semiconductor / ferromagnetic semiconductor / semiconductor junction. The current-current correlation function is calculated by use of the quantum scattering theory and we compare it with the local hidden variable theory. We also discuss the influence of an imperfect spin filter and derive the condition to see the violation of Bell's inequality experimentally.Comment: 6 pages, 4 figures, submitted to J. Phys. Soc. Jp

Kochen-Specker theorem for a single qubit using positive operator-valued measures

A proof of the Kochen-Specker theorem for a single two-level system is presented. It employs five eight-element positive operator-valued measures and a simple algebraic reasoning based on the geometry of the dodecahedron.Comment: REVTeX4, 4 pages, 2 figure

Problems and Aspects of Energy-Driven Wavefunction Collapse Models

Four problematic circumstances are considered, involving models which describe dynamical wavefunction collapse toward energy eigenstates, for which it is shown that wavefunction collapse of macroscopic objects does not work properly. In one case, a common particle position measuring situation, the apparatus evolves to a superposition of macroscopically distinguishable states (does not collapse to one of them as it should) because each such particle/apparatus/environment state has precisely the same energy spectrum. Second, assuming an experiment takes place involving collapse to one of two possible outcomes which is permanently recorded, it is shown in general that this can only happen in the unlikely case that the two apparatus states corresponding to the two outcomes have disjoint energy spectra. Next, the progressive narrowing of the energy spectrum due to the collapse mechanism is considered. This has the effect of broadening the time evolution of objects as the universe evolves. Two examples, one involving a precessing spin, the other involving creation of an excited state followed by its decay, are presented in the form of paradoxes. In both examples, the microscopic behavior predicted by standard quantum theory is significantly altered under energy-driven collapse, but this alteration is not observed by an apparatus when it is included in the quantum description. The resolution involves recognition that the statevector describing the apparatus does not collapse, but evolves to a superposition of macroscopically different states.Comment: 17 page

Quantum inseparability as local pseudomixture

We show how to decompose any density matrix of the simplest binary composite systems, whether separable or not, in terms of only product vectors. We determine for all cases the minimal number of product vectors needed for such a decomposition. Separable states correspond to mixing from one to four pure product states. Inseparable states can be described as pseudomixtures of four or five pure product states, and can be made separable by mixing them with one or two pure product states.Comment: 5 pages late

Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels

Two separated observers, by applying local operations to a supply of not-too-impure entangled states ({\em e.g.} singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity ({\em e.g.} near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transfrom one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield $D(M)$ of pure singlets ($\ket{\Psi^-}$) distillable from mixed states $M$, showing $D(M)>0$ if \bra{\Psi^-}M\ket{\Psi^-}>\half.Comment: 4 pages (revtex) plus 1 figure (postscript). See also http://vesta.physics.ucla.edu/~smolin/ . Replaced to correct interchanged $\sigma_x$ and $\sigma_z$ near top of column 2, page

Quantum Nonlocality in Two-Photon Experiments at Berkeley

We review some of our experiments performed over the past few years on two-photon interference. These include a test of Bell's inequalities, a study of the complementarity principle, an application of EPR correlations for dispersion-free time-measurements, and an experiment to demonstrate the superluminal nature of the tunneling process. The nonlocal character of the quantum world is brought out clearly by these experiments. As we explain, however, quantum nonlocality is not inconsistent with Einstein causality.Comment: 16 pages including 24 figure

Mixed State Entanglement and Quantum Error Correction

Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbi- trary quantum state $|\xi\rangle$ can be transmitted at some rate Q through a noisy channel $\chi$ without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state $\hat{M}(\chi)$ (obtained by sharing halves of EPR pairs through a channel $\chi$) yields a QECC on $\chi$ with rate $Q=D$, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts $D_1(M)$ and $D_2(M)$ that can be locally distilled from it by EPPs using one- and two-way classical communication respectively, and give an exact expression for $E(M)$ when $M$ is Bell-diagonal. While EPPs require classical communica- tion, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way com- munication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve an asymptotic $Q$ (or $D$) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single- error-correcting quantum block code. We prove that {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.Comment: Resubmission with various corrections and expansions. See also http://vesta.physics.ucla.edu/~smolin/ for related papers and information. 82 pages latex including 19 postscript figures included using psfig macro