180 research outputs found

### Betting on the Outcomes of Measurements: A Bayesian Theory of Quantum Probability

We develop a systematic approach to quantum probability as a theory of
rational betting in quantum gambles. In these games of chance the agent is
betting in advance on the outcomes of several (finitely many) incompatible
measurements. One of the measurements is subsequently chosen and performed and
the money placed on the other measurements is returned to the agent. We show
how the rules of rational betting imply all the interesting features of quantum
probability, even in such finite gambles. These include the uncertainty
principle and the violation of Bell's inequality among others. Quantum gambles
are closely related to quantum logic and provide a new semantics to it. We
conclude with a philosophical discussion on the interpretation of quantum
mechanics.Comment: 21 pages, 2 figure

### Macroscopic objects in quantum mechanics: A combinatorial approach

Why we do not see large macroscopic objects in entangled states? There are
two ways to approach this question. The first is dynamic: the coupling of a
large object to its environment cause any entanglement to decrease
considerably. The second approach, which is discussed in this paper, puts the
stress on the difficulty to observe a large scale entanglement. As the number
of particles n grows we need an ever more precise knowledge of the state, and
an ever more carefully designed experiment, in order to recognize entanglement.
To develop this point we consider a family of observables, called witnesses,
which are designed to detect entanglement. A witness W distinguishes all the
separable (unentangled) states from some entangled states. If we normalize the
witness W to satisfy |tr(W\rho)| \leq 1 for all separable states \rho, then the
efficiency of W depends on the size of its maximal eigenvalue in absolute
value; that is, its operator norm ||W||. It is known that there are witnesses
on the space of n qbits for which ||W|| is exponential in n. However, we
conjecture that for a large majority of n-qbit witnesses ||W|| \leq O(\sqrt{n
logn}). Thus, in a non ideal measurement, which includes errors, the largest
eigenvalue of a typical witness lies below the threshold of detection. We prove
this conjecture for the family of extremal witnesses introduced by Werner and
Wolf (Phys. Rev. A 64, 032112 (2001)).Comment: RevTeX, 14 pages, some additions to the published version: A second
conjecture added, discussion expanded, and references adde

### Probability and Nonlocality in Many Minds Interpretations of Quantum Mechanics

We argue that a certain type of many minds (and many worlds) interpretations
of quantum mechanics due to Lockwood (and Deutsch) do not provide a coherent
interpretation of the quantum mechanical probabilistic algorithm. By contrast,
in Albert and Loewer's version of the many minds interpretation there is a
coherent interpretation of the quantum mechanical probabilities. We consider
Albert and Loewer's probability interpretation in the context of Bell-type and
GHZ-type states and argue that it exhibits a certain form of nonlocality which
is, however, much weaker than Bell's nonlocality.Comment: 22 pages, last section rewritten, to appear in British Journal for
the Philosophy of Scienc

### Generalizing Tsirelson's bound on Bell inequalities using a min-max principle

Bounds on the norm of quantum operators associated with classical Bell-type
inequalities can be derived from their maximal eigenvalues. This quantitative
method enables detailed predictions of the maximal violations of Bell-type
inequalities.Comment: 4 pages, 2 figures, RevTeX4, replaced with published versio

### New Bell inequalities for the singlet state: Going beyond the Grothendieck bound

Contemporary versions of Bell's argument against local hidden variable (LHV)
theories are based on the Clauser Horne Shimony and Holt (CHSH) inequality, and
various attempts to generalize it. The amount of violation of these
inequalities cannot exceed the bound set by the Grothendieck constants.
However, if we go back to the original derivation by Bell, and use the perfect
anti-correlation embodied in the singlet spin state, we can go beyond these
bounds. In this paper we derive two-particle Bell inequalities for traceless
two-outcome observables, whose violation in the singlet spin state go beyond
the Grothendieck constants both for the two and three dimensional cases.
Moreover, creating a higher dimensional analog of perfect correlations, and
applying a recent result of Alon and his associates (Invent. Math. 163 499
(2006)) we prove that there are two-particle Bell inequalities for traceless
two-outcome observables whose violation increases to infinity as the dimension
and number of measurements grow. Technically these result are possible because
perfect correlations (or anti-correlations) allow us to transport the indices
of the inequality from the edges of a bipartite graph to those of the complete
graph. Finally, it is shown how to apply these results to mixed Werner states,
provided that the noise does not exceed 20%.Comment: 18 pages, two figures, some corrections and additional references,
published versio

### Testing the bounds on quantum probabilities

Bounds on quantum probabilities and expectation values are derived for
experimental setups associated with Bell-type inequalities. In analogy to the
classical bounds, the quantum limits are experimentally testable and therefore
serve as criteria for the validity of quantum mechanics.Comment: 9 pages, Revte

### Effects and Propositions

The quantum logical and quantum information-theoretic traditions have exerted
an especially powerful influence on Bub's thinking about the conceptual
foundations of quantum mechanics. This paper discusses both the quantum logical
and information-theoretic traditions from the point of view of their
representational frameworks. I argue that it is at this level, at the level of
its framework, that the quantum logical tradition has retained its centrality
to Bub's thought. It is further argued that there is implicit in the quantum
information-theoretic tradition a set of ideas that mark a genuinely new
alternative to the framework of quantum logic. These ideas are of considerable
interest for the philosophy of quantum mechanics, a claim which I defend with
an extended discussion of their application to our understanding of the
philosophical significance of the no hidden variable theorem of Kochen and
Specker.Comment: Presented to the 2007 conference, New Directions in the Foundations
of Physic

### Geometry of quantum correlations

Consider the set Q of quantum correlation vectors for two observers, each
with two possible binary measurements. Quadric (hyperbolic) inequalities which
are satisfied by every vector in Q are proved, and equality holds on a two
dimensional manifold consisting of the local boxes, and all the quantum
correlation vectors that maximally violate the Clauser, Horne, Shimony, and
Holt (CHSH) inequality. The quadric inequalities are tightly related to CHSH,
they are their iterated versions (equation 20). Consequently, it is proved that
Q is contained in a hyperbolic cube whose axes lie along the non-local
(Popescu, Rohrlich) boxes. As an application, a tight constraint on the rate of
local boxes that must be present in every quantum correlation is derived. The
inequalities allow testing the validity of quantum mechanics on the basis of
data available from experiments which test the violation of CHSH. It is noted
how these results can be generalized to the case of n sites, each with two
possible binary measurements.Comment: Published version, slight change in titl

### New optimal tests of quantum nonlocality

We explore correlation polytopes to derive a set of all Boole-Bell type
conditions of possible classical experience which are both maximal and
complete. These are compared with the respective quantum expressions for the
Greenberger-Horne-Zeilinger (GHZ) case and for two particles with spin state
measurements along three directions.Comment: 10 page

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