arXiv subjects

James D. Malley

Publications and source records attributed to James D. Malley.

6 recordsLinked to original sources

arXiv2015

POVMs and the Two Theorems of Naimark and Sz.-Nagy

In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be so resolved, with the resolution respecting all linear sums. Crucially, both resolutions return the correct Born probabilities for the original observables. Here, an alternative proof of the Sz.-Nagy result is given using elementary inner product spaces. A version of the resolution is then shown to respect all products of observables on the base space. Practical and theoretical consequences are indicated. For example, quantum statistical inference problems that involve any algebraic functionals can now be studied using classical statistical methods over commuting observables. The estimation of quantum states is a problem of this type. Further, as theoretical objects, classical and quantum systems are now distinguished by only more or less degrees of freedom.

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arXiv2014

A Simplified Basis for Bell-Kochen-Specker Theorems

We show that a reduced form of the structural requirements for deterministic hidden variables used in Bell-Kochen-Specker theorems is already sufficient for the no-go results. Those requirements are captured by the following principle: an observable takes a spectral value x if and only if the spectral projector associated with x takes the value 1. We show that the only if part of this condition suffices. The proof identifies an important structural feature behind the no-go results; namely, if at least one projector is assigned the value 1 in any resolution of the identity, then at most one is.

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arXiv2007

Update for "The collapse of Bell determinism"

We update our paper: The collapse of Bell determinism (Physics Letters A, 359 (2006): 122-125; available online 16 June 2006). First, we point out that Olivier Brunet, using lattice theoretic methods, has recently, and quite independently, derived the core technical lemma of our paper; see: A priori knowledge and the Kochen-Specker theorem (Physics Letters A, available online 5 January 2007). He has also kindly pointed out a misstep in the last line of one of our results. We discuss the correction and comment on slightly revised notions of how Bell determinism collapses. We also correct a few typos in the original paper.

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arXiv2006

The Collapse of Bell Determinism

The Bell-Kochen-Specker conditions (BKS) for a deterministic noncontextual hidden-variable model are wonderfully simple to state, deal with just one-dimensional projectors on a Hilbert space H and make no reference to a probabilistic phase space or quantum system. They only ask for an assignment of zero or one to every projector such that the assignment respects orthogonal resolutions of the identity. Various no-go results in the literature show that the pair of statements {BKS is valid; dim H greater than or equal to 3} are inconsistent. Here we show, more radically, that the pair actually contradicts the dimensionality of the space itself, by implying that there can exist at most a single one-dimensional projector acting on H. Our derivation involves only elementary inner product spaces. It is non-probabilistic, inequality-free, state independent, does not use entanglement, and is simultaneously valid in all dimensions three or greater.

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arXiv2005

Noncommuting Observables and Local Realism

A standard approach in the foundations of quantum mechanics studies local realism and hidden variables models exclusively in terms of violations of Bell-like inequalities. Thus quantum nonlocality is tied to the celebrated no-go theorems, and these comprise a long list that includes the Kochen-Specker and Bell theorems, as well as elegant refinements by Mermin, Peres, Hardy, GHZ, and many others. Typically entanglement or carefully prepared multipartite systems have been considered essential for violations of local realism and for understanding quantum nonlocality. Here we show, to the contrary, that sharp violations of local realism arise almost everywhere without entanglement. The pivotal fact driving these violations is just the noncommutativity of quantum observables. We demonstrate how violations of local realism occur for arbitrary noncommuting projectors, and for arbitrary quantum pure states. Finally, we point to elementary tests for local realism, using single particles and without reference to entanglement, thus avoiding experimental loopholes and efficiency issues that continue to bedevil the Bell inequality related tests.

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arXiv2004

All quantum observables in a hidden-variables model must commute simultaneously

Under a standard set of assumptions for a hidden-variables model for quantum events, we show that all observables must commute simultaneously. And, despite Bell's complaint that a key condition of von Neumann's was quite unrealistic, we show these these conditions are entirely equivalent to those later introduced by Bell, Kochen and Specker. As is it known that these conditions are also equivalent to those under which the Bell-Clauser-Horne inequalities are derived, we see that any experimental violations of the inequalities demonstrate only that quantum observables do not commute.The same conclusion applies to the collection of elegant inequality-free no-go proofs of Greenberger-Horne-Zeilinger, Mermin and Peres. Otherwise expressed, the usual hidden-variable models have assumptions that are collectively too strong to be interesting, and hence need modification or deletion.

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