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Itamar Pitowsky

Publications and source records attributed to Itamar Pitowsky.

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Universality of state-independent violation of correlation inequalities for noncontextual theories

We show that the state-independent violation of inequalities for noncontextual hidden variable theories introduced in [Phys. Rev. Lett. 101, 210401 (2008)] is universal, i.e., occurs for any quantum mechanical system in which noncontextuality is meaningful. We describe a method to obtain state-independent violations for any system of dimension d > 2. This universality proves that, according to quantum mechanics, there are no "classical" states.

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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.

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Two dogmas about quantum mechanics

We argue that the intractable part of the measurement problem -- the 'big' measurement problem -- is a pseudo-problem that depends for its legitimacy on the acceptance of two dogmas. The first dogma is John Bell's assertion that measurement should never be introduced as a primitive process in a fundamental mechanical theory like classical or quantum mechanics, but should always be open to a complete analysis, in principle, of how the individual outcomes come about dynamically. The second dogma is the view that the quantum state has an ontological significance analogous to the significance of the classical state as the 'truthmaker' for propositions about the occurrence and non-occurrence of events, i.e., that the quantum state is a representation of physical reality. We show how both dogmas can be rejected in a realist information-theoretic interpretation of quantum mechanics as an alternative to the Everett interpretation. The Everettian, too, regards the 'big' measurement problem as a pseudo-problem, because the Everettian rejects the assumption that measurements have definite outcomes, in the sense that one particular outcome, as opposed to other possible outcomes, actually occurs in a quantum measurement process. By contrast with the Everettians, we accept that measurements have definite outcomes. By contrast with the Bohmians and the GRW 'collapse' theorists who add structure to the theory and propose dynamical solutions to the 'big' measurement problem, we take the problem to arise from the failure to see the significance of Hilbert space as a new kinematic framework for the physics of an indeterministic universe, in the sense that Hilbert space imposes kinematic (i.e., pre-dynamic) objective probabilistic constraints on correlations between events.

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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%.

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A quantum secret ballot

The paper concerns the protection of the secrecy of ballots, so that the identity of the voters cannot be matched with their vote. To achieve this we use an entangled quantum state to represent the ballots. Each ballot includes the identity of the voter, explicitly marked on the "envelope" containing it. Measuring the content of the envelope yields a random number which reveals no information about the vote. However, the outcome of the elections can be unambiguously decided after adding the random numbers from all envelopes. We consider a few versions of the protocol and their complexity of implementation.

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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(Wρ)| \leq 1 for all separable states ρ, 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)).

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Grover's Quantum Search Algorithm and Diophantine Approximation

In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can find a single marked object in a database of size N by using only O(N^{1/2}) queries of the oracle that identifies the object. His result was generalized to the case of finding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, M<K, our task is to determine which is the case. We show how to solve this problem with a high probability of success using only iterations of Grover's basic step (and no other algorithm). Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation m < (2N^{1/2})/(K^{1/2}-M^{1/2}) obtains. This bound sharpens previous results and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an orbit of an ergodic automorphism of the torus. We comment on situations where the algorithm may be slow, and note the similarity between these cases and the problem of small divisors in classical mechanics.

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Quantum mechanics as a theory of probability

We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it. The construction proceeds in the following steps: (a) Axioms for the algebra of events are introduced following Birkhoff and von Neumann. All axioms, except the one that expresses the uncertainty principle, are shared with the classical event space. The only models for the set of axioms are lattices of subspaces of inner product spaces over a field K. (b) Another axiom due to Soler forces K to be the field of real, or complex numbers, or the quaternions. We suggest a probabilistic reading of Soler's axiom. (c) Gleason's theorem fully characterizes the probability measures on the algebra of events, so that Born's rule is derived. (d) Gleason's theorem is equivalent to the existence of a certain finite set of rays, with a particular orthogonality graph (Wondergraph). Consequently, all aspects of quantum probability can be derived from rational probability assignments to finite "quantum gambles". We apply the approach to the analysis of entanglement, Bell inequalities, and the quantum theory of macroscopic objects. We also discuss the relation of the present approach to quantum logic, realism and truth, and the measurement problem.

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Generalizations of Kochen and Specker's Theorem and the Effectiveness of Gleason's Theorem

Kochen and Specker's theorem can be seen as a consequence of Gleason's theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason's theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated.

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Most Bell Operators do not Significantly Violate Locality

The worst violation of Bell's inequality for $n$ qbits is of size $2^{\frac{n-1}{2}}$ and it is obtained by a specific operator acting on a specific state. We show, to the contrary, that for a vast majority of Bell operators the worst violation is bounded by $O((n\log n)^{1/2})$, below experimental detection. With respect to the extremal operators, introduced by Werner and Wolf [Phys. Rev. A 64, 032112 (2001)], we show that a large majority of them have a norm bounded by $O(n^{1/2})$.

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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.

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