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International Journal of Modern Physics B

Quantum Nondeterministic Computation based on Statistics Superselection Rules

Giuseppe Castagnoli

Published Apr 20, 1997• 1 citation
Publisher

Abstract

Quantum states which obey certain symmetry superselection rules under identical particles permutation can be interpreted as computational states satisfying corresponding Boolean predicates. Given the NP-complete problem of testing the satisfiability of a generic Boolean predicate P, we investigate the possibility of achieving quantum nondeterministic computation by deriving, from P, a physical situation in which the computational states satisfy Piff they satisfy a special fermion statistics.

DOI
10.1142/s0217979297000678

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Quantum states which obey certain symmetry superselection rules under identical particles permutation can be interpreted as computational states satisfying corresponding Boolean predicates. Given the NP-complete problem of testing the satisfiability of a generic Boolean predicate P, we investigate the possibility of achieving quantum nondeterministic computation by deriving, from P, a physical situation in which the computational states satisfy Piff they satisfy a special fermion statistics.
The permutation symmetry of quantum mechanics is widely thought to imply a sort of metaphysical underdetermination about the identity of particles. Despite claims to the contrary, this implication does not hold in the more fundamental quantum field theory, where an ontology of particles is not generally available. Although permutations are often defined as acting on particles, a more general account of permutation symmetry can be formulated using superselection theory. As a result, permutation symmetry applies even in field theories with no particle interpretation. The quantum mechanical account of permutations acting on particles is recovered as a special case.
Abstract The empirical rule that systems of identical particles always obey either Bose or Fermi statistics is customarily imposed on the theory by adding it to the axioms of nonrelativistic quantum mechanics, with the result that other statistical behaviors are excluded a priori. A more general approach is to ask what other many-particle statistics are consistent with the indistinguishability of identical particles. This strategy offers a way to discuss possible violations of the Pauli Exclusion Principle, and it leads to some interesting issues related to preparation of states and a superselection rule arising from invariance under the permutation group.
The anticommutation of Fermionic fields raises the problem of simulating the evolution of Fermionic systems by means of commuting quantum systems, say qubits. We tackle the issue considering local Fermionic modes as the elementary systems of an operational probabilistic theory. We show that the locality of Fermionic operations, namely operations on systems that are not causally connected must commute, implies the parity superselection rule, which inhibits the superposition of states with an even and an odd number of excitations. Accordingly we derive the largest probabilistic theory compatible with the parity superselection constraint and show that it lacks two distinctive traits of quantum theory, the local tomography and the monogamy of the entanglement. We generalize the notion of superselection rule to general probabilistic theories as sets of linear constraints on the convex set of states and prove a trade-off between the cardinality of the superselection rule and the degree of holism of the resulting theory. Within this scenario the Fermionic quantum theory and the real quantum theory can be regarded as superselected versions of the usual quantum theory. These results are published in Ref. [7] http://iopscience.iop.org/0295-5075/107/2/20009 and Ref. [8] http://www.worldscientific.com/doi/abs/10.1142/S0217751X14300257. In the last three decades the relation between Fermionic systems and other quantum systems has been throughly investigated from both the computational and the physical point of view. In particular the puzzling anti-commuting nature of the Fermionic systems casts a shadow on the possibility of simulating the physical evolution of a bunch of Fermionic systems by means of commuting quantum systems—say qubits. This issue was raised by R. P. Feynman in 1982 [9], when in his seminal work on physical computation he wondered about the possibility of simulating Fermions by local quantum systems in interaction—what we would call nowadays a quantum computer. The problem is that of encoding the evolution of Fermionic fields onto localized quantum systems. A well-known encoding of N Fermionic systems into N qubits is given by the Jordan-Wigner transform (JWT) [14]. Such an encoding, based on the identification between the Fock space of N Fermions and the Hilbert space of N qubits, provides a -algebra isomorphism between the Fermionic anticommuting algebra and the commuting algebra of qubits. Such a correspondence has been a valuable instrument in modern solid state physics for solving the one dimensional XY spin-chains [18, 15] and then for the understanding of superconductivity and the quantum Hall effect. Moreover, a time-adaptive JWT has been introduced in Ref. [19], which allows to contract Fermionic unitary circuits with the same complexity as for the corresponding spin model. In quantum information science the JWT has been used to extend to the Fermionic case notions as entanglement [1], the entropic area law [24], and universal computation [5]. More recently the JWT, which originally regards one dimensional chains of spin-1=2 systems, has been generalized to any spin [3] and lattice [13] dimension. Despite its computational power, the JWT fails to solve completely the issue established by Feynman: physically local Fermionic operations are mapped into nonlocal quantum ones and viceversa. As noticed by many authors this can lead to ambiguities in defining the partial trace [16, 4, 17, 10], and in assessing
We discuss the possibility of observing quantum nonlocality using the so-called mode entanglement, analyzing the differences between different types of particles in this context. We first discuss the role of coherent states in such experiments, and we comment on the existence of coherent states in nature. The discussion of coherent states naturally raises questions about the role of particle statistics in this problem. Although the Pauli exclusion principle precludes coherent states with a large number of fermionic particles, we find that a large number of fermionic coherent states, each containing at most one particle, can be used to achieve the same effect as a bosonic coherent state for the purposes of this problem. The discussion of superselection rules arises naturally in this context, because their applicability to a given situation prohibits the use of coherent states. This limitation particularly affects the scenario that we propose for detecting the mode entanglement of fermionic particles.