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arXiv:1806.02631

Excluded states in entangled systems: technical and conceptual aspects

Pedro Sancho

Published Jun 7, 2018• 0 citations
arXivHTMLPDF

Abstract

The impossibility of ascribing definite states to the constituents of an entangled system restricts the scope of Pauli's principle in this context. We analyze the conceptual and physical aspects of the problem by studying the actual scope of the principle and the possibility of extending the concept of exclusion in correlated systems. When the entanglement is weak, as in the archetypical case of the Helium atom, the principle can be applied in an approximated way. The concept of non-complete set of properties plays a crucial role in the argument, which also clarifies the physical meaning of the atomic quantum numbers; they are a multi-particle property, not an one-electron one. In contrast, for strong entanglement the excluded states are independent of the principle. We describe some of these states, many times determined by symmetries of the multi-fermion state.

DOI
10.48550/arxiv.1806.02631

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The impossibility of ascribing definite states to the constituents of an entangled system restricts the scope of Pauli's principle in this context. We analyze the conceptual and physical aspects of the problem by studying the actual scope of the principle and the possibility of extending the concept of exclusion in correlated systems. When the entanglement is weak, as in the archetypical case of the Helium atom, the principle can be applied in an approximated way. The concept of non-complete set of properties plays a crucial role in the argument, which also clarifies the physical meaning of the atomic quantum numbers; they are a multi-particle property, not an one-electron one. In contrast, for strong entanglement the excluded states are independent of the principle. We describe some of these states, many times determined by symmetries of the multi-fermion state.
We review the main aspects of entanglement and we prove some general theorems concerning entangled states. In order to satisfy the symmetry requirements for systems containing identical constituents one is unavoidably led to consider entangled states. One could then naively conclude that the identity gives rise, by itself, to peculiar and problematic aspects of physical systems. This is by no means the case: the symmetrization or antisymmetrization of factorized states is an absolutely benign procedure and does not pose any specific conceptual problem apart from the one of the impossibility of attributing their elemetary constituents to one or another of two far-away systems. We briefly reconsider the same problem within the context of recently proposed models of dynamical reduction based on nonlinear and stochastic modifications of the standard evolution.
The characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.
Abstract Section I.—The exclusion principle of Pauli was introduced into the old quantum theory as an empirical fact that had been brought to light in the ordering of spectra. The new mechanics has provided some sort of explanation of the principle, for in a closed system with many electrons the mathematically possible stationary states can be separated into a number of groups, with the property that transitions between stationary states in different groups cannot occur. One of these group is made up of the stationary states corresponding occur. One of the these groups is made up of the stationary states corresponding to wave functions that are antisymmetrical in the co-ordinates of the electrons. Apart from accidental degeneracies, the stationary states in different groups have different energy values. The exclusion principle then states that only energy values belonging to the antisymmertical group are found in nature. The exclusion principle has had great success not only in explaining the spectra of helium and of more complicated atoms, but also, under the form of the Fermi-Dirac statistics, in accounting for metallic conduction and ferromagnetism. In all these phenomena we are dealing with systems is stationary states, possessing energy values which are discrete, although they may lie very close together. Now, as was first emphasised by Oppenheimer, we must also use antisymmetrical wave-functions to describe aperiodic phenomena, such as the collision between an electron and an atom. If we do not, we obtain probabilities for the formation of atoms whose wave-funtions are not antisymmetrical, as we shall show in section 4, where we consider the collision between an electron and a helium atom. A helium atom described by a symmetrical wave-function would show a singlet series in palce of the observed triplets and triplet series in place of the observed singlets. The wave-functions of open systems are essentially degenerate; the symmetical and antisymmetrical solution are not separated from one another by a finite energy difference; but for any arbitrary value of the energy (and of the other integrals of the motion) we can form a symmetrical and an antisymmetrical solution. This is somewhat fundamental difference between open and closed systems. For closed systems containing two electrons there exist only the symmetrical and the antisymmetical solution; but for open systems we might take any combination of the two. In fact, to describe an observable phenomenon such as a collision, the wave-function that it would first occur to us to use is a combination of the two. To fix our ideas we shall discuss the collision between two electrons. Our arguments could equally well be applied to the collision between an electron and a hydrogen atom, the problem originally discussed by Born; but the former is the simpler case, and perhaps illustrates our theory better.
Why is it that particles with half-integer spin are Fermi particles whereas particles with integer spin are Bose particles? An explanation has been worked out by Pauli from complicated arguments from quantum field theory and relativity. He has shown that the two must necessarily go together … but we have not been able to reproduce his arguments on an elementary level. This probably means we do not have a complete understanding of the fundamental principle involved… R. P. Feynman, R.B. Leighton and M. Sands, Feynman Lectures on Physics , Volume 3, Chapter 4, Section 1, Addison-Wesley, Reading, MA (1963) Introduction Particles with half-integer angular momentum obey the Pauli exclusion principle (PEP) – a restriction that a non-degenerate single-particle quantum state can have occupation number of only 0 or 1. This restriction was announced by W. Pauli in 1924 for which, in 1945, he received the Nobel Prize in Physics. Soon after Pauli, the exclusion principle was generalized by P. Dirac and E. Fermi who – independently – integrated it into quantum mechanics. As a consequence half-integer spin particles are called Fermi–Dirac particles or fermions . PEP applies to electrons, protons, neutrons, neutrinos, quarks – and their antiparticles – as well as composite fermions such as He 3 atoms. Thermodynamic properties of metals and semiconductors are largely determined by electron (fermion) behavior.
In this chapter, we study the properties of quantum entanglement, and more generally the way correlations can appear in quantum mechanics. Quantum entanglement is an important notion that we have already discussed, for instance in the context of the Von Neumann chain or of the Schrödinger cat, but here we give more details on its properties. In classical physics, the notion of correlation is well known. It hinges on the calculation of probabilities and on linear averages over a number of possibilities. A distribution gives the probability of having the first system in a some given state and the second in another state. If this distribution is not a product, the two systems are correlated. If it is a product, they are uncorrelated; measuring the properties of one system does not bring any information on the other. This is in particular the case if the state of each of the two systems is perfectly defined (which also defines the state of the whole system perfectly well). The notion of correlation between sub-systems therefore stems from the multiplicity of possible states of the whole system; fluctuations of this state are necessary to give its full meaning to the classical notion of correlation. In quantum mechanics, the situation is different: as we have seen (in particular in Chapter 4), even a physical system that is perfectly defined by a given state vector already contains fluctuations.