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Oxford University Press eBooks

Entangled states

J. C. Garrison, R. Y. Chiao

Published Jun 5, 2008• 0 citations
Publisher

Abstract

A state of two distinguishable particles is “separable” if the wavefunction is a product of single-particle wavefunctions. Schrödinger introduced the term “entangled” to describe a two-particle state that is not separable. This notion is extended to all pairs of distinguishable quantum systems by using tensor products of Hilbert spaces and the Schmidt decomposition. Equivalent operational definitions are expressed in terms of correlation functions representing measurements. A state of two indistinguishable particles is kinematically separable if the product function satisfies Bose or Fermi statistics, otherwise it is kinematically entangled. Alternatively, a two-particle state is dynamically separable if the wavefunction has the minimal form required by Bose or Fermi statistics, and dynamically entangled otherwise. For photons, the role of the missing position-space wave function is played by a detection amplitude directly related to counting rates.

DOI
10.1093/acprof:oso/9780198508861.003.0007

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A state of two distinguishable particles is “separable” if the wavefunction is a product of single-particle wavefunctions. Schrödinger introduced the term “entangled” to describe a two-particle state that is not separable. This notion is extended to all pairs of distinguishable quantum systems by using tensor products of Hilbert spaces and the Schmidt decomposition. Equivalent operational definitions are expressed in terms of correlation functions representing measurements. A state of two indistinguishable particles is kinematically separable if the product function satisfies Bose or Fermi statistics, otherwise it is kinematically entangled. Alternatively, a two-particle state is dynamically separable if the wavefunction has the minimal form required by Bose or Fermi statistics, and dynamically entangled otherwise. For photons, the role of the missing position-space wave function is played by a detection amplitude directly related to counting rates.
Entanglement can be understood as an extraordinary degree of correlation between states of quantum systems – a correlation that cannot be given an explanation in terms of something like a common cause. Entanglement can occur between two or more quantum systems, and the most interesting case is when these correlations occur between systems that are space-like separated, meaning that changes made to one system are immediately correlated with changes in a distant system even though there is no time for a signal to travel between them. In this case one says that quantum entanglement leads to non-local correlations, or non-locality. More precisely, entanglement can be defined in the following way. Consider two particles, A and B, whose (pure) states can be represented by the state vectors ψA and ψB. Instead of representing the state of each particle individually, one can represent the composite two-particle system by another wavefunction, Ψ AB. If the two particles are unentangled , then the composite state is just the tensor product of the states of the components: Ψ AB = ψ A ⊗ ψ B; the state is then said to be factorable (or separable). If the particles are entangled, however, then the state of the composite system cannot be written as such a product of a definite state for A and a definite state for B. This is how an entangled state is defined for pure states: a state is entangled if and only if it cannot be factored: Ψ AB ≠ ψ A ⊗ ψ B.
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.
Up to now we have limited ourselves to states of a single particle. In the present chapter we shall introduce the description of two-particle states. Once this case is understood, it will be easy to generalize to any number of particles. States of two (or more) particles lead to very rich configurations called entangled states. A remarkable feature is that two entangled quantum particles, even at arbitrarily large spatial separations, continue to form a single entity and no classical probabilistic model is able to reproduce the correlation between particles. In the first section we shall present the essential mathematical formalism, that of the tensor product. This will permit us in Section 6.2 to describe quantum mixtures using the state operator formalism. Section 6.3 is devoted to the study of important physical consequences like the Bell inequalities and interference experiments involving entangled states, which will lead us to a deeper understanding of quantum physics. Finally, in the last section we shall briefly review applications to measurement theory and quantum information theory. The latter is undergoing rapid development at present and has applications to quantum computing, cryptography, and teleportation. The tensor product of two vector spaces Definition and properties of the tensor product We wish to construct the space of states of two physical systems which we assume initially to be completely independent. Let ℌ N 1 and ℌ M 2 be the spaces of states of the two systems, of dimension N and M , respectively.
Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so that a first subspace is more entangled than a second, if the Schmidt string of the second majorizes the Schmidt string of the first. The idea is applied to the antisymmetric and symmetric tensor products of a finite-dimensional Hilbert space with itself, and also to the tensor product of an angular momentum jjj with a spin 1∕21∕21∕2. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.
We relate the notion of entanglement for quantum systems composed of two identical constituents to the impossibility of attributing a complete set of properties to both particles. This implies definite constraints on the mathematical form of the state vector associated with the whole system. We then analyze separately the cases of fermion and boson systems, and we show how the consideration of both the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition of the global state vector and the von Neumann entropy of the one-particle reduced density operators can supply us with a consistent criterion for detecting entanglement. In particular, the consideration of the von Neumann entropy is particularly useful in deciding whether the correlations of the considered states are simply due to the indistinguishability of the particles involved or are a genuine manifestation of the entanglement. The treatment leads to a full clarification of the subtle aspects of entanglement of two identical constituents which have been a source of embarrassment and of serious misunderstandings in the recent literature.
Quantum mechanics of composite systems, gives rise to certain special states called entangled states. A physical system, that is in an entangled state displays an intricate correlation between its subsystems. There are also some composite quantum states (classically correlated states or separable states) that are not entangled. It is generally claimed, often without a rigorous proof to support, that these intricate correlations of an entangled state cannot occur in a classical system. This expository article, provides an elementary proof that entangled states cannot arise in the setting of classical mechanics. In addition, a detailed description of the origin of entanglement in quantum systems is included. The mathematical concepts that are necessary for this purpose are presented. The absence of entanglement in the classical setting is due to the fact that every pure classical state of a composite system is a product state, that is, a tensor product of two pure states of the subsystems. In contrast, there are pure composite quantum states that cannot be expressed in the form of a product state or even by a convex sum of product states. Roughly speaking, this is because classical states are positive valued functions on the phase-space while quantum states are positive linear operators. The structure of the tensor product between two commutative spaces of scalar valued functions is drastically different from that of the tensor product between two non-commutative spaces of linear operators. In other words, entanglement is a non-commutative phenomenon.