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arXiv:quant-ph/0410245

On the relative quantum entanglement with respect to tensor product structure

Xulin Liu, C. P. Sun

Published Oct 29, 2004• 4 citations
arXivHTMLPDF

Abstract

Mathematical foundation of the novel concept of quantum tensor product by Zanardi et al is rigorously established. The concept of relative quantum entanglement is naturally introduced and its meaning is made clear both mathematically and physically. For a finite or an infinite dimensional vector space $W$ the so called tensor product partition (TPP) is introduced on $End(W)$, the set of endmorphisms of $W$, and a natural correspondence is constructed between the set of TPP's of $End(W)$ and the set of tensor product structures (TPS's) of $W$. As a byproduct, it is shown that an arbitrarily given wave function belonging to an n-dimensional Hilbert space, $n$ being not a prime number, can be interpreted as a separable state with respect to some man-made TPS, and thus a quantum entangled state of a many-body system with respect to the "God-given" TPS can be regarded as a quantum state without entanglement in some sense. The concept of standard set of observables is also introduced to probe the underlying structure of the object TPP and to establish its connection with practical physical measurement.

DOI
10.48550/arxiv.quant-ph/0410245

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Mathematical foundation of the novel concept of quantum tensor product by Zanardi et al is rigorously established. The concept of relative quantum entanglement is naturally introduced and its meaning is made clear both mathematically and physically. For a finite or an infinite dimensional vector space WWW the so called tensor product partition (TPP) is introduced on End(W)End(W)End(W), the set of endmorphisms of WWW, and a natural correspondence is constructed between the set of TPP's of End(W)End(W)End(W) and the set of tensor product structures (TPS's) of WWW. As a byproduct, it is shown that an arbitrarily given wave function belonging to an n-dimensional Hilbert space, nnn being not a prime number, can be interpreted as a separable state with respect to some man-made TPS, and thus a quantum entangled state of a many-body system with respect to the "God-given" TPS can be regarded as a quantum state without entanglement in some sense. The concept of standard set of observables is also introduced to probe the underlying structure of the object TPP and to establish its connection with practical physical measurement.
In early days of quantum theory it was believed that the results of measurements performed on two distant physical systems should be uncorrelated thus their quantum state should be separable it means described by a simple tensor product of the individual local state vectors or a tensor product of individual local density operators. It was shown many years ago by EPR that two systems which interacted in the past and separated afterwards had to be described in most cases by particular non-separable states which are called entangled. It was noticed by Zanardi et al. that a Hilbert space of possible state vectors of compound physical system could be partitioned in different way by introducing various tensor product structures induced by the experimentally accessible observables (interactions and measurements). Therefore a separable state in one partition could become entangled in different partition. In this sense the entanglement was relative to a particular set of experimental capabilities. Continuing this line of thought Torre et al. claimed to prove that for any separable state there existed strongly correlated physical observables therefore all quantum states were entangled. The same claims one may find in several recent papers. In this paper we will discuss in operational way the differences existing between separable, non-separable and entangled states and we will show that the conclusions of Torre et al. were unjustified. A sufficient condition for entanglement is the violation of BI-CHSH and/or steering inequalities. Since there exist experiments outside the quantum physics violating these inequalities therefore in the operational approach one cannot say that the entanglement is an exclusive quantum phenomenon.
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One of the crucial differences between mathematical models of classical and quantum mechanics is the use of the tensor product of the state spaces of subsystems as the state space of the corresponding composite system. (To describe an ensemble of classical composite systems one uses random variables taking values in the Cartesian product of the state spaces of subsystems.) We show that, nevertheless, it is possible to establish a natural correspondence between the classical and quantum probabilistic descriptions of composite systems. Quantum averages for composite systems (including entangled) can be represented as averages with respect to classical random fields. It is essentially what Albert Einstein was dreamed of. Quantum mechanics is represented as classical statistical mechanics with infinite-dimensional phase space. While the mathematical construction is completely rigorous, its physical interpretation is a complicated problem (which will not be discussed in this paper).
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.
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.