Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:physics/0701246

Further de-Entangling Entanglement

Elemér Elad Rosinger

Published Jan 22, 2007• 8 citations
arXivHTMLPDF

Abstract

A recent general model of entanglement, [5], that goes much beyond the usual one based on tensor products of vector spaces is further developed here. It is shown that the usual Cartesian product can be seen as two extreme particular instances of non-entanglement. Also the recent approach to entanglement in [8] is incorporated in the general model in [5]. The idea pursued is that entanglement is by now far too important a phenomenon in Quantum Mechanics, in order to be left confined to its present exclusive modelling by tensor products. Once this is realized, it turns out that one can quite easily de-entangle entanglement from tensor products, and in fact, one can do so in a large variety of ways. Within such general settings the issue of entanglement becomes connected with the issue of "System versus Subsystem" in General Systems Theory, where synthesizing subsystems into a system may often be less difficult than identifying subsystems in a system.

DOI
10.48550/arxiv.physics/0701246

Related papers

A General Scheme of Entanglement

Elemér Elad Rosinger

Entanglement is a well known fundamental resource in quantum information. Here the following question is addressed : which are the deeper roots of entanglement that may help in its better understanding and use ? The answer is that one can reproduce the phenomenon of entanglement in a far more general and simple way, a way that goes much beyond the usual one which is limited to the framework of tensor products of vector spaces. In this general approach to entanglement presented here - and much unlike in the particular setup of tensor products of vector spaces - the spaces involved can be rather arbitrary sets, just as in the case of Cartesian products. In particular, they need not even have any algebraic structure. Thus they do not have to be vector spaces, groups or even semigroups.

arXiv:physics/0701116•Jan 10, 2007•8 citations

Entanglement

Teiko Heinosaari, Mário Ziman

In Section 2.4 we learned that a composite quantum system, composed of two subsystems A and B , is associated with a tensor product space HA ? HB . Composite quantum systems have already played an important role in many instances in this text, especially in open systems and measurement models. In this chapter we dive deeper into tensor product spaces, discovering some strange implications of the quantum theory of composite quantum systems. As a result of its tensor product structure, quantum theory has embedded into it the phenomenon of entanglement , which is often seen as the foremost quantum feature. As early as in 1935 it puzzled Erwin Schrödinger, who introduced the German term Verschränkung [127] and Albert Einstein, who, together with Boris Podolski and Nathan Rosen, used the specific example of an entangled state to argue that quantum theory is incomplete [58]. In the 1960s John Bell demonstrated that the consequences of entanglement are incompatible with the Einstein– Podolsky–Rosen model of local realism [11], [12]. In recent years entanglement has been recognized as the resource for quantum information processing. Nowadays entanglement theory is a highly developed subject, and we can present only the mathematical basics. For the reader with a deeper interest in entanglement we recommend the recent reviews by Horodecki et al . [80] and Plenio et al . [118], which also cover the quantum information aspects of entanglement. To simplify our discussion we will assume that the Hilbert spaces are finite dimensional.

Cambridge University Press eBooks•Dec 15, 2011•2 citations

Group Invariant Entanglements in Generalized Tensor Products

Elemér Elad Rosinger

The group invariance of entanglement is obtained within a very general and simple setup of the latter, given by a recently introduced considerably extended concept of tensor products. This general approach to entanglement - unlike the usual one given in the particular setup of tensor products of vector spaces - turns out not to need any specific algebraic structure. The resulting advantage is that, entanglement being in fact defined by a negation, its presence in a general setup increases the chances of its manifestations, thus also its availability as a resource.

arXiv:0808.0095•Prespacetime journal•Aug 1, 2008•6 citations

Quantum entanglement

Ingemar Bengtsson, Karol Życzkowski

Entanglement is not one but rather the characteristic trait of quantum mechanics. Erwin Schrödinger Introducing entanglement So far, when working in a Hilbert space that is a tensor product of the form H = H A ⊗ H B , we were really interested in only one of the factors; the other factor played the role of an ancilla describing an environment outside our control. Now the perspective changes: we are interested in a situation where there are two masters. The fate of both subsystems are of equal importance, although they may be sitting in two different laboratories. The operations performed independently in the two laboratories are described using operators of the form Φ A ⊗ 1 and 1 ⊗ Φ B , respectively, but due perhaps to past history, the global state of the system may not be a product state. In general, it may be described by an arbitrary density operator ρ acting on the composite Hilbert space H . The peculiarities of this situation were highlighted in 1935 by Einstein, Podolsky and Rosen (1935). Their basic observation was that if the global state of the system is chosen suitably then it is possible to change, and to some extent to choose, the state assignment in laboratory A by performing operations in laboratory B . The physicists in laboratory A will be unaware of this until they are told, but they can check in retrospect that the experiments they performed were consistent with the state assignment arrived at from afar – even though there was an element of choice in arriving at that state assignment.

Cambridge University Press eBooks•May 11, 2006•1 citation

Further on Non-Cartesian Systems

Elemér Elad Rosinger

A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones. ”History is written with the feet...” Ex-Chairman Mao ”Science in not done scientifically, since it is mostly done by non-scientists... ” Anonymous 1 1. Preliminaries As seen in [7], there is a natural division of physical systems in Cartesian

•Dec 14, 2010•2 citations

Entanglement beyond subsystems

Lorenza Viola, Howard Barnum, Emanuel H. Knill +2 more

We present a notion of generalized entanglement which goes beyond the conventional definition based on quantum subsystems. This is accomplished by directly defining entanglement as a property of quantum states relative to a distinguished set of observables singled out by Physics. While recovering standard entanglement as a special case, our notion allows for substantially broader generality and flexibility, being applicable, in particular, to situations where existing tools are not directly useful.

arXiv:quant-ph/0403044•Mar 4, 2004•0 citations
A recent general model of entanglement, [5], that goes much beyond the usual one based on tensor products of vector spaces is further developed here. It is shown that the usual Cartesian product can be seen as two extreme particular instances of non-entanglement. Also the recent approach to entanglement in [8] is incorporated in the general model in [5]. The idea pursued is that entanglement is by now far too important a phenomenon in Quantum Mechanics, in order to be left confined to its present exclusive modelling by tensor products. Once this is realized, it turns out that one can quite easily de-entangle entanglement from tensor products, and in fact, one can do so in a large variety of ways. Within such general settings the issue of entanglement becomes connected with the issue of "System versus Subsystem" in General Systems Theory, where synthesizing subsystems into a system may often be less difficult than identifying subsystems in a system.
Entanglement is a well known fundamental resource in quantum information. Here the following question is addressed : which are the deeper roots of entanglement that may help in its better understanding and use ? The answer is that one can reproduce the phenomenon of entanglement in a far more general and simple way, a way that goes much beyond the usual one which is limited to the framework of tensor products of vector spaces. In this general approach to entanglement presented here - and much unlike in the particular setup of tensor products of vector spaces - the spaces involved can be rather arbitrary sets, just as in the case of Cartesian products. In particular, they need not even have any algebraic structure. Thus they do not have to be vector spaces, groups or even semigroups.
In Section 2.4 we learned that a composite quantum system, composed of two subsystems A and B , is associated with a tensor product space HA ? HB . Composite quantum systems have already played an important role in many instances in this text, especially in open systems and measurement models. In this chapter we dive deeper into tensor product spaces, discovering some strange implications of the quantum theory of composite quantum systems. As a result of its tensor product structure, quantum theory has embedded into it the phenomenon of entanglement , which is often seen as the foremost quantum feature. As early as in 1935 it puzzled Erwin Schrödinger, who introduced the German term Verschränkung [127] and Albert Einstein, who, together with Boris Podolski and Nathan Rosen, used the specific example of an entangled state to argue that quantum theory is incomplete [58]. In the 1960s John Bell demonstrated that the consequences of entanglement are incompatible with the Einstein– Podolsky–Rosen model of local realism [11], [12]. In recent years entanglement has been recognized as the resource for quantum information processing. Nowadays entanglement theory is a highly developed subject, and we can present only the mathematical basics. For the reader with a deeper interest in entanglement we recommend the recent reviews by Horodecki et al . [80] and Plenio et al . [118], which also cover the quantum information aspects of entanglement. To simplify our discussion we will assume that the Hilbert spaces are finite dimensional.
The group invariance of entanglement is obtained within a very general and simple setup of the latter, given by a recently introduced considerably extended concept of tensor products. This general approach to entanglement - unlike the usual one given in the particular setup of tensor products of vector spaces - turns out not to need any specific algebraic structure. The resulting advantage is that, entanglement being in fact defined by a negation, its presence in a general setup increases the chances of its manifestations, thus also its availability as a resource.
Entanglement is not one but rather the characteristic trait of quantum mechanics. Erwin Schrödinger Introducing entanglement So far, when working in a Hilbert space that is a tensor product of the form H = H A ⊗ H B , we were really interested in only one of the factors; the other factor played the role of an ancilla describing an environment outside our control. Now the perspective changes: we are interested in a situation where there are two masters. The fate of both subsystems are of equal importance, although they may be sitting in two different laboratories. The operations performed independently in the two laboratories are described using operators of the form Φ A ⊗ 1 and 1 ⊗ Φ B , respectively, but due perhaps to past history, the global state of the system may not be a product state. In general, it may be described by an arbitrary density operator ρ acting on the composite Hilbert space H . The peculiarities of this situation were highlighted in 1935 by Einstein, Podolsky and Rosen (1935). Their basic observation was that if the global state of the system is chosen suitably then it is possible to change, and to some extent to choose, the state assignment in laboratory A by performing operations in laboratory B . The physicists in laboratory A will be unaware of this until they are told, but they can check in retrospect that the experiments they performed were consistent with the state assignment arrived at from afar – even though there was an element of choice in arriving at that state assignment.
A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones. ”History is written with the feet...” Ex-Chairman Mao ”Science in not done scientifically, since it is mostly done by non-scientists... ” Anonymous 1 1. Preliminaries As seen in [7], there is a natural division of physical systems in Cartesian
We present a notion of generalized entanglement which goes beyond the conventional definition based on quantum subsystems. This is accomplished by directly defining entanglement as a property of quantum states relative to a distinguished set of observables singled out by Physics. While recovering standard entanglement as a special case, our notion allows for substantially broader generality and flexibility, being applicable, in particular, to situations where existing tools are not directly useful.