The role of transposition and CPT operation for entanglement
Introduction
If in quantum theory we are dealing with two independent systems we can distinguish between separable states, i.e., states that are linear combination of factoring states and their complement, the entangled states. If the two independent systems are in an entangled state this can possibly be observed by the violation of the Bell inequality: we can find operators, whose expectation values are smaller than 1 in all separable states, and this is also to be expected by arguing on classical probabilistic grounds. But nevertheless there exist also other states in which their expectation value can be larger than 1 and therefore these states are entangled. Special operators of this type are given by the CHSH inequality [1], and operators of this form suffice to distinguish between pure separable and pure entangled states [2] but not necessarily if the states are mixed. But there is also another characterization of separable states offered in [3]: entangled states do not necessarily stay positive if we apply a positive but not completely positive map on only one of the systems, and it is always possible to find such a positive map for a given entangled state. The proof for this statement is given in [3] for finite dimensional algebras. For infinite algebras this characterization remains valid insofar, as a state that violates the positivity condition cannot be separable.
In fact infinite algebras should be taken into account for several reasons. Entanglement was already observed on macroscopic systems [4], [5]. Also we can expect that entanglement of crystals is possible and shows interesting features. On the other hand, any subsystem can never completely be decoupled from its surrounding, therefore even for photon experiments we should be aware that Alice and Bob are concerned with the infinite algebras of photon fields restricted to local measurements. Nevertheless the entanglement will be observed by a finite number of measurements that can be expressed by projections in an appropriately chosen finite dimensional subalgebra. The key feature is the fact that entanglement is monotonic for increasing algebras, the state reduced to a smaller algebra has less entanglement [6]. Therefore, it seems a natural question to study how far the entanglement can be proved by examining whether positive maps on the infinite algebras act as entanglement witnesses and if so, how these positive maps on the infinite algebras act on finite dimensional subalgebras.
Differently from completely positive maps for which we know how they can be constructed [7] not too much is known for positive maps. In fact entangled states can be used to construct positive maps [8]. On the other hand, there are some popular positive maps: from the mathematical point of view we can consider the transposed map for type I algebras, T(A)=At, and the map implemented by the modular conjugation [9], [10]. We will argue that for type I algebras they are equivalent so that the modular conjugation serves as an extension of the transposition to those algebras (type II and III) where the transposition is not well defined. The description will turn out to be useful also for considering the restriction of the positive map to subalgebras. This is important because we take as definition of entanglement of infinite algebras, that the state restricted to some sufficiently large subalgebras is entangled [6].
Also in physics we are familiar with maps that are positive but not completely positive: the time reversal in non-relativistic quantum mechanics satisfies this requirement and was already used in [11] to characterize non separable Gauss-states. Its generalization to relativistic quantum field theory is the CPT-map [12]. Combined with the result of Bisognano–Wichmann [13] that relates the CPT-map to the modular conjugation this allows to prove that for all causally separated diamond algebras the vacuum state violates positivity in the sense of [3] and is therefore in the definition of [3] distillably entangled.
In Section 2 we will repeat the definition of the modular conjugation and relate it to the transposition. In Section 3, respectively, Section 4 we will give its explicit form as we can observe it for quasifree states of bosons, respectively, fermions. In Section 5 we will use the modular conjugation to formulate an inequality that is powerful enough to distinguish between pure separable and pure entangled states but differs from the CHSH inequality. Finally, in Section 6 we will apply this inequality to relativistic quantum field theory and prove that locally separated regions are always distillably entangled in a dense set of states that includes the vacuum.
Section snippets
The transposition map vs. the modular conjugation
Linear maps between two algebras and are positive, if they preserve positivity They are completely positive if the extension of Λ to is also positive for all full complex matrix algebras Mn. If or is an Abelian algebra then a positive map is automatically completely positive. Therefore, the existence of positive maps that are not completely positive is a typical quantum phenomenon. In [3] this quantum phenomenon was successfully
The transposition for bosons in quasifree states
Already in [11] the transposition was used to determine whether a Gaussian state of two particles is separable or not. The method can be generalized to analyse quasifree states over the CCR algebra.
The CCR algebra is built over a one particle Hilbert space by considering the creation and annihilation operators b(f) and with satisfying A quasifree gauge invariant state is given by Positivity of ω requires A⩾1.The state
The transposition for fermions in quasifree states
A similar method is applicable if we are considering the entanglement of fermions in quasifree states. Again we describe the system by creation and annihilation operators, now obeying anticommutation rules If we therefore consider the subalgebras corresponding to orthogonal projections , i.e., and similarly , then the even polynomials in creation and annihilation operators will commute, when PQ=0. The odd
The transposition implies an expectation inequality
Though every γ in the equivalence class of transpositions indicates whether the state is distillably entangled, this non-positivity of ω∘(γ⊗1) is felt in general by different operators. Therefore, it is useful to express this fact already by examining some special expectation values. This also brings us in close contact with the Bell inequality [19] or the description that the set of separable states can be characterized by its tangent functionals [3], [20], [21].
Again we use the γ that is
The operator inequality for relativistic quantum field theory
The above operator inequality can also be applied to prove, that local regions in relativistic quantum field theories, that are causally separated and therefore independent, are entangled in the vacuum state. Already in [23] this question was raised and as characteristic for this entanglement the CHSH inequality was used, but could not supply us with a complete answer. In [6] we studied the entanglement quantitatively, where the amount of entanglement was measured by the entanglement of
Conclusion
Translating the transposition in the language of the GNS representation in a map combining the modular conjugation with an arbitrary isomorphism between algebra and commutant we have extended the definition of the transposition to infinite algebras. This transposition serves as entanglement witness also for subalgebras of the complete system. Especially in quasifree states of bosons as well as of fermions it can be expressed already on the level of the two point functions offering a handy
Acknowledgements
It is a pleasure to thank Walter Thirring for his interest and many useful suggestions for improvements on the manuscript.
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