Abstract
Entanglement measures quantify the amount of entanglement between parts of a system, but a considerable part of the literature in Quantum Information Theory has focussed on quantum systems with finitely many degrees of freedom. In this volume, we will focus on the question whether qualitatively new features can arise due to the presence of infinitely many degrees of freedom.
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- 1.
This stands for “local operations and classical communications”. In this volume, we will actually use an even broader class.
- 2.
In classical physics, if \(\mu \) is a measure on a product phase space \(X=X_A \times X_B\) which is, say, absolutely continuous relative to the Lebesgue measure, then we can approximate it with arbitrary precision by sums of product measures \(\sum _i \mu _{Ai} \times \mu _{Bi}\) (e.g. on the dense subspace of smooth observables).
- 3.
It may or may not be possible/desirable to also have other properties such as convexity under convex linear combinations.
- 4.
In fact, as shown in [26], entanglement measures that are well-behaved in the type I-setting can become ill-defined for type III, as is the case e.g. for the “entanglement of formation”. [26] has also shown that the entanglement entropy \(E_R(\rho _0)\) behaves well under a “nuclearity condition”, a technique to which we will come back in the body of the volume.
- 5.
In the body of this volume we will distinguish, for technical reasons, the expectation value function of a statistical operator \(\omega ( \ . \ ) = \text {Tr}(\ . \ \rho )\) and the statistical operator \(\rho \) itself.
- 6.
Formula (1.13) below suggests that the upper bound can be improved to \(C_\infty (\delta ) e^{-mr(1-\delta )}\) for each \(\delta >0\).
- 7.
We cannot put \(\kappa \) or \(\delta \) to zero, since the asymptotic bound holds, roughly speaking, when \(1/(\delta \kappa ) \lesssim mr\).
- 8.
It is defined as \(H_n(\rho ) = \frac{1}{1-n} \,\text {ln}\, \text {Tr}\rho ^n \ .\)
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Hollands, S., Sanders, K. (2018). Introduction. In: Entanglement Measures and Their Properties in Quantum Field Theory. SpringerBriefs in Mathematical Physics, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-319-94902-4_1
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