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Abstract The observables associated with a quantum system S form a non-commutative algebra A S . It is assumed that a density matrix ρ can be determined from the expectation values of observables. But A S admits inner automorphisms a ↦ u a u − 1 , a , u ∈ A S , u * u = u u * = 1 , so that its individual elements can be identified only up to unitary transformations. So since Tr ρ ( uau *) = Tr( u * ρu ) a , only the spectrum of ρ , or its characteristic polynomial, can be determined in quantum mechanics. In local quantum field theory, ρ cannot be determined at all, as we shall explain. However, abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables in abelian algebras A M ⊂ A S ( M for measurement, S for system). We study the uncertainties in extending ρ | A M to ρ | A S (the determination of which means measurement of A S ) and devise a protocol to determine ρ | A S ≡ ρ by determining ρ | A M for different choices of A M . The problem we formulate and study is a generalization of the Kadison–Singer theorem. We give an example where the system S is a particle on a circle and the experiment measures the abelian algebra of a magnetic field B coupled to S . The measurement of B gives information about the state ρ of the system S due to operator mixing. Associated uncertainty principles for von Neumann entropy are discussed in the appendix, adapting the earlier work by Białynicki-Birula and Mycielski (1975 Commun. Math. Phys. 44 129) to the present case.
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An alternative method for extracting the von Neumann entropy from Rényi entropies
A bstract An alternative method is presented for extracting the von Neumann entropy − Tr( ρ ln ρ ) from Tr( ρ n ) for integer n in a quantum system with density matrix ρ . Instead of relying on direct analytic continuation in n , the method uses a generating function − Tr{ ρ ln[(1 − zρ )/(1 − z )]} of an auxiliary complex variable z . The generating function has a Taylor series that is absolutely convergent within |z| < 1, and may be analytically continued in z to z = −∞ where it gives the von Neumann entropy. As an example, we use the method to calculate analytically the CFT entanglement entropy of two intervals in the small cross ratio limit, reproducing a result that Calabrese et al. obtained by direct analytic continuation in n . Further examples are provided by numerical calculations of the entanglement entropy of two intervals for general cross ratios, and of one interval at finite temperature and finite interval length.
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- PAR ID:
- 10228361
- Date Published:
- Journal Name:
- Journal of High Energy Physics
- Volume:
- 2021
- Issue:
- 1
- ISSN:
- 1029-8479
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/JHEP01(2021)042
