We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A V acting on the Hilbert space completion of V and prove that the isomorphism class of A V does not depend on the choice of the scalar product on V . We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V , the map W → A W gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of A V . Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known c = 1 unitary vertex operator algebras, the moonshine vertex operator algebra, together with their coset and orbifold subalgebras, turn out to be strongly local. We give various applications of our results. In particular we show that the even shorter Moonshine vertex operator algebra is strongly local and that the automorphism group of the corresponding conformal net is the Baby Monster group. We prove that a construction of Fredenhagen and Jörß gives back the strongly local vertex operator algebra V from the conformal net A V and give conditions on a conformal net A implying that A = A V for some strongly local vertex operator algebra V .if S ⊂ B(H) is self-adjoint then S ′ is a self-adjoint subalgebra of B(H) which is also unital, i.e. 1 H ∈ S ′ .A self-adjoint subalgebra M ⊂ B(H) is called a von Neumann algebra if M = M ′′ . Accordingly, (S ∪ S * ) ′ is a von Neumann algebra for all subsets S ⊂ B(H) and W * (S) ≡ (S ∪ S * ) ′′ is the smallest von Neumann algebra containing S.A von Neumann algebra M is said to be a factor if M ′ ∩ M = C1 H , i.e. M has a trivial center. B(H) is a factor for any Hilbert space H. Its isomorphism class as an abstract complex * -algebra only depends on the Hilbertian dimension of H. A von Neumann algebra M isomorphic to some B(H) (here H is not necessarily the same Hilbert space on which M acts) is called a type I factor. If H has dimension n ∈ Z >0 then M is called a type I n factor while if H is infinite-dimensional then M is called a type I ∞ factor.There exist factors which are not of type I. They are divided in two families: the type II factors (type II 1 or type II ∞ ) and type III factors (type III λ , λ ∈ [0, 1], cf.[23]).If M and N are von Neumann algebras and N ⊂ M then N is called a von Neumann subalgebra of M. If M is a factor then a von Neumann subalgebra N ⊂ M which is also a factor is called a subfactor. The theory of subfactors is a central topic in the theory of operator algebras and in its applications to quantum field theory. Subfactor theory was initiated in the seminal work [56] where V. Jones introduced and studied an index [M : N] for type II 1 factors. Subfac...
We exhibit an infinite family of triplets of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, F (a, b). However, in the main result of the paper we also prove that for any values of the parameters (a, b), the standard basis and F (a, b) cannot be extended to a MUB-quartet. The main novelty lies in the method of proof which may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.
A Möbius covariant net of von Neumann algebras on S 1 is diffeomorphism covariant if its Möbius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4-regular net such an extension is unique: the local algebras together with the Möbius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Möbius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff + (S 1 ).
A game is played by a team of two -say Alice and Bob -in which the value of a random variable x is revealed to Alice only, who cannot freely communicate with Bob. Instead, she is given a quantum n-level system, respectively a classical n-state system, which she can put in possession of Bob in any state she wishes. We evaluate how successfully they managed to store and recover the value of x by requiring Bob to specify a value z and giving a reward of value f (x, z) to the team.We show that whatever the probability distribution of x and the reward function f are, when using a quantum n-level system, the maximum expected reward obtainable with the best possible team strategy is equal to that obtainable with the use of a classical n-state system. The proof relies on mixed discriminants of positive matrices and -perhaps surprisingly -an application of the Supply-Demand Theorem for bipartite graphs.As a corollary, we get an infinite set of new, dimension dependent inequalities regarding positive operator valued measures and density operators on complex n-space.As a further corollary, we see that the greatest value, with respect to a given distribution of x, of the mutual information I(x; z) that is obtainable using an n-level quantum system equals the analogous maximum for a classical n-state system.
Abstract:We continue the analysis of the set of locally normal KMS states w.r.t. the translation group for a local conformal net A of von Neumann algebras on R. In the first part we have proved the uniqueness of the KMS state on every completely rational net. In this second part, we exhibit several (non-rational) conformal nets which admit continuously many primary KMS states. We give a complete classification of the KMS states on the U (1)-current net and on the Virasoro net Vir 1 with the central charge c = 1, whilst for the Virasoro net Vir c with c > 1 we exhibit a (possibly incomplete) list of continuously many primary KMS states. To this end, we provide a variation of the ArakiHaag-Kastler-Takesaki theorem within the locally normal system framework: if there is an inclusion of split nets A ⊂ B and A is the fixed point of B w.r.t. a compact gauge group, then any locally normal, primary KMS state on A extends to a locally normal, primary state on B, KMS w.r.t. a perturbed translation. Concerning the non-local case, we show that the free Fermi model admits a unique KMS state.
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