Journal of Mathematical Physics, since 1960, publishes some of the best papers from outstanding mathematicians and physicists. It was the first journal in the field of mathematical physics and publishes research that connects the application of mathematics to problems in physics, as well as illustrates the development of mathematical methods suitable for such applications and for the formulation of physical theories.
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We show a fundamental limitation in the description of quantum manybody mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.

A novel class of Nbody problems is identified, with N an arbitrary positive integer (N ≥ 2). These models are characterized by Newtonian (“accelerations equal forces”) equations of motion describing N equal pointparticles moving in the complex zplane. These highly nonlinear equations feature N arbitrary coupling constants, yet they can be solved by algebraic operations and if all the N coupling constants are real and rational the corresponding Nbody problem is isochronous: its generic solutions are all completely periodic with an overall period T independent of the initial data (but many solutions feature subperiods T/p with p integer). It is moreover shown that these models are Hamiltonian.

In 1932, Dirac proposed a formulation in terms of multitime wave functions as candidate for relativistic manyparticle quantum mechanics. A wellknown consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spincoupling. Under suitable assumptions on the differentiability of possible solutions, we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills the physically desirable Poincaré invariance. We conclude that in this sense, Dirac’s multitime formalism does not allow to model interaction by multiplication operators, and briefly point out several promising approaches to interacting models one can instead pursue.

In this paper, we will review the coadjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant–Kirillov–Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1form, settling the issues about its very definition and explicit computation. Moreover, the fibration of coadjoint orbits, seen as spaces of mixed states, is also discussed.

The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂ^{d1} ⊗ ℂ^{d2} are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.