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Register a new uservon Neumanns Minimax Theorem for Continuous Quantum Games
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The concept of a classical player, corresponding to a classical random variable, is extended to include quantum random variables in the form of self adjoint operators on infinite dimensional Hilbert space. A quantum version of Von Neumanns Minimax theorem for infinite dimensional (or continuous) games is proved.
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Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumanns theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumanns. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
Using the Trotter-Kato theorem we prove the convergence of the unitary dynamics generated by an increasingly singular Hamiltonian in the case of a single field coupling. The limit dynamics is a quantum stochastic evolution of Hudson-Parthasarathy type, and we establish in the process a graph limit convergence of the pre-limit Hamiltonian operators to the Chebotarev-Gregoratti-von Waldenfels Hamiltonian generating the quantum Ito evolution.
We consider the general physical situation of a quantum system $H_0$ interacting with a chain of exterior systems $bigotimes_N H$, one after the other, during a small interval of time $h$ and following some Hamiltonian $H$ on $H_0 otimes H$. We discuss the passage to the limit to continuous interactions ($h to 0$) in a setup which allows to compute the limit of this Hamiltonian evolution in a single state space: a continuous field of exterior systems $otimes_{R} H$. Surprisingly, the passage to the limit shows the necessity for 3 different time scales in $H$. The limit evolution equation is shown to spontaneously produce quantum noises terms: we obtain a quantum Langevin equation as limit of the Hamiltonian evolution. For the very first time, these quantum Langevin equations are obtained as the effective limit from repeated to continuous interactions and not only as a model. These results justify the usual quantum Langevin equations considered in continual quantum measurement or in quantum optics. We show that the three time scales correspond to the normal regime, the weak coupling limit and the low density limit. Our approach allows to consider these two physical limits altogether for the first time. Their combination produces an effective Hamiltonian on the small system, which had never been described before. We apply these results to give an Hamiltonian description of the von Neumann measurement. We also consider the approximation of continuous time quantum master equations by discrete time ones. In particular we show how any Lindblad generator is obtained as the limit of completely positive maps.
We give a mathematically rigorous derivation of Ehrenfests equations for the evolution of position and momentum expectation values, under general and natural assumptions which include atomic and molecular Hamiltonians with Coulomb interactions.
Given $n times n$ real symmetric matrices $A_1, dots, A_m$, the following {it spectral minimax} property holds: $$min_{X in mathbf{Delta}_n} max_{y in S_m} sum_{i=1}^m y_iA_i bullet X=max_{y in S_m} min_{X in mathbf{Delta}_n} sum_{i=1}^m y_iA_i bullet X,$$ where $S_m$ is the simplex and $mathbf{Delta}_n$ the spectraplex. For diagonal $A_i$s this reduces to the classic minimax.
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