We discuss a scaling limit of the spectral distribution of the adjacency operator (or Laplacian) on the Johnson graph J(v,d) with respect to the Gibbs state associated with the graph. (The adjacency operator on J(v,d), whose vertices consist of the d-subsets of a v-set, gives us the Bernoulli–Laplace diffusion model.) We compute the limit distribution and its moments exactly in the situation of infinite degree (v,d→∞) and zero temperature (β→∞) limit where the three parameters v, d and β keep appropriate scaling balances. A method of quantum decomposition of an adjacency operator plays a key role for expressing the limit moments in terms of a creation operator, an annihilation operator and a number operator on a suitable Hilbert space. Using this expression, we analyze the limit moments in detail in combinatorial and analytical ways. In our previous work [Hora, A., Probab. Theory Relat. Fields, 118: 115–130 (2000)], a partial solution was given where v=2d and some additional constraints of scaling were assumed. In this note, we remove all such restrictions.

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In this paper we study a one-dimensional quantum random walk with the Hadamard transformation which is often called the Hadamard walk. We construct the Hadamard walk using a transition matrix on probability amplitude and give some results on symmetry of probability distributions for the Hadamard walk.

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On one-dimensional two-way infinite quantum lattice system, a property of translationally invariant stationary states with nonvanishing current expectation is investigated. We consider GNS representation with respect to such a state, on which we have a group of space-time translation unitary operators. We show that spectrum of the unitary operators, energy-momentum spectrum with respect to the state, has a singularity at the origin.

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For given two regions of the lattice, assume that our state has a pure-state restriction for each of them. We discuss whether its restrictions to the intersection and the union regions are pure states or not. It is immediate to see that the answer is affirmative for any tensor-product systems. (We, however, include its proof for the completeness sake.) In this note, we consider mostly CAR systems. We show that the assertion above holds for any such a state of finite-dimensional CAR systems. For infinite-dimensional CAR systems, assuming additionally the product properties between each of the regions and its complement region on the states satisfying the standing assumption, we can show the purity on the intersection region for them. However, the latter part of the assertion will not always hold for the infinite-dimensional CAR systems unlike for the finite-dimensional CAR or any dimensional tensor-product systems. We establish the criterions for the purity and the non-purity on the union region for those states of the infinite-dimensional CAR systems.

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A general method for obtaining a vacuum spectral distribution of the adjacency matrix of a star graph is established within the framework of quantum probability theory. The spectral distribution tends asymptotically to the Bernoulli distribution as the number of leaves of a star graph tends to the infinity.

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The aim of this paper is to show some properties of translation-invariant quantum Markov states introduced by Accardi and Frigerio. The KMS condition and the mean entropies of translation-invariant quantum Markov states are also considered.

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We show that the Boolean and the Fermi convolutions are minimal in the sense that the subset of partitions involved in the corresponding moment-cumulant formula cannot be decreased without violating some natural requirements for convolution laws.

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The Heisenberg uncertainty relation for measurement noise and disturbance states that any position measurement with noise ε brings the momentum disturbance not less than hbar⁄2ε. However, this relation holds only for restricted class of measurements. Here, we discuss universally valid uncertainty relations for measurement noise and disturbance, which hold for all the possible quantum measurements.

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