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Register a new userUnfrustrated Qudit Chains and their Ground States
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We investigate chains of d dimensional quantum spins (qudits) on a line with generic nearest neighbor interactions without translational invariance. We find the conditions under which these systems are not frustrated, i.e. when the ground states are also the common ground states of all the local terms in the Hamiltonians. The states of a quantum spin chain are naturally represented in the Matrix Product States (MPS) framework. Using imaginary time evolution in the MPS ansatz, we numerically investigate the range of parameters in which we expect the ground states to be highly entangled and find them hard to approximate using our MPS method.
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The existence of a spectral gap above the ground state has far-reaching consequences for the low-energy physics of a quantum many-body system. A recent work of Movassagh [R. Movassagh, PRL 119 (2017), 220504] shows that a spatially random local quantum Hamiltonian is generically gapless. Here we observe that a gap is more common for translation-invariant quantum spin chains, more specifically, that these are gapped with a positive probability if the interaction is of small rank. This is in line with a previous analysis of the spin-$1/2$ case by Bravyi and Gosset. The Hamiltonians are constructed by selecting a single projection of sufficiently small rank at random, and then translating it across the entire chain. By the rank assumption, the resulting Hamiltonians are automatically frustration-free and this fact plays a key role in our analysis.
Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an $N$-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary $SU(N)$-invariant counterpart of the $N$-level state space $mathfrak{P}_N$, i.e., the unitary orbit space $mathfrak{P}_N/SU(N)$. It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of $mathfrak{P}_N/SU(N)$. To illustrate the general situation, a detailed description of $mathfrak{P}_N/SU(N)$ for low-level systems: qubit $(N=2),,$ qutrit $(N=3),,$ quatrit $(N=4),$ - will be given.
We consider an open quantum system, with dissipation applied only to a part of its degrees of freedom, evolving via a quantum Markov dynamics. We demonstrate that, in the Zeno regime of large dissipation, the relaxation of the quantum system towards a pure quantum state is linked to the evolution of a classical Markov process towards a single absorbing state. The rates of the associated classical Markov process are determined by the original quantum dynamics. Extension of this correspondence to absorbing states with internal structure allows us to establish a general criterion for having a Zeno-limit nonequilibrium stationary state of arbitrary finite rank. An application of this criterion is illustrated in the case of an open XXZ spin-1/2 chain dissipatively coupled at its edges to baths with fixed and different polarizations. For this system, we find exact nonequilibrium steady-state solutions of ranks 1 and 2.
We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers $j=1/2, 1, 3/2, ...$, introduced by Miyazaki {it et al.}, each of which the walkers wave function has $2j+1$ components and hopping range at each time step is $2j$. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konnos density function. Since Konnos density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the $(2j+1)$-component model can have $2j+1$ pikes, when $2j+1$ is even. When $j$ becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the $j to infty$ limit.
We investigate genuine multipartite nonlocality of pure permutationally invariant multimode Gaussian states of continuous variable systems, as detected by the violation of Svetlichny inequality. We identify the phase space settings leading to the largest violation of the inequality when using displaced parity measurements, distinguishing our results between the cases of even and odd total number of modes. We further consider pseudospin measurements and show that, for three-mode states with asymptotically large squeezing degree, particular settings of these measurements allow one to approach the maximum violation of Svetlichny inequality allowed by quantum mechanics. This indicates that the strongest manifestation of genuine multipartite quantum nonlocality is in principle verifiable on Gaussian states.
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