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Register a new userKraus representation of destruction of states for one qudit
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Added by
Kordian A. Smolinski
Publication date
2003
fields
Physics
and research's language is
English
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Quantum operations arise naturally in many fields of quantum information theory and quantum computing. One of the simplest example of quantum operation is the von Neumann-Lueders measurement. Destruction of states in quantum mechanics can be treated as a supplement to the von Neumann-Lueders measurement [P. Caban, J. Rembielinski, K. A. Smolinski and Z. Walczak, J. Phys. A 35, 3265 (2002)]. We show that destruction of states in one-qudit system is a quantum operation by finding its Kraus representation.
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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 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.
A description of destruction of states on the grounds of quantum mechanics rather than quantum field theory is proposed. Several kinds of maps called supertraces are defined and used to describe the destruction procedure. The introduced algorithm can be treated as a supplement to the von Neumann-Lueders measurement. The discussed formalism may be helpful in a description of EPR type experiments and in quantum information theory.
We address the problem of unambiguously identifying the state of a probe qudit with the state of one of d reference qudits. The reference states are assumed pure and linearly independent but we have no knowledge of them. The state of the probe qudit is assumed to coincide equally likely with either one of the d unknown reference states. We derive the optimum measurement strategy that maximizes the success probability of unambiguous identification and find that the optimum strategy is a generalized measurement. We give both the measurement operators and the optimum success probability explicitly. Technically, the problem we solve amounts to the optimum unambiguous discrimination of d known mixed quantum states.
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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