Parameterizing Qudit States

• URL: http://arxiv.org/abs/2108.12499v1
• Date: Fri, 27 Aug 2021 20:55:25 GMT
• Title: Parameterizing Qudit States
• Authors: Arsen Khvedelidze, Dimitar Mladenov and Astghik Torosyan
• Abstract summary: We will discuss the problem of explicit parameterization of state space of an $N$-level quantum system. It will be demonstrated that the combination of well-known methods of the invariant theory and convex geometry provides useful parameterization for the elements of $mathfrakP_N/SU(N)$
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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