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arXiv:1805.12274

Necessary and sufficient condition for the existence of Schmidt decomposition in multipartite Hilbert spaces

Spandan Das, Goutam Paul

Published May 31, 2018• 0 citations
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Abstract

Pati (Physics Letters A, 2000) derived a sufficient condition for the existence of Schmidt decomposition in tripartite Hilbert spaces. In this paper, we show that the condition is erroneous by demonstrating some counter-examples. Moreover, we suitably modify the condition and provide a correctness proof. We also show for the first time how this can be generalized to $n$-partite Hilbert spaces for any $n\geq 2$. Finally, we prove that this condition is also a necessary condition.

DOI
10.48550/arxiv.1805.12274

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Pati (Physics Letters A, 2000) derived a sufficient condition for the existence of Schmidt decomposition in tripartite Hilbert spaces. In this paper, we show that the condition is erroneous by demonstrating some counter-examples. Moreover, we suitably modify the condition and provide a correctness proof. We also show for the first time how this can be generalized to nnn-partite Hilbert spaces for any n≥2n\geq 2n≥2. Finally, we prove that this condition is also a necessary condition.
In this report, we discuss a necessary and sufficient condition for the existence of Schmidt decomposition of a pure state in a multipartite Hilbert space. Schmidt decomposition exists for any pure state in a bipartite Hilbert space. It is extremely useful since it reduces the number of terms needed to represent a pure state using orthonormal bases of the respective parts. However, for 3 or more partite Hilbert spaces, Schmidt decomposition does not exist in general for an arbitrary pure state. In this thesis, we derive a condition that states whether a state in an n-partite Hilbert space is Schmidt-decomposable or not, i.e., whether we can represent it through the orthonormal bases of the respective parts using less number of terms that is required in general.
For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system under certain condition. If the partial inner product of a basis (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled basis, then the Schmidt decomposition for a tripartite system exists. In this case the reduced density matrix of each of the subsystem has equal spectrum in the Schmidt basis.
It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we derive several generally valid purity-type formulas for one-party reductions of rank-1 operators, and we point out relevant relations between the Schmidt decomposition and the Bloch representation of bipartite pure states.
Multipartite quantum states that cannot be uniquely determined by their reduced states of all proper subsets of the parties exhibit some inherit `high-order' correlation. This paper elaborates this issue by giving necessary and sufficient conditions for a pure multipartite state to be locally undetermined, and moreover, characterizing precisely all the pure states sharing the same set of reduced states with it. Interestingly, local determinability of pure states is closely related to a generalized notion of Schmidt decomposition. Furthermore, we find that locally undetermined states have some applications to the well-known consensus problem in distributed computation. To be specific, given some physically separated agents, when communication between them, either classical or quantum, is unreliable and they are not allowed to use local ancillary quantum systems, then there exists a totally correct and completely fault-tolerant protocol for them to reach a consensus if and only if they share a priori a locally undetermined quantum state.