Abstract
Entanglement is a central feature of quantum mechanical systems that lies at the heart of most quantum information processing tasks. In this chapter, based on lectures by K.R. Parthasarathy, we examine the important question of quantifying entanglement in bipartite quantum systems. We begin with a brief review of the mathematical framework of quantum states and probabilities and formally define the notion of entanglement in composite Hilbert spaces via the Schmidt number and Schmidt rank. For pure bipartite states in a finite-dimensional Hilbert space, we study the question of how many subspaces exist in which all unit vectors are of a certain minimal Schmidt number k. For mixed states, we describe the role of k-positive maps in identifying states whose Schmidt rank exceeds k. Finally, we compute the Schmidt rank of an interesting class of states called generalized Werner states.
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Notes
- 1.
In the bra-ket notation, \(\text{Tr}[\vert u\rangle \langle v\vert ] \equiv \langle v\vert u\rangle\).
- 2.
Since deterministic state changes are effected by unitary operators in quantum theory, unitaries are often referred to as quantum gates in the quantum computing literature.
- 3.
A remark on notation: we use \(\mathbb{I}\) to denote the identity map \(\mathbb{I}: \mathcal{B}(\mathcal{H}) \rightarrow \mathcal{B}(\mathcal{H})\) , with \(\mathbb{I}(\rho ) =\rho\) for any \(\rho \in \mathcal{B}(\mathcal{H})\).
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Gupta, V.P., Mandayam, P., Sunder, V.S. (2015). Entanglement in Bipartite Quantum States. In: The Functional Analysis of Quantum Information Theory. Lecture Notes in Physics, vol 902. Springer, Cham. https://doi.org/10.1007/978-3-319-16718-3_2
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DOI: https://doi.org/10.1007/978-3-319-16718-3_2
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16717-6
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