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arXiv:quant-ph/0301015 · Physical Review A

Lower bound on entanglement of formation for the qubit-qudit system

E. Gerjuoy

Published May 23, 2003• 57 citations
arXivHTMLPDF

Abstract

Wootters [Phys. Rev. Lett. $80,$ 2245 (1998)] has derived a closed formula for the entanglement of formation (EOF) of an arbitrary mixed state in a system of two qubits. There is no known closed-form expression for the EOF of an arbitrary mixed state in any system more complicated than two qubits. This paper, via a relatively straightforward generalization of Wootters' original derivation, obtains a closed-form lower bound on the EOF of an arbitary mixed state of a system composed of a qubit and a qudit (a d-level quantum system, with $d>~3).$ The derivation of the lower bound is detailed for a system composed of a qubit and a qutrit $(d=3);$ the generalization to $d>3$ then follows readily.

DOI
10.1103/physreva.67.052308

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Wootters [Phys. Rev. Lett. 80,80,80, 2245 (1998)] has derived a closed formula for the entanglement of formation (EOF) of an arbitrary mixed state in a system of two qubits. There is no known closed-form expression for the EOF of an arbitrary mixed state in any system more complicated than two qubits. This paper, via a relatively straightforward generalization of Wootters' original derivation, obtains a closed-form lower bound on the EOF of an arbitary mixed state of a system composed of a qubit and a qudit (a d-level quantum system, with d> 3).d>~3).d> 3). The derivation of the lower bound is detailed for a system composed of a qubit and a qutrit (d=3);(d=3);(d=3); the generalization to d>3d>3d>3 then follows readily.
We present observable lower and upper bounds for the entanglement of formation (EOF) and quantum discord (QD), which facilitates estimates of EOF and QD for arbitrary experimental unknown states in finite-dimensional bipartite systems. These bounds can be easily obtained by a few experimental measurements on a twofold copy $\ensuremath{\varrho}\ensuremath{\bigotimes}\ensuremath{\varrho}$ of the mixed states. Based on our results, we use the experimental measurement data of the real experiment given by Schmid et al. [Phys. Rev. Lett. 101, 260505 (2008)] to obtain the lower and upper bounds of EOF and QD for the experimental unknown state.
The entanglement of formation (EOF), which quantifies the required minimally physical resources to prepare a quantum state, plays important roles in many quantum information processes. Due to the convex-roof definition, it is hard to find the explicit analytic formulae of EOF, except for some special quantum systems. The correlation matrix of a quantum state is closely related to entanglement property. In this paper, we use the Lagrange-multiplier approach to obtain an analytic lower bound of EOF for arbitrary dimensional bipartite states, which improves the previous conclusions. Numerical example shows that the bound is also valid for some PPT entangled states.
By investigating the convex property of the function RRR, which appeared when computing the entanglement of formation for isotropic states [Phys. Rev. Lett. 85, 2625 (2000)] and in a tight lower bound of entanglement of formation for arbitrary bipartite mixed states [Phys. Rev. Lett. 95, 210501 (2005)], we show analytically that the results in these papers are valid not only for dimensions two and three but for any dimensions.
Entanglement cost is the asymptotic rate of Bell pairs required to prepare a quantum state. Its regularized definition has made exact evaluation difficult, even for isotropic states (i.e., depolarized maximally entangled states). In this work, we determine the entanglement cost of every qubit isotropic state and, more generally, every two-qubit Bell-diagonal state. Our proof uses a family of suitably tuned qubit semigroups to transform a general log-Sobolev entropy bound into supporting lines of Wootters' function. A recent exact tensorization theorem of Dong et al. [arXiv:2606.17729] extends these bounds to arbitrarily many copies, yielding a general lower bound on entanglement cost. Matching this bound with the entanglement of formation also determines the exact cost for a broader class of two-qubit states. For qudit isotropic states, we derive another general lower bound from the full depolarizing 2→32\to 32→3 norm. This bound substantially improves on the PPT-relative entropy of entanglement and nearly matches the entanglement of formation, leaving a gap of at most 2%2\%2% of log⁡d\log dlogd that vanishes as ddd grows. Finally, the exact qubit results and qudit bounds extend to the entanglement cost of preparing quantum depolarizing channels under both parallel and sequential strategies.