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Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange’s identity and wedge product

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Abstract

Concurrence, introduced by Hill and Wootters (Phys Rev Lett 78:5022, 1997), provides an important measure of entanglement for a general pair of qubits that is faithful: strictly positive for entangled states and vanishing for all separable states. Such a measure captures the entire content of entanglement, providing necessary and sufficient conditions for separability. We present an extension of concurrence to multiparticle pure states in arbitrary dimensions by a new framework using the Lagrange’s identity and wedge product representation of separability conditions, which coincides with the “I-concurrence” of Rungta et al. (Phys Rev A 64:042315, 2001) who proposed by extending Wootters’s spin-flip operator to a so-called universal inverter superoperator. Our framework exposes an inherent geometry of entanglement and may be useful for the further extensions to mixed and continuous variable states.

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Acknowledgements

Bhaskara is thankful to Oliver Knill for pointing out the Binet–Cauchy identity. This work was supported by the National Initiative on Undergraduate Science (NIUS) undertaken by the Homi Bhabha Centre for Science Education, Tata Institute of Fundamental Research (HBCSE–TIFR), Mumbai, India. The authors acknowledge Vijay Singh and Praveen Pathak for continuous encouragement.

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Correspondence to Prasanta K. Panigrahi.

Appendix: Proof of Lagrange’s identity

Appendix: Proof of Lagrange’s identity

Consider

$$\begin{aligned} RHS&=||\overrightarrow{a}\wedge \overrightarrow{b}||^2\\ {}&=\sum _{i=1}^{m-1} \sum _{j=i+1}^m |a_i b_j - a_j b_i|^2 \\&=\frac{1}{2} \sum _{i=1}^{m} \sum _{j=1}^{m} ~ |a_i b_j - a_j b_i|^2 \\&=\frac{1}{2} \sum _{i=1}^{m} \sum _{j=1}^{m} ~ (a_{i}b_{j}-a_{j}b_{i})(\overline{a}_{i}\overline{b}_{j}-\overline{a}_{j}\overline{b}_{i}) \\&=\frac{1}{2} \sum _{i=1}^{m} \sum _{j=1}^{m}~(|a_{i}|^2|b_{j}|^2-2Re(a_ib_j \overline{a}_{j} \overline{b}_{i})+|a_{j}|^2|b_{i}|^2) \\&=\left( \sum _{i=1}^m |a_i|^2\right) \left( \sum _{j=1}^m |b_j|^2\right) -Re\sum _{i=1}^m \sum _{j=1}^m (a_ib_j\overline{a}_{j} \overline{b}_{i}) \\&=\left( \sum _{i=1}^m |a_i|^2\right) \left( \sum _{j=1}^m |b_j|^2\right) - \left| \sum _{i=1}^m a_i \overline{b}_{i}\right| ^2 \\&=\Vert \overrightarrow{a}\Vert ^2 \Vert \overrightarrow{b}\Vert ^2- |\overrightarrow{a} \cdot \overrightarrow{b}|^2=LHS. \end{aligned}$$

Hence the identity.

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Bhaskara, V.S., Panigrahi, P.K. Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange’s identity and wedge product. Quantum Inf Process 16, 118 (2017). https://doi.org/10.1007/s11128-017-1568-0

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