There is method of a Schmidt decomposition, negativity, purification etc to quantify entanglement of quantum state. What is the advantage of local operations and classical communication (LOCC)? And why we transform one entangled state to other?
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3$\begingroup$ It may be useful for readers to know that LOCC stands for "local operations and classical communication" $\endgroup$– Clara Díaz SanchezCommented Dec 18, 2019 at 11:10
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3$\begingroup$ "Schmidt decomposition, negativity, purification etc" typically all quantify entanglement up to LOCC. You should try to make your question more focused. $\endgroup$– Norbert SchuchCommented Dec 19, 2019 at 13:37
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2 Answers
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Measures and ideas adopted in information theory, as in science more generally, are adopted because of their usefulness in capturing some wide-ranging aspect of an informational or physical scenario in a succinct way, or in a way which makes clear physical sense. In the present example LOCC has a clear physical sense, and it is also useful because if we adopt this as a concept when thinking about measures of entanglement then it turns out we have a useful concept which clarifies our thinking. Measures of entanglement which otherwise might appear to be distinct turn out to be equivalent up to LOCC and this is a highly simplifying equivalence: one which brings many protocols or measures into one class while maintaining a useful distinction from other classes.
answered Dec 26, 2022 at 11:23
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What is charming about LOCC operations is that they have operational significance. The form the broadest class of operations where communication parties act locally on their systems and are moreover able to send classical messages.
The "entanglement of formation" gets its operational significance via this class. The (regularized) entanglement of formation of a bipartite state is the optimal asymptotic conversion rate when producing systems in this state from pure maximally entangled systems. See link.
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1$\begingroup$ But aren't all entanglement measures by definition LOCC monotones, including the ones listed in the question? $\endgroup$ Commented Dec 19, 2019 at 13:37
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1$\begingroup$ @NorbertSchuch is completely right -- one usually should demand LOCC monotonicity of a functional used as entanglement measure. My remark above heads in a different direction. The entanglement of formation turns out to be the optimal rate in an information theoretic optimization problem where LOCC is the allowed protocol class. $\endgroup$– GisbertCommented Dec 19, 2019 at 16:31