Axiomatic approach

The aim of the axiomatic approach to entanglement measures is to find, classify and study all functions that capture our intuitive notion of what it means to measure entanglement. The approach sets out axioms, i.e. properties, that an entanglement measure should or should not satisfy. This intuitive notion may be based on more practical grounds such as operational definitions. The most striking applications of the axiomatic approach are upper and lower bounds on operational measures such as distillable entanglement, entanglement cost and most recently distillable key. == Properties == The following is a list of properties that have been studied in the context of [[entanglement measures]]. $E(\backslash rho)$ denotes an entanglement measure. '''Norm:''' normalised on maximally entangled states For all $\backslash psi=\backslash frac\{1\}\{\backslash sqrt\{\backslash dim\; A\}\}\backslash sum\_i\; |i\backslash rangle^A|i\backslash rangle^B$, $E(|\backslash psi\backslash rangle\; \backslash langle\; \backslash psi\; |)=\backslash log\; \backslash dim\; A$ with $\backslash \{\; |i\backslash rangle^A\backslash \}$ an orthonormal basis. '''Van Sep:''' vanishing on separable states For all separable states $\backslash rho$, $E(\backslash rho)=0$ . '''PPT Mon:''' PPT monotone For all PPT operations $\backslash rho\; \backslash rightarrow\; \backslash \{p\_i,\; \backslash rho\_i\backslash \},\; E(\backslash rho)\; \backslash geq\; \backslash sum\_i\; p\_i\; E(\backslash rho\_i)$. '''SEP Mon:''' SEP monotone For all separable operations $\backslash rho\; \backslash rightarrow\; \backslash \{p\_i,\; \backslash rho\_i\backslash \},\; E(\backslash rho)\; \backslash geq\; \backslash sum\_i\; p\_i\; E(\backslash rho\_i)$. '''LOCC Mon:''' LOCC monotone For all LOCC operations $\backslash rho\; \backslash rightarrow\; \backslash \{p\_i,\; \backslash rho\_i\backslash \},\; E(\backslash rho)\; \backslash geq\; \backslash sum\_i\; p\_i\; E(\backslash rho\_i)$. '''Loc Mon:''' local monotone For all (strictly) local instruments (i.e. an instrument that acts either on A or B) $\backslash rho\; \backslash rightarrow\; \backslash \{p\_i,\; \backslash rho\_i\backslash \},\; E(\backslash rho)\; \backslash geq\; \backslash sum\_i\; p\_i\; E(\backslash rho\_i)$. '''LOq Mon:''' LOq monotone For all LOq operations $\backslash rho\; \backslash rightarrow\; \backslash \{p\_i,\; \backslash rho\_i\backslash \},\; E(\backslash rho)\; \backslash geq\; \backslash sum\_i\; p\_i\; E(\backslash rho\_i)$. '''As Cont:''' asymptotic continuity There are $c,\; c\text{'}\backslash geq\; 0$ s.th. for all $\backslash rho,\; \backslash sigma$ with $\backslash delta(\backslash rho,\; \backslash sigma)\; \backslash leq\; \backslash epsilon,\; |E(\backslash rho)-E(\backslash sigma)|\backslash leq\; c\; \backslash epsilon\; \backslash log\; d\; +\; c\text{'}$. '''As Cont Pure:''' asympt. cont. near pure states There are $c,\; c\text{'}\backslash geq\; 0$ s.th. for all $\backslash rho,\; \backslash sigma=|\backslash psi\backslash rangle\; \backslash langle\; \backslash psi|$ with $\backslash delta(\backslash rho,\; \backslash sigma)\; \backslash leq\; \backslash epsilon,\; |E(\backslash rho)-E(\backslash sigma)|\backslash leq\; c\; \backslash epsilon\; \backslash log\; d\; +\; c\text{'}$. '''Conv:''' convex For all $\backslash rho,\; \backslash sigma$ and $p\; \backslash in\; [0,1]$ , $pE(\backslash rho)+(1-p)E(\backslash sigma)\; \backslash geq\; E(p\; \backslash rho+(1-p)\backslash sigma)$. '''Conv Pure:''' convex on pure states For all $\backslash \{p\_i,\; |\backslash psi\_i\; \backslash rangle\backslash \}$ with $p\_i\; \backslash geq\; 0$ and $\backslash sum\_i\; p\_i\; =1,\; \backslash sum\_i\; p\_i\; E(\backslash rho\_i)\; \backslash geq\; E(\backslash rho)$. '''Strong Super:''' superadditive For all $\backslash rho^\{AA\text{'}BB\text{'}\},\; E(\backslash rho^\{AA\text{'},BB\text{'}\})\backslash geq\; E(\backslash rho^\{AB\})+E(\backslash rho^\{A\text{'}B\text{'}\})$. '''Add:''' additive For all $\backslash rho,\; \backslash sigma\; ,\; E(\backslash rho\; \backslash otimes\; \backslash sigma)=\; E(\backslash rho)+E(\backslash sigma)$. '''Ext (Add i.i.d.):''' extensive For all $\backslash rho$ and $N,\; NE(\backslash rho)=\; E(\backslash rho^\{\backslash otimes\; N\})$. '''Sub Add:''' subadditive For all $\backslash rho,\; \backslash sigma,\; E(\backslash rho\backslash otimes\; \backslash sigma)\backslash leq\; E(\backslash rho)+E(\backslash sigma)$. '''Sub Add i.i.d.''' subadditive i.i.d. For all $\backslash rho$ and $m,\; n\; ,\; E(\backslash rho^\{\backslash otimes\; (m+n)\})\backslash leq\; E(\backslash rho^\{\backslash otimes\; m\})+E(\backslash rho^\{\backslash otimes\; n\})$. '''Regu:''' regularisable For all $\backslash rho$, the limit $E^\backslash infty(\backslash rho)=\backslash lim\_\{n\; \backslash rightarrow\; \backslash infty\}\; \backslash frac\{E(\backslash rho^\{\backslash otimes\; n\})\}\{n\}$ exists. '''Non Lock:''' not lockable There is $c\backslash geq\; 0$ s.th. for all $\backslash rho^\{AA\text{'}B\}$, $E(\backslash rho^\{AA\text{'}B\})\; \backslash leq\; E(\backslash rho^\{AB\})+\; c\; \backslash log\; rank\; \backslash rho^\{A\text{'}\}$. [[Category:Handbook of Quantum Information]] [[Category:Entanglement]]