Entanglement of formation

For a pure bipartite quantum state $|\backslash psi\backslash rangle\_\{AB\}$, using Schmidt decomposition, we see that the reduced density matrices of systems A and B, $\backslash rho\_A$ and $\backslash rho\_B$, have the same spectrum. The von Neumann entropy $S(\backslash rho\_A)=S(\backslash rho\_B)$ of the reduced density matrix can be used to measure the entanglement of the state $|\backslash psi\backslash rangle\_\{AB\}$. We denote this kind of measure as $E\_\{f\}(|\backslash psi\backslash rangle\_\{AB\})=S(\backslash rho\_A)=S(\backslash rho\_B)$, and call it the entanglement entropy. This is also known as the entanglement of formation of a pure state.

For a mixed bipartite state $\backslash rho\_\{AB\}$, a natural generalization is to consider all the ensemble realizations of the mixed state.

We define the entanglement of formation for mixed states by minimizing over all these ensemble realizations,

:$E\_f\; (\backslash rho\_\{AB\})=\; \backslash inf\backslash left\backslash \{\; \backslash sum\_i\; p\_i\; E\_f(|\backslash psi\_i\backslash rangle\_\{AB\})\backslash right\backslash \}$, where the infimum is taken over all the possible ways in which one can decompose $\backslash rho\_\{AB\}$ into pure states $\backslash rho\_\{AB\}=\backslash sum\_i\; p\_i\; |\backslash psi\_i\; \backslash rangle\; \backslash langle\; \backslash psi\_i|\_\{AB\}$.

This kind of extension of a quantity defined on some set (here the pure states) to its convex hull (here the mixed states) is called a convex roof construction.

Entanglement of formation quantifies how much entanglement (measured in ebits) is necessary, on average, to prepare the state. The measure clearly coincides with entanglement entropy for pure states. It is zero for all separable states and non-zero for all entangled states. By construction, $E\_f$ is convex.

Entanglement of formation is known to be a non-additive measure of entanglement.{{cite journal |title=Quantum entanglement |first1=Ryszard |last1=Horodecki |first2=Pawel |last2=Horodecki |first3=Michal |last3=Horodecki |first4=Karol |last4=Horodecki |year=2009 |journal=Rev. Mod. Phys. |volume=81 |pages=907–908 |doi=10.1103/RevModPhys.81.865 |arxiv=quant-ph/0702225}} That is, there are bipartite quantum states $\backslash rho\_\{AB\},\; \backslash sigma\_\{AB\}$ such that the entanglement of formation of the joint state $\backslash rho\_\{AB\}\backslash otimes\backslash sigma\_\{AB\}$ is smaller than the sum of the individual states' entanglement, i. e., . Note that for other states (for example pure or separable states) equality holds.

Furthermore, it has been shown that the regularized entanglement of formation equals the entanglement cost. That is, for large $n$ the entanglement of formation of $n$ copies of a state $\backslash rho$ divided by $n$ converges to the entanglement cost{{cite journal |title=The asymptotic entanglement cost of preparing a quantum state |first1=Patrick M. |last1=Hayden |first2=Michal |last2=Horodecki |first3=Barbara M. |last3=Terhal |arxiv=quant-ph/0008134 |journal=J. Phys. A: Math. Gen. |volume=34 |number=35 |pages=6891–6898 |year=200 |doi=10.1088/0305-4470/34/35/314}}

:$\backslash lim\_\{n\backslash to\backslash infty\}\; E\_f(\backslash rho^\{\backslash otimes\; n\})/n\; =\; E\_c(\backslash rho)$

The non-additivity of $E\_f$ thus implies that there are quantum states for which there is a “bulk discount” when preparing them from pure states by local operations: it is cheaper, on average, to prepare many together than each one separately.

For states of two qubits, the entanglement of formation has a close relationship with concurrence. For a given state $\backslash rho\_\{AB\}$, its entanglement of formation $E\_f\; (\backslash rho\_\{AB\})$ is related to its concurrence $C$:

:$E\_f=h\backslash left(\backslash frac\{1+\backslash sqrt\{1-C^2\}\}\{2\}\backslash right)$

where $h(x)$ is the Shannon entropy function,

:$h(x)\; =\; -x\; \backslash log\_2\; x\; -(1-x)\; \backslash log\_2\; (1-x).$

{{Reflist|refs=

{{cite journal | last=Hill | first=Scott | last2=Wootters | first2=William K. | title=Entanglement of a Pair of Quantum Bits | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=78 | issue=26 | date=1997-06-30 | issn=0031-9007 | doi=10.1103/physrevlett.78.5022 | pages=5022–5025|arxiv=quant-ph/9703041}}

{{cite journal | last=Wootters | first=William K. | title=Entanglement of Formation of an Arbitrary State of Two Qubits | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=80 | issue=10 | date=1998-03-09 | issn=0031-9007 | doi=10.1103/physrevlett.80.2245 | pages=2245–2248| arxiv=quant-ph/9709029 }}

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{{DEFAULTSORT:Measure of entanglement}}

Category:Quantum information theory