| entanglement of formation | |
|---|---|
| Name | Entanglement of formation |
| Field | Quantum information theory |
| Introduced | 1996 |
| Introduced by | William K. Wootters et al. |
| Related | Entanglement measure, Concurrence (quantum computing) |
entanglement of formation
Entanglement of formation is an entanglement measure that quantifies the minimal quantum resources required to prepare a given mixed quantum state of a composite system. It captures the asymptotic cost, in units of ebits, to create the state from pure maximally entangled states under local operations and classical communication (LOCC). The measure is central in quantum information theory because it connects operational preparation costs with mathematical descriptions of quantum entanglement.
The entanglement of formation (E_F) of a bipartite density operator ρ_{AB} is defined as the convex-roof extension of the entanglement entropy for pure states. For a pure state |ψ⟩_{AB}, the entanglement is given by the von Neumann entropy S(ρ_A) where ρ_A = Tr_B |ψ⟩⟨ψ|. For a mixed state, E_F(ρ_{AB}) is the infimum of the average pure-state entanglement over all ensemble decompositions ρ = Σ_i p_i |ψ_i⟩⟨ψ_i|. This operationally corresponds to the minimal average number of Bell state (or ebit) preparations required per copy if one allows for stochastic ensemble preparation protocols. The concept is closely related to resource theories formalized by groups such as NIST and theoretical frameworks developed by researchers at IBM Research and universities including MIT and Caltech.
Formally, for ρ_{AB}, E_F(ρ_{AB}) = inf_{ {p_i,|ψ_i⟩} } Σ_i p_i S(Tr_B |ψ_i⟩⟨ψ_i| ), where the infimum runs over all ensembles {p_i, |ψ_i⟩} realizing ρ_{AB}. Here S(σ) = -Tr(σ log σ) is the von Neumann entropy. The definition uses the convex roof construction common in entanglement theory and in studies of mixed-state quantum tomography and optimization. For two-qubit systems, an analytic formula exists that expresses E_F in terms of concurrence (quantum computing) C(ρ), a polynomial invariant related to the spin-flip operation; Wootters provided a closed form: E_F(ρ) = h((1+√{1-C^2})/2) where h is the binary entropy. This closed form links E_F to explicit algebraic operations on the density matrix used in many quantum computing characterizations.
E_F is nonincreasing under LOCC (entanglement monotone), convex, and reduces to the entropy of entanglement on pure states. It is zero for separable states and positive for entangled states. Computability is challenging: the convex-roof optimization is generally nonconvex and high-dimensional, making exact evaluation NP-hard in some settings. For special cases — notably two-qubit systems and certain symmetric states such as Werner states and isotropic states — analytic or semi-analytic solutions exist. Numerical approaches include semidefinite programming methods developed in quantum information research groups, heuristics using gradient descent on complex projective space, and exploitation of symmetry by groups like those at Perimeter Institute and Institute for Quantum Computing.
Known exact results include Wootters' formula for two qubits and expressions for rank-2 two-qudit states under symmetry constraints. For Werner states and isotropic states in d×d systems, E_F can often be bounded tightly using entanglement measures such as negativity and relative entropy of entanglement. Example calculations demonstrate how E_F interpolates between separable mixtures (E_F = 0) and maximally entangled pure states (E_F = log d). The measure has been computed in experimental contexts for photonic systems using state reconstruction via quantum state tomography at laboratories such as Max Planck Institute of Quantum Optics and University of Vienna's IQOQI.
E_F is related to several other measures: it provides an upper bound on the entanglement cost and is bounded below by the distillable entanglement under LOCC in general; in many cases the entanglement cost equals E_F in the asymptotic regime. The entanglement of formation is generally larger than or equal to measures like relative entropy of entanglement and can be compared to logarithmic negativity and concurrence. The phenomenon of irreversibility in entanglement transformations — established in work by Vidal and Jonathan— connects E_F to entanglement dilution and concentration protocols. Connections to the quantum discord and more recent resource measures illustrate subtle distinctions between classical and quantum correlations.
E_F serves as a benchmark in tasks such as entanglement sharing, entanglement distribution over quantum channels (including noisy channels studied by Charles H. Bennett and collaborators), and in quantifying resources for quantum teleportation and entanglement-assisted communication. It informs error thresholds for quantum error correction codes and benchmark protocols in quantum key distribution implementations. In theoretical studies, E_F appears in analyses of multipartite entanglement scaling, entanglement area laws in many-body systems (connected to research at Harvard University and Princeton University), and in resource-theoretic formulations that underpin proposals for quantum networks and distributed quantum computation pursued by institutions like Google Quantum AI and Microsoft Quantum.
Category:Quantum information theory Category:Entanglement measures