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In quantum mechanics, entanglement measures quantify how much entanglement is contained in a quantum state. Formally it is any nonnegative real function of a state which can not increase under local operations and classical communication (LOCC) (so called monotonicity), and is zero for separable states.

There are as well as abstractly defined measures such as ones based on convex roof construction (e.g. concurrence and entanglement of formation) or based on distance from set of separable states such as relative entropy of entanglement .

Often the abstract measures are bounds for operational measures. For example relative entropy of entanglement is an upper bound for distillable entanglement and distillable key.

One of typical applications of abstract EM's is to show that certain task can not be achieved by means of LOCC. One does it by showing that if the task could be done, then some EM would increase.

EM's are not linearly ordered that is there exists entanglement measures $E_{1}$ and $E_{2}$ two states $\rho$ and $\sigma$ such that

$EM_{1}(\rho )<EM_{2}(\rho )\quad and\quad EM_{1}(\sigma )>EM_{2}(\sigma )$

Different entanglement measures determine different types of entanglement. All EMs for pure states are classified.

Entanglement measures are also studied and classified according to their properties, e.g. additivity, convexity and continuity. This approach to entanglement measures is known as axiomatic approach.

Bipartite case

An ebit is one unit of bipartite entanglement, the amount of entanglement that is contained in a maximally entangled two-qubit state (Bell state).

If a state is said to have X ebits of entanglement (quantified by some entanglement measure) it has the same amount of entanglement (in that measure) as X Bell states. If a task requires Y ebits, it can be done with Y or more Bell states, but not with fewer.

Maximally entangled states in d × d dimensions have log₂(d) ebits.

Entanglement of formation

The entanglement of formation is an entanglement measure for bipartite quantum states.

It is defined as

$E_{f}(\rho )=\min _{\mathcal {E}}p_{i}E_{E}(|\psi _{i}\rangle )$

where the minimization is over all ensembles of pure states ${\mathcal {E}}=\{(p_{i},|\psi _{i}\rangle )\}$ that realizes the given state, $\rho =\sum _{i}p_{i}|\psi _{i}\rangle \langle \psi _{i}|$, and $E_{E}(|\psi \rangle )$ is the entropy of entanglement which is defined for pure states. This kind of extension of a quantity defined on pure states to mixed states is called a convex roof construction.

Entanglement of formation quantifies how many bell states are needed per copy of to prepare many copies of $\rho$ using the following specific LOCC procedure:

• For each copy, select which pure state $|\phi _{i}\rangle$ to prepare from a probability distribution $q_{i}$.
• For each of the different $|\phi _{i}\rangle$, prepare the required number of copies from bell states.
• Discard the information about which copy is in which pure state.

It is not known if the entanglement of formation is equal to the entanglement cost in general. However, the entanglement cost is equal to the regularization of the entanglement of formation,

$E_{c}(\rho )=\lim _{n\to \infty }{\frac {1}{n}}E_{f}(\rho ^{\otimes n}).$

Operational entanglement measures

Entanglement cost

Entanglement cost is an entanglement measure that aims to quantify how many ebits are required to prepare a copy of a state using only LOCC operations. Many copies can be prepared at the same time and the entanglement cost therefore quantifies how many ebits are required per copy of the state. The preparation is allowed to be approximate, as long as the approximation can be made arbitrary good by preparing many copies at a time.

Formal definition

Let $P_{+}$ be the projector onto a Bell state, $P_{+}:=|\Phi ^{+}\rangle \langle \Phi ^{+}|$, where $|\Phi ^{+}\rangle =(|00\rangle +|11\rangle )/{\sqrt {2}}$. The entanglement cost aims to quantify the rate m/n at which it is possible to convert $P_{+}^{\otimes m}$ into $\rho ^{\otimes n}$ with a LOCC operation $\Lambda$. Since it is usually impossible to perform this exactly, we settle for $\Lambda (P_{+}^{\otimes m})\approx \rho ^{\otimes n}$ and let the quality of the approximation be quantified by a distance measure $D(\Lambda (P_{+}^{\otimes m}),\rho ^{\otimes n})$ which can be either the Bures distance, the trace distance or another suitable distance. The entanglement cost E_C is then the infimum of all possible rates m/n such that the approximation can be made arbitrarily good by choosing m and n large enough. This can be formulated mathematically as [quant-ph/0008134]

$E_{c}(\rho )=\inf\{E\mid \forall \epsilon >0,\delta >0,\exists m,n,\Lambda ,|E-{\frac {m}{n}}|\leq \delta {\text{ and }}D(\Lambda (P_{+}^{\otimes m}),\rho ^{\otimes n})\leq \epsilon \}$

Relations to other entanglement measures

The entanglement cost has been show to be equal to the regularization of the entanglement of formation,

$E_{C}(\rho )=\lim _{n\to \infty }{\frac {1}{n}}E_{f}(\rho ^{\otimes n})$

If the entanglement of formation turns out to be additive, the entanglement cost will be equal to the entanglement of formation.