3. CAPÍTULO 3: DESCRIPCIÓN DE LA ARQUITECTURA
3.4. VISTA LOGICA
As discussed in details in Section 1.2.2 of Chapter 1, in any resource theory, any measure of the resources must satisfy three axioms. For the resource theory of entanglement, these can be translated into the followings:
1. For any measure of entanglementE, a quantum state ˆσ is separable if and only E(ˆσ) = 0.
2. For any LOCC operation O ∈ OLOCC acting on quantum states, the measureE has to be strongly monotonic in the following sense. Suppose that ˆ% is transformed to an ensemble of states { ˆ%i} with the corresponding probabilities {pi} upon the action of O. Then, it must be true that,
ˆ
%7−→O X
i
pi%ˆi ←→ E(ˆ%) >X
i
piE(ˆ%i), (2.38)
meaning that entanglement should not increase on average via LOCC operations.
3. A measure E of entanglement must be convex, i.e., suppose that { ˆ%i} is an ensemble of quantum states with the corresponding probabilities {pi}, then it must hold true that
E(X
i
pi%ˆi) 6X
i
piE(ˆ%i), (2.39)
meaning that the average entanglement should not increase by mixing.
With these axioms at hand, any functional from the set of quantum states of compound sys-tems to nonnegative real numbers satisfying the axioms can be called a proper and faithful entanglement measure. We also recall that there exists a very fundamental preorder on the set of entangled states, namely, the majorization preorder [7], that provides the convertibility condition of entangled states. Naturally, we would like to have measures of entanglement that respect the majorization criterion. Below, we briefly review some popular existing measures of entanglement.
Entanglement of formation
Definition 2.3.1. [52] Given a quantum state ˆ%AB, the entanglement of formation of ˆ%AB is defined as Eq. (1.117), is the entropy of entanglement of the pure state |ψiiABhψi|, and the minimization is performed over all possible decompositions of ˆ%AB into ensembles {(pi, |ψiiABhψi|)}.
The entanglement of formation has a simple interpretation, as it characterizes the ensemble decomposition of the state which delivers the minimum average local entropies. Equivalently, due to the fact that the entanglement of bipartite pure states is best quantified via their local entropies, named the entropy of entanglement Ee, it characterizes the ensemble decomposition of the state which delivers the minimum average pure state entanglement. Using the fact that Ef does not increase on average under LOCC, we infer that Alice and Bob who wish to create
ˆ
%AB with Ef( ˆ%AB) must initially share at least an equal amount of entanglement stored in a collection of pure entangled states. By making use of the majorization theorem for entangled state transformation [7,53], one can show that the converse is also true, i.e., Alice and Bob who have access to Ef amount of pure state entanglement, say in Bell states, can generate ˆ%AB with the same amount of entanglement of formation vial LOCC. Hence,Efoperationally characterizes the asymptotic rate at which bipartite Bell states have to be consumed to generate an entangled quantum state [52]. This measure, as one could guess, is very difficult to be measured or calculated because, in general, it is extremely hard to obtain all the possible decompositions of a multipartite quantum state. However, Brand˜ao [54] showed that any entanglement witness provides a lower bound to entanglement of formation. Consequently, while the measure itself cannot be directly measured, it can be bounded below within experiments by a witnessing procedure.
Negativity
Definition 2.3.2. [55, 56] Given a quantum state ˆ%AB, its negativity is defined as
En( ˆ%AB) =
in which
It should be noted that negativity only quantifies NPT-entanglement, because i.e., the state remains positive under PT operation, we have
that the first axiom is violated: there exist entangled states (PPT entangled states) for which En( ˆ%AB) = 0. According to our discussion in Chapter1under the Section1.2.2, such quantities are called pseudomeasures or sometimes entanglement monotones.
Despite not being a faithful measure of entanglement, due to being simply calculable and the vast use of NPT entangled states in quantum information science, negativity is a very popular and useful pseudomeasure. Moreover, Brand˜ao [54] showed that negativity can be obtained as
En( ˆ%AB) = max{0, − min
Wˆ
Tr ˆW ˆ%TABB}, (2.42) where the minimization is performed over all sub-normalized NPT entanglement witnesses satisfying 0 6 ˆW 6 ˆ1.
Robustness
Definition 2.3.3. [57] Given a quantum state ˆ%AB, its robustness relative to a separable state10 ˆ
σAB is defined as
Err( ˆ%AB||ˆσAB) = min
s {s ∈ R+|ˆ%AB(s) = 1
1 + s( ˆ%AB+ sˆσAB) ∈ Ssep}. (2.43) The robustness of a quantum state ˆ%AB is then given by
Er( ˆ%AB) = min
ˆ σAB∈Ssep
Err( ˆ%AB||ˆσAB). (2.44)
Robustness also has a very simple geometrical and physical interpretation. Note that the state ˆ
%AB(s) in Eq.2.43is the convex mixture of a point inside the separable set Ssepand the state the entanglement of which has to be quantified. Then, the parameter s characterizes the minimum amount by which one should move along the line segment connecting the two points to pass the separability border. At this point, one can easily verify that Axiom1of faithfulness is satisfied for Err: if ˆ%AB is separable, then obviously for any point of Ssep one has s = 0. On the other hand, if ˆ%AB is entangled, i.e. it is exterior to the closed and bounded set of separable states, then s > 0. The quantity Er( ˆ%AB) is just the optimal value over all separable elements so to make the measure independent of the reference separable point.
Physically, Err( ˆ%AB) just means how much of a given separable state has to be mixed with the original state so that the mixing procedure removes all the entanglement. Hence, one can think of ˆσAB as a noise added to the state ˆ%AB. Equation (2.44) thus only gives the optimal
10In fact, this can be replaced by a generic quantum state.
Figure 2.6: The scheme of a semiquantum nonlocal game. Charlie asks the players quantum questions while the players return classical answers. The shared state between the players helps them to obtain a maximum pay-off in the game. Here, we allow LOCC operations to be applied to the shared state and quantum questions, and introduce a device-independent measure of entanglement.
value for the mixture over all possible noises. Interestingly, it can be shown that a generalized form of the robustness can also be mapped onto a witnessing procedure [54].