ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

24 1. Characterizing entanglement
not all entanglement measures satisfy Ineq. (1.45). In particular, the entanglement
of formation, Eq. (1.35), fails to fulfill the task, and this fact led CKW to define,
for qubit systems, a new measure of bipartite entanglement consistent with the
quantitative monogamy constraint expressed by Ineq. (1.45).
1.4.2.1. Entanglement of two qubits. For arbitrary states of two qubits, the entanglement
of formation, Eq. (1.35), has been explicitly computed by Wootters [273],
and reads
EF (ϱ) = F[C(ϱ)] , (1.46)
with F(x) = H[(1 + √ 1 − x 2 )/2] and H(x) = −x log 2 x − (1 − x) log 2(1 − x). The
quantity C(ϱ) is called the concurrence [113] of the state ϱ and is defined as
C(ϱ) = max{0, λ1 − λ2 − λ3 − λ4} , (1.47)
where the {λi}’s are the eigenvalues of the matrix ϱ(σy⊗σy)ϱ ∗ (σy⊗σy) in decreasing
order, σy is the Pauli spin matrix and the star denotes complex conjugation in the
computational basis {|ij〉 = |i〉 ⊗ |j〉, i, j = 0, 1}. Because F(x) is a monotonic
convex function of x ∈ [0, 1], the concurrence C(ϱ) and its square, the tangle [59]
τ(ϱ) = C 2 (ϱ) , (1.48)
are proper entanglement monotones as well. On pure states, they are monotonically
increasing functions of the entropy of entanglement, Eq. (1.25).
The concurrence coincides (for pure qubit states) with another entanglement
monotone, the negativity [283], defined in Eq. (1.40), which properly quantifies
entanglement of two qubits as PPT criterion [178, 118] is necessary and sufficient
for separability (see Sec. 1.3.2.1). On the other hand, the tangle is equal (for pure
states |ψ〉) to the linear entropy of entanglement EL, defined as the linear entropy
SL(ϱA) = 1 − TrAϱ 2 A , Eq. (1.2), of the reduced state ϱA = TrB|ψ〉〈ψ| of one party.
1.4.3. Residual tripartite entanglement
After this survey, we can now recall the crucial result that, for three qubits, the
desired measure E such that the CKW inequality (1.45) is satisfied is exactly the
tangle [59] τ, Eq. (1.48). The general definition of the tangle, needed e.g. to compute
the leftmost term in Ineq. (1.45) for mixed states, involves a convex roof analogous
to that defined in Eq. (1.25), namely
τ(ϱ) = min
{pi,ψi}
pi τ(|ψi〉〈ψi|) . (1.49)
i
With this general definition, which implies that the tangle is a convex measure on
the set of density matrices, it was sufficient for CKW to prove Ineq. (1.45) only for
pure states of three qubits, to have it satisfied for free by mixed states as well [59].
Once one has established a monogamy inequality like Ineq. (1.45), the following
natural step is to study the difference between the leftmost quantity and the rightmost
one, and to interpret this difference as the residual entanglement, not stored in
couplewise correlations, that hence quantifies the genuine tripartite entanglement
shared by the three qubits. The emerging measure
τ A|B|C
3 = τ A|(BC) − τ A|B C − τ A|C B , (1.50)
known as the three-way tangle [59], has indeed some nice features. For pure states, it
is invariant under permutations of any two qubits, and more remarkably it has been