The Mathematics of Entanglement - Summer 2013 30 May, 2013

Three Qubit Entanglement Polytopes

Lecturer: Michael Walter Lecture 10

Last time we talked about SLOCC (stochastic LOCC), where we can post-select on particularoutcomes

Given a class of states that can be interconverted by SLOCC into |ψ〉 other by SLOCC, Cψ ={|φ〉 : |φ〉 ↔ |ψ〉}, a result by Dur-Vidal-Cirac says that

Cψ := {(A⊗B ⊗ C)|ψ〉/‖ . . . ‖ : A,B,C ∈ SL(d)} (1)

For three qubits there is a simple classification of all possible types of entanglement. Apartfrom product states, and states with only bipartite entanglement, the two classes have the followingrepresentative states:

|GHZ〉 =1

2(|000〉+ |111〉) (2)

and

|W 〉 =1

2(|001〉+ |010〉+ |100〉) (3)

The class of SLOCC operations forms a group:

G = {A⊗B ⊗ C : A,B,C ∈ SL(d)}. (4)

A easy to check fact is that SL(d) = {eX : tr(X) = 0}. Therefore

G = {eA ⊗ eB ⊗ eC = eA⊗I⊗I+I⊗B⊗I+I⊗I⊗C : tr(A) = tr(B) = tr(C) = 0}. (5)

We denote CψABCby G.|ψ〉ABC .

1 Quantum Marginal Problem for Three Parties

What are the possible ρA, ρB, ρC compatible with a pure state |φ〉ABC ∈ CψABC? We say before

that this only depends on the spectra λA, λB and λC , as one can always apply local unitaries andchange the basis.

For example, for the W class the set of compatible spectra is given by the equation λAmax +λBmax + λCmax ≥ 2.

Let us start with a simpler problem, namely given a state |ψ〉ABC , does there exist a state inG.|ψ〉ABC with ρA = ρB = ρC = I/d? This is equivalent to

tr(ρAA) = tr(ρBB) = tr(ρCC) = 0, (6)

for all Hermitian traceless matrices A,B,C, which in turn is equivalent to

tr(ρAA) + tr(ρBB) + tr(ρCC) = 0, (7)

for all Hermitian traceless matrices A,B,C. We can write it as

〈ψABC |A⊗ I ⊗ I + I ⊗B ⊗ I + I ⊗ I ⊗ C|ψABC〉 = 0. (8)

10-1

Thus the norm of the state |ψABC〉 should not change (to 1st order) when we apply an infinitesimalSLOCC operation.

Let us look at

∂

∂t‖etA ⊗ etB ⊗ etC |ψABC〉‖ =

∂

∂t〈ψABC |etA ⊗ etB ⊗ etC |ψABC〉

= 2〈ψABC |A⊗ I ⊗ I + I ⊗B ⊗ I + I ⊗ I ⊗ C|ψABC〉. (9)

So from Eq. (??), if |ψABC〉 is the closest point to the origin in G.|φABC〉, then ρA = ρB =ρC = I/d.

What happens when there is no point in the class with ρA = ρB = ρC = I/d. That seemsstrange, as it implies by the above that there is no closest point to the origin. But indeed this isthe case for the |W 〉 class, for example. Consider

(ε0; 01/ε)⊗ (ε0′01/ε)⊗ (ε0′01/ε)|W 〉 = ε|W 〉 (10)

and when ε goes to zero, one approaches the origin. However the limit is not in G.|W 〉.

Theorem 1. The following are equivalent:

• There exists a closest point to 0 in G.|φABC〉.

• There exists a quantum state in G.|φABC〉 with ρA = ρB = ρC = I/d.

• G.|φABC〉 is closed.

10-2