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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)
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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.
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