The Mathematics of Entanglement - Summer 2013 28 May, 2013

Introduction to the Quantum Marginal Problem

Lecturer: Matthias Christandl Lecture 5

In the picture is the cover of a a book by Godel, Escher and Bach. You see it projects B, G orE depending from where you project the light on. Is it possible to project any triple of letters inthis way? It turns out that the answer is no. For example, by geometric considerations there is noway of projecting A everywhere.

The goal of this lecture is to introduce a quantum version of this problem!

1 The Quantum Marginal Problem or Quantum RepresentabilityProblem

Consider a set of n particles each with d-dimensions. The state lives in (Cd)n. We considerdifferent subsets of the particles Si {1, . . . , N} and suppose we are given quantum states in eachof this sets: Si . The question we want to address is whether they are compatible, i.e. does thereexist a quantum state {1,...,n} such that for all Si,

trSci () = Si , (1)

with Sci the complement of Si in {1, . . . , n}.

1.1 Physical Motivation

This is a interesting problem from a mathematical point of view, but it is also a prominent problemin the context of condensed matter physics and quantum chemistry. Consider a nearest-neighboursHamiltonian on a line H =

i hi,i+1, where hi,i+1 := hi,i+1I{1,...,n} i,i+1 only acts on qubits i and

i + 1. A quantity of interest is the groundenergy of the model, given by the minimum eigenvalueof H. We can write it variationally as

Eg = min||H | = min

{1,...,n}tr({1,...,n}) (2)

since the set of quantum states is convex and the extremal points are the pure states. Continuing,

Eg = min{1,...,n}

tr({1,...,n})

= min

i

tr(hi,i+1 I{1,...,n} i,i+1

)= min

i

tr(hi,i+1i,i+1). (3)

Therefore

Eg = min{i,i+1}compatible

i

tr(hi,i+1i,i+1), (4)

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where the minimization is over {i,i+1} which are compatible.Observe that the initial maximization is over |

(Cd)n

, which has dimension dn. In contrast,the minimization in Eq. (4) is over nd variables. Therefore if we could solve the compatibilityproblem we could solve the original one in a much more efficient way. Unfortunately this is nota good strategy and in fact one can show that the compatibility problem is computationally hard(NP-hard and even QMA-hard).

There is an interesting connection of the representability problem with quantum entropies. Aimportant inequality of the von Neumann entropy of three subsystems ABC is strong subadditivity:

S(AB) + S(BC) S(B) + S(ABC). (5)

Clearly this inequality puts restrictions on compatible states. More interestingly, one can also useresults from the quantum marginal problem to give a proof of the inequality.

2 Quantum Marginals for 3 Parties

A particular case of the marginal problem is the following: given three quantum states A, B andC , are they compatible? In this case it is that the answer is yes, just consider ABC = ABC .

But what if we require that the global state ABC is pure? I.e. we would like to have a purestate |ABC such that

trAB (| |ABC) = C , trAC (| |ABC) = B, trBC (| |ABC) = A. (6)

Then just taking the tensor product of the reduced states is not an option any more.Example 1: A = B = C = I/2 are compatible, with the GHZ state (|0, 0, 0 + |1, 1, 1)/

2

being a possible extension. A = B = C = I/2 are compatible, with the GHZ state (|0, 0, 0 +|1, 1, 1)/

2 being a possible extension.

Example 2: Suppose A, B and C are compatible. Are A = UAAU

A,

B = UBBU

B, and

C = UCCUC compatible too? The answer is yes. Indeed if |ABC was an extension of A, B

and C , then UA UB UC |ABC is an extension of A, B and C .Therefore we see that the property of A, B, C being compatible only depends on the spectra

A, B and C of A, B and C . Here A := (A,1, . . . , A,d) with A,i the eigenvalues of A.

2.1 Warm-Up: 2 Parties

Given A and B, are they compatible?A useful way of writing a bipartite pure state |AB is in its Schmidt form:

|AB =i

si |ei |fi (7)

for orthogonal basis {|ei} and {|fi} o A and B, respectively. The numbers {si} are called Schmidtvalues of |AB. The reductions of |AB are

A =i

s2i |ei ei| (8)

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andA =

i

s2i |fi fi| (9)

Therefore we see that the eigenvalues of A and B are equal and given by {s2i }.Going back to the compatibility question, we then see from the discussion above that A and

B are compatible if, and only if, they have the same spectrum.

2.2 3 Parties of Qubits

Consider A, B and C each acting on C2. Then since A = (A,1, 1A,1), the compatible regionis a subset of R3. This have a simple algebraic characterization (shown in []):

Amax + Bmax 1 + Cmax (10)

plus all possible cyclic permutations of the labels.

5-3

The Quantum Marginal Problem or Quantum Representability ProblemPhysical Motivation

Quantum Marginals for 3 PartiesWarm-Up: 2 Parties3 Parties of Qubits