1. Quantum Hammersley-Clifford Theorem Winton Brown CRM-Workshop on quantum information in Quantum many body physics 2011

2. Motivations The Hammersley-Clifford theorem is a standard representation theorem for positive classical Markov networks. Recently, quantum Markov networks have been of interest in relation to quantum belief propagation (QBP) and Markov entropy decomposition (MED) approximation methods. Connections to other problems in QIS

3. Conditional Mutual Information Mutual Information Conditional Mutual Information Strong subadditivity Markov Condition (classical)  where

4. Markov Networks A B C Def : A Markov network is probability distribution, ρ, defined on a graph G, such that for any division of G into regions A, B and C such that B separates A and C, ρ A and ρ C are independent conditioned on ρ B For every B separating A and C

5. Hammersley-Clifford Theorem (classical) Thm: A positive probability distribution, p, is a Markov Network on a graph G iff p factorizes over the complete subgraphs (cliques) of G.  for traceless X and Y that do not lie on the same clique Proof:  Let From conditional independence  Done.

6. Quantum Hammersley-Clifford Theorem  such that Now let so where Hayden, et. al. Commun. Math. Phys., 246(2):359-374, 2004 For quantum states with:

7. Quantum Hammersley-Clifford Theorem   Now H decomposes just as in the classical case for traceless X and Y that do not lie on the same clique But, must show terms commute! to show

8. Quantum Hammersley-Clifford Theorem For each division into regions A and C separated by B: There exist terms with such that

9. Hammersley-Clifford Theroem (quantum) Since the commutators of each pair of terms in K AB and K BC have different support, their commutators can not cancel. Thus, implies: Then their commutator must be a genuine 3-body operator on ABC. If two genuine 2-body operators share support only on B

10. 1 2 3 If G contains only 2-vertex cliques then a boundary can always be drawn so that Two-Vertex Cliques can not be cancelled by any other terms. Thus, implies

11. 1 2 3 Two-Vertex Cliques If there is a single-body term then one need only consider the tree surrounding the vertex. The Hammersley-Clifford decomposition has been proved to hold on trees. Hastings, Poulin 2011 Thus all positive quantum Markov networks with 2-vertex cliques, are factorizable into commuting operators on the cliques of the graph.

12. Three-vertex cliques X X X X XX X X Z Y Z Y A1A1 A4A4 A3A3 A2A2

13. X X X X XX X X Z Y Z Y A4A4 A3A3 A2A2 A1A1 Cut 1 Counter-Example

14. X X X X XX X X Z Y Z Y A4A4 A3A3 A2A2 A1A1 Cut 1 Counter-Example Yields a positive quantum Markov network which can not be factorized into commuting terms on its cliques! But factorizability can be recovered by course-graining.

15. PEPS Each bond indicates a completely entangled state If Λ is unitary, then the PEPS is a Markov network. Apply a linear map Λ to each site to obtain the PEPS Under what conditions can the reverse be shown?

16. PEPS For a non-degenerate eigenstate of quantum Markov network. Markov Properties  Entanglement Area Law Hammersy-Cliffors Decomposition  PEPS representation of fixed bond dimension Thus: For non-degenerate quantum Markov networks with Hammsley-Clifford decomposition each eigenstate is a PEPS of fixed bond dimension. Open Problem: show under what conditions quantum Markov networks which are pure states have a Hammersley-Clifford decomposition.

17. PEPS Non-factorizable pure state quantum Markov network U1U1 U 2 =U 1 * Thus any state of the form is a quantum Markov network. Let U 1 be sqrt of SWAP, then | ψ> can not be specified by projectors on the cliques. Unitary invariants of completely entangled states Bell pair network graph

18. Conclusions The Hammersley-Clifford Theorem generalizes to quantum Markov network when restricted to lattices containing only two-vertex cliques. Counter examples for positive Markov networks can be constructed for graphs with three-vertex cliques and for pure states rectangular graphs. Whether counterexamples exist that can’t be course-grained into factorizable networks is an open question.