Quantum Hammersley-Clifford Theorem

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 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, ρA and ρC are independent conditioned on ρB B A C 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. Proof: Let From conditional independence  for traceless X and Y that do not lie on the same clique   Done.

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

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) If two genuine 2-body operators share support only on B Then their commutator must be a genuine 3-body operator on ABC. Since the commutators of each pair of terms in KAB and KBC have different support, their commutators can not cancel. Thus, implies:

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

11. Two-Vertex Cliques If there is a single-body term then one need only consider the tree surrounding the vertex. 3 The Hammersley-Clifford decomposition has been proved to hold on trees. 2 1 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 A1 X X X X Z A4 Y Y A2 Z X X X X A3

13. Counter-Example A1 X X X X Z A4 Y Y A2 Z X X X X A3 Cut 1

14. Counter-Example Cut 1 A1 X X X X Z A4 Y Y A2 Z X X X X A3 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 Apply a linear map Λ to each site to obtain the PEPS If Λ is unitary, then the PEPS is a Markov network. 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 U1 U2=U1* Unitary invariants of completely entangled states network graph Bell pair Thus any state of the form is a quantum Markov network. Let U1 be sqrt of SWAP, then |ψ> can not be specified by projectors on the cliques.

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.