Entanglement Loss Along RG Flows Entanglement and Quantum Phase Transitions José Ignacio Latorre Dept. ECM, Universitat de Barcelona Newton Institute, Cambridge, August 2004

Entanglement in Quantum Critical Phenomena G. Vidal, J. I. Latorre, E. Rico, A. Kitaev. Phys. Rev. Lett. 90 (2003) Ground State Entanglement in Quantum Spin Chains J. I. Latorre, E. Rico, G. Vidal. Quant. Inf. & Comp. 4 (2004) 48 Adiabatic Quantum Computation and Quantum Phase Transitions J. I. Latorre, R. Orús, PRA, quant-ph/ Universality of Entanglement and Quantum Computation Complexity R. Orús, J. I. Latorre, Phys. Rev. A69 (2004) , quant-ph/ Fine-Grained Entanglement Loss along Renormalization Group Flows J. I. Latorre, C.A. Lütken, E. Rico, G. Vidal. quant-ph/

Entanglement loss along RG flows Introduction Scaling of entropy Entanglement loss along RG flows Preview of new results

HEP Black hole entropy Conformal field theory Condensed Matter Spin networks Extensions of DMRG Quantum Information Entanglement theory Efficient simulation Scaling of entropy

Entanglement measures for many-qubit systems Few-qubit systems  Formation, Distillation, Schmidt coefficients,…  N=3, tangle (for GHZ-ness) out of 5 invariants  Bell inequalities, correlators based measures  Entropy, negativity, concurrence,… Many-qubit systems  Scaling of correlators  Concurrence does not scan the system  We need a measure that obeys scaling and does not depend on the particular operator content of a theory Reznik’s talk

Reduced density matrix entropy Schmidt decomposition A B  =min(dim H A, dim H B ) is the Schmidt number

The Schmidt number relates to entanglement Let’s compute the von Neumann entropy of the reduced density matrix  =1 corresponds to a product state Large  implies large superpositions e-bit

Maximum Entropy for N-qubits Strong subadditivity theorem implies concavity on a chain of spins SLSL S L-M S L+M S max =N

n→  -party entanglement Goal: Analyze S L as a function of L for relevant theories S L measures the quantum correlations with the rest of the system Ground state reduced density matrix entropy

Note that ground state reduced density matrix entropy S L Measures the entanglement corresponding to the block spins correlations with the rest of the chain Depends only the ground state, not on the operator content of the theory (Relates to the energy-momentum tensor!!) Scans different scales in the system: Is sensitive to scaling!! Has been discussed in other branches of theoretical physics  Black hole entropy  Field Theory entanglement, conformal field theory  No condensed matter computations

Scaling of entropy for spin chains XY model Quantum Ising model in a transverse magnetic field Heisenbeg model

XY plane massive fermion massive scalar Quantum phase transitions occur at T=0.

Jordan-Wigner transformation to spinless fermions Lieb, Schultz, Mattis (1961) Espectrum of the XY model

Fourier plus Bogoliubov transformation For γ=0, E k =λ-cos(2πk/n)

Coordinate space correlators can be reconstructed

Some intuition The XY chain reduces to a gaussian hamiltonian We have the exact form of the vacuum We can compute exact correlators The partial trace of N-L does not imply interaction Each k mode becomes a mixed state L

Universality of scaling of entanglement entropy At the quantum phase transition point Quantum Ising c=1/2 free fermion XY c=1/2 free fermion XX c=1 free boson Heisenberg c=1 free boson Universality Logarithmic scaling of entropy controled by the central charge

Conformal Field Theory  A theory is defined through the Operator Product Expansion  In d=1+1, the conformal group is infinite dimensional: the structure of “descendants” is fixed the theory is defined by C ijk and h i Scaling dimensions=anomalous dimensions Structure constants Central charge Stress tensor

Away from criticality Saturation of entanglement Quantum Ising

Connection with previous results  Srednicki ’93 (entanglement entropy)  Fiola, Preskill, Strominger, Trivedi ’94 (fine-grained entropy)  Callan, Holzey, Larsen, Wilczeck ’94 (geometric entropy) Poor performance of DMRG at criticality Area law for entanglement entropy A B S A = S B → Area Law Entropy comes from the entanglement of modes at each side of the boundary Entanglement depends on the connectivity! Schmidt decomposition

Entanglement bonds Area law Area law in d>1+1 does not depend on the mass Valence bond representation of ground state Plenio’s talk Verstraete’s talk

Entanglement in higher dimensions, “Area Law”, for free theories c 1 is an anomaly!!!! Von Neumann entropy captures a most elementary counting of degrees of freedom Trace anomalies Kabat – Strassler

Is entropy scheme dependent is d>1+1? Yes No c 1 =1/6 bosons c 1 =1/12 fermionic component

Entanglement along quantum computation Spin chains are slightly entangled → Vidal’s theorem  Schmidt decomposition  If  Then The register can be classically represented in an efficient way! All one- and two-qubit gates actions are also efficiently simulated!! A B  max(  AB )  poly(n) << e n Quantum speed-up needs large entanglement !!!

The idea for an efficient representation of states is to store and manipulate information on entanglement, not on the coefficients!! Low entanglement iff α i =1,…,  and  << e n Representation is efficient Single qubit gates involve only local update Two-qubit gates involve only local update Impressive performance when simulating d=1+1 quantum systems! Holy Grail=Extension to higher dimensions Cirac,Verstraete - Vidal

Entanglement in Shor’s algorithm (Orús) rr r small = easy = small entanglement no need for QM r large = hard = large entanglement QM exponential speed-up

Entanglement and 3-SAT  3-SAT 3-SAT is NP-complete K-SAT is hard for k > SAT with m clauses: easy-hard-easy around m=4.2n  Exact Cover A clause is accepted if 001 or 010 or 100 Exact Cover is NP-complete For every clause, one out of eight options is rejected instance

Adiabatic quantum evolution (Farhi-Goldstone-Gutmann) H(s(t)) = (1-s(t)) H 0 + s(t) H p Inicial hamiltonianProblem hamiltonian s(0)=0 s(T)=1 t Adiabatic theorem: if E1E1 E0E0 E t g min

Adiabatic quantum evolution for exact cover |0> |1> (|0>+|1>) …. (|0>+|1>)

Typical gap for an instance of Exact Cover

Scaling is consistent with gap ~ 1/n If correct, all NP problems could be solved efficiently! Be cautious

Scaling of entropy for Exact Cover A quantum computer passes nearby a quantum phase transition!

n=6-20 qubits 300 instances n/2 partition S ~.2 n Entropy seems to scale maximally!

Scaling of entropy of entanglement summary Non-critical spin chainsS ~ ct Critical spin chainsS ~ log 2 n Spin networks in d-dimensions “Area Law” S ~ n d-1/d NP-complete problemsS ~ n

What has Quantum Information achieved? “Cleaned” our understanding of entropy Rephrased limitations of DMRG Focused on entanglement Represent and manipulate states through their entanglement Opened road to efficient simulations in d>1+1 Next?

Entanglement loss along RG RG flow = loss of information RG flow = loss of Quantum information 1.Global loss of entanglement along RG 2.Monotonic loss of entanglement along RG 3.Fine-grained loss of entanglement along RG

Global loss of entanglement along RG Monotonic loss of entanglement along RG + c-theorem S L UV  S L IR

Majorization theory Entropy provides a modest sense of ordering among probability distributions Muirhead (1903), Hardy, Littlewood, Pólya,…, Dalton Consider such that p are probabilities, P permutations d cumulants are ordered D is a doubly stochastic matrix

Fine-grained loss of entanglement L LL  L t  L t’ t t’ RG  1   ’ 1  1 +  2   ’ 1 +  ’ 2  1 +  2 +  3   ’ 1 +  ’ 2 +  ’ 3 …….. Strict majorization !!!

Recent sets of results I Lütken, R. Orús, E. Rico, G. Vidal, J.I.L. Analytical majorization along Rg Exact results for XX and QI chains based on Calabrese-Cardy hep-th/ , Peschel cond-mat/ Efficient computations in theories with c  1/2,1 Exact eigenvalues, equal spacing Exact majorization along RG Detailed partition function

Recent sets of results II R. Orús, E. Rico, J. Vidal, J.I.L. Lipkin model Full connectivity (simplex) → symmetric states → S L <Log L Entropy scaling characterizes a phase diagram as in XY + c=1/2 !!! Underlying field theory? SUSY? Effective connectivity of d=1  =1 11

Conclusion: A fresh new view on RG RG on Hamiltonians Wilsonian Exact Renormalization Group RG on correlators Flow on parameters from the OPE RG on states Majorization controls RG flows? Lütken, Rico, Vidal, JIL Cirac, Verstraete, Orús, Rico, JIL The vacuum by itself may reflect irreversibility through a loss of entanglement RG irreversibility would relate to a loss of quantum information