Measures of Entanglement at Quantum Phase Transitions

1. Measures of Entanglement at Quantum Phase Transitions G. Morandi F. Ortolani E. Ercolessi C. Degli Esposti Boschi L. Campos Venuti S. Pasini M. Roncaglia Condensed Matter Theory Group in Bologna Open Systems & Quantum Information Milano, 10 Marzo 2006

2. Spin chains are natural candidates as quantum devices QUBITS • Entanglement is a resource for: teleportation dense coding quantum cryptography quantum computation • Strong quantum fluctuations in low-dimensional quantum systems at T=0 • The Entanglement can give another perspective for understanding Quantum Phase Transitions Open Systems & Quantum Information Milano, 10 Marzo 2006

3. Entanglement is a property of a state, not of an Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled. A B • Direct product states • Nonzero correlations at T=0revealentanglement • 2-qubit states Product states Maximally entangled (Bell states) Open Systems & Quantum Information Milano, 10 Marzo 2006

4. Block entropy B A • Reduced density matrix for the subsystem A • Von Neumann entropy • For a 1+1 Dcritical system Off-critical CFT with central charge c l= block size [ See P.Calabrese and J.Cardy, JSTAT P06002 (2004).] Open Systems & Quantum Information Milano, 10 Marzo 2006

5. RG flow UV fixed point IR fixed point RG flow UV fixed point Renormalization Group (RG) • c-theorem: (Zamolodchikov, 1986) • Massive theory (off critical) • Block entropy saturation Irreversibility of RG trajectories Loss of entanglement Open Systems & Quantum Information Milano, 10 Marzo 2006

6. Local Entropy: when the subsystem A is a single site. • Applied to the extended Hubbard model • The local entropy depends only on the average double occupancy • The entropy is maximal at the phase transition lines • (equipartition) [ S.Gu, S.Deng, Y.Li, H.Lin, PRL 93, 86402 (2004).] Open Systems & Quantum Information Milano, 10 Marzo 2006

7. Bond-charge Hubbard model • (half-filling, x=1) • Critical points: U=-4, U=0 • Negativity • Mutual information • Some indicators show • singularities at transition points, while others don’t. [ A.Anfossi et al., PRL 95, 056402 (2005).] Open Systems & Quantum Information Milano, 10 Marzo 2006

8. Ising model in transverse field • Critical point: l=1 • The concurrence measures the entanglement between two sites after having traced out the remaining sites. • The transition is signaled by the first derivative of the concurrence, which diverges logarithmically (specific heat). [ A.Osterloh, et al., Nature 416, 608 (2002).] Open Systems & Quantum Information Milano, 10 Marzo 2006

9. Concurrence For a 2-qubit pure state the concurrence is (Wootters, 1998) if • Is maximal for the Bell states and zero for product states For a 2-qubit mixed state in a spin ½ system Open Systems & Quantum Information Milano, 10 Marzo 2006

10. Ising model in transverse field 2D classical Ising model CFTwith central chargec=1/2 Critical point Jordan-Wigner transformation Exactly solvable fermion model Open Systems & Quantum Information Milano, 10 Marzo 2006

11. Near the transition (h=1): S1 has the same singularity as Local (single site) entropy: Local measures of entanglement based on the 2-site density matrix depend on 2-point functions Nearest-neighbour concurrence inherits logarithmic singularity Accidental cancellation of the leading singularity may occur, as for the concurrence at distance 2 sites Open Systems & Quantum Information Milano, 10 Marzo 2006

12. Seeking for QPT point Alternative: FSS of magnetization Standard route: PRG First excited state needed C. Hamer, M. Barber, J. Phys. A: Math. Gen. (1981) 247. Exact scaling function in the critical region Crossing points: Shift term Open Systems & Quantum Information Milano, 10 Marzo 2006

13. Quantum phase transitions (QPT’s) Let • First order: discontinuity in (level crossing) • Second order: diverges for some • At criticality the correlation length diverges • GS energy: scaling hypothesis • Differentiating w.r.t. g Open Systems & Quantum Information Milano, 10 Marzo 2006

14. The singular term appears in every reduced density • matrix containing the sites connected by . • Local algebrahypothesis: every local quantity can be expanded • in terms of the scaling fields permitted by the symmetries. • Any local measure of entanglement contains the singularity • of the most relevant term. • Warning: accidental cancellations may occur depending on • the specific functional form next to leading singularity • The best suited operator for detecting and classifying QPT’s • is V , that naturally contains . Moreover, FSS at criticality Open Systems & Quantum Information Milano, 10 Marzo 2006

15. In this case (sine-Gordon) Spin 1 l-D model l D l =Ising-like D = single ion Phase Diagram • Symmetries: U(1)xZ2 Around the c=1 line: Critical exponents Open Systems & Quantum Information Milano, 10 Marzo 2006

16. Two-sites density matrix contains the same leading singularity Derivative The same for Crossing effect • What about local measures • of entanglement? Using symmetries: Single-site entropy [ L.Campos Venuti, et. al., PRA 73, 010303(R) (2006).] Open Systems & Quantum Information Milano, 10 Marzo 2006

17. [ F.Verstraete, M.Popp, J.I.Cirac, PRL 92, 27901 (2004).] Localizable Entanglement • LE is the maximum amount of entanglement that can • be localized on two q-bits by localmeasurements. j i N+2 particle state • Maximum over alllocal measurement basis = probability of getting is a measure of entanglement (concurrence) Open Systems & Quantum Information Milano, 10 Marzo 2006

18. [ L. Campos Venuti, M. Roncaglia, PRL 94, 207207 (2005).] Calculating the LE requires finding an optimal basis, which is a formidable task in general However, using symmetries some maximal (optimal) basis are easily found and the LE takes a manageable form Spin 1/2 Spin 1 • Ising model • Quantum XXZ chain • MPS (AKLT) LE = max of correlation LE = string correlations 1 • : • The lower bound is attained • The LE shows that spin 1 are • perfect quantum channels but is insensitive to phase transitions. Open Systems & Quantum Information Milano, 10 Marzo 2006

19. A spin-1 model: AKLT =Bell state Optimal basis: • Infinite entanglement length but finite correlation length • Actually in S=1 case LE is related to string correlation Typical configurations Open Systems & Quantum Information Milano, 10 Marzo 2006

20. Conclusions • Low-dimensional systems are good candidates for Quantum Information devices. • Several local measures of entanglement have been proposed recently for the detection and classification of QPT. (nonsystematic approach) • Apart from accidental cancellations all the scaling properties of local entanglement come from the most relevant (RG) scaling operator. • The most natural local quantity is , where g is the driving parameter • across the QPT. • it shows a crossing effect • it is unique and generally applicable Advantages: • Localizable Entanglement  It is related to some already known correlation functions. It promotes S=1 chains as perfect quantum channels. • Open problem: Hard to define entanglement for multipartite systems, • separating genuine quantum correlations and classical ones. References: L.Campos Venuti, C.Degli Esposti Boschi, M.Roncaglia, A.Scaramucci, PRA 73, 010303(R) (2006). L.Campos Venuti and M. Roncaglia, PRL 94, 207207 (2005). Open Systems & Quantum Information Milano, 10 Marzo 2006

21. The End Open Systems & Quantum Information Milano, 10 Marzo 2006