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

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

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

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

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

Open Systems & Quantum Information Milano, 10 Marzo 2006 [ S.Gu, S.Deng, Y.Li, H.Lin, PRL 93, (2004).] 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)

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

Open Systems & Quantum Information Milano, 10 Marzo 2006 [ A.Osterloh, et al., Nature 416, 608 (2002).] Ising model in transverse field Critical point: =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).

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

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

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

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

Open Systems & Quantum Information Milano, 10 Marzo 2006 The singular term appears in every reduced density matrix containing the sites connected by. Local algebra hypothesis: 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. The best suited operator for detecting and classifying QPT’s is V, that naturally contains. Moreover, FSS at criticality Warning: accidental cancellations may occur depending on the specific functional form next to leading singularity

Open Systems & Quantum Information Milano, 10 Marzo 2006 Spin 1  D model D =Ising-like D = single ion In this case Phase Diagram Symmetries: U(1)xZ 2 Around the c=1 line: (sine-Gordon) Critical exponents

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

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

Open Systems & Quantum Information Milano, 10 Marzo 2006 LE = max of correlationLE = string correlations [ L. Campos Venuti, M. Roncaglia, PRL 94, (2005).] Ising model Quantum XXZ chain MPS (AKLT) 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 1/2Spin 1 1 The LE shows that spin 1 are perfect quantum channels but is insensitive to phase transitions. : The lower bound is attained

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

Open Systems & Quantum Information Milano, 10 Marzo 2006 Conclusions References: L.Campos Venuti, C.Degli Esposti Boschi, M.Roncaglia, A.Scaramucci, PRA 73, (R) (2006). L.Campos Venuti and M. Roncaglia, PRL 94, (2005). Localizable Entanglement  It is related to some already known correlation functions. It promotes S=1 chains as perfect quantum channels. 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: 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) Open problem: Hard to define entanglement for multipartite systems, separating genuine quantum correlations and classical ones. Apart from accidental cancellations all the scaling properties of local entanglement come from the most relevant (RG) scaling operator.

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