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

Open Systems & Quantum Information Entanglement is a resource for: teleportation dense coding quantum cryptography quantum computation QUBITS Spin chains are natural candidates as quantum devices 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

Open Systems & Quantum Information 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=0 reveal entanglement 2-qubit states Product states Maximally entangled (Bell states) Open Systems & Quantum Information Milano, 10 Marzo 2006

Open Systems & Quantum Information Block entropy B A Reduced density matrix for the subsystem A Von Neumann entropy For a 1+1 D critical 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

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

Open Systems & Quantum Information 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

Open Systems & Quantum Information 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

Open Systems & Quantum Information 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

Open Systems & Quantum Information 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

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

Open Systems & Quantum Information 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

Open Systems & Quantum Information 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

Open Systems & Quantum Information 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

Open Systems & Quantum Information 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. 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

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

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

Open Systems & Quantum Information [ 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 local measurements. j i N+2 particle state Maximum over all local measurement basis = probability of getting is a measure of entanglement (concurrence) Open Systems & Quantum Information Milano, 10 Marzo 2006

Open Systems & Quantum Information [ 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

Open Systems & Quantum Information 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

Open Systems & Quantum Information 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

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