Quantum information approach to the description of quantum phase transitions O. Casta˜ nos Instituto de Ciencias Nucleares, UNAM Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Guy Paic and the ICN 1996-1998, new development plan of the ICN. Creation of the Department: High energy physics. 2001-2002, Guy agreed to come to Mexico at the ICN. C´ atedra Patrimonial de Excelencia Nivel II (CONACyT). Purpose: Create a laboratory to support measurements and test of detectors mainly related with the ALICE experiment. April 2003 to March 2005 Got a position in June 2005. Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Achivements in the two years The laboratory was equiped: to develop and test detectors Members 1 researcher, 2 posdocs, 3 PhD students, and 1 M. Sc. student Construction of a electronic card to characterize the scintillators for the ACORDE detector Design of an emulator of signals to test the data acquisition system of ALICE Several simulations related with the V0 detector and the analysis of data of ALICE. Design of a very high momentum particle identification detector for ALICE Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Guy Paic and the ICN Silver Juchiman Award Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Content Quantum phase transitions Information concepts Fidelity and Fidelity Susceptibility Entanglement Linear and von Neumann Entropies Conclusions Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Quantum phase transitions Typically they are driven by purely quantum fluctuations Characterized by the vanishing, in the thermodynamic limit, of the energy gap Sudden change, non analytical, in the ground state properties of a system Classically they are determined by the stability properties of the potential energy surface, the order is determined by the Ehrenfest classification This can be extended to the quantum case: Expectation value of the Hamiltonian with respect to a variational function Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Phase transitions Family of potentials V = V ( x,c ) , with x = ( x 1 , · · · x n ) and c = ( c 1 ,c 2 , · · · ,c k ) . Equilibrium and stability properties: ∂ 2 V ∂V = 0, > 0. ∂x j ∂x j ∂x k State equation: x ( p ) = x ( p ) ( c 1 ,c 2 , · · · ,c k ) A phase transition occurs when the point x ( p ) ( c ) cross the separatrix of the physical system. The separatrix is the union of the bifurcation and Maxwell sets. Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Separatrix Ground state energy for a system of N particles � H � = E ( x α ,c j ) → E = E ( x α ,c j ) N with α = 1, · · · n and j = 1,2, · · · ,k . ∂ E Bifurcation and Maxwell sets: ∂x k = 0 ∂ 2 E ˛ E i,j = x ( p ) ( c ) , ˛ ∂x i ∂x j ˛ � � ∂ E ( p ) − ∂ E ( p + 1 ) E ( p + 1 ) , E ( p ) = δc j = 0. ∂c j ∂c j Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Quantum phase transitions A finite temperature, a quantum system is a mixture of pure states, where each one occurs with probability P k = 1/Z exp (− βE k ) , κ B T and the partition function Z = � 1 with β = i exp (− βE i ) . The expectation value of an operator is given in terms of the density operator � � ^ P i � ψ i | ^ O | ψ i � = Tr ( ρ ^ O � = O ) . i At T = 0 only the ground state contributes For T � = 0 , the quantum state is determined by the condition of minimum free energy instead of minimum energy. Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Energy and information Since 1961, from the Landauer principle, is known the mantra: information is physical The reason the Maxwell demon cannot violate the second law: in order to observe a molecule, it must first forget the results of previous observations. Forgetting results, or discarding information, is thermodynamically costly ( ∆S e = k B ln 2 ) Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Hamiltonian Model The Ising model for two spins 1/2 or qubits ∗ “ ” H = σ ( 1 ) σ ( 2 ) σ ( 1 ) + σ ( 2 ) + B 0 , z z z z where the coupling of the qubits has been taken to be the unity. The σ ( i ) are z Pauli matrices and B 0 is a magnetic field. In terms of the total angular momentum, the Hamiltonian can be written H = 2 ^ J 2 z − 1 + 2B 0 ^ J z , where 2J z = σ ( 1 ) + σ ( 2 ) . z z ∗ J. Zhang, X. Peng, N. Rajendran, and D. Suter, Phys. Rev. Latt. 100 , 100501 (2008) Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Solution Energies and eigenstates = { − 1, − 1, 1 − 2B 0 , 1 + 2B 0 } , E | σ 1 ,σ 2 � = {| + , − � , | − , + � , | − , − � , | + , + � } . Semiclassical solution H = cos 2 θ − 2B 0 cos θ , where the variational state is given by s 1 − cos 2 θ | j = 1, θ � = 1 − cos θ | 1,0 � + 1 + cos θ | 1,1 � + | 1, − 1 � . 2 2 2 Critical points θ c : { 0, π, arccos B 0 } . Energies and eigenstates − B 2 E = { 1 − 2B 0 , 1 + 2B 0 , 0 , − 1 } , | θ c � {| 1, − 1 � , | 1,1 � , | 1,0 � , | 0,0 � } . = Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Energies and fidelity E 6 4 2 Out[53]= B 0 � 3 � 2 � 1 1 2 3 � 2 � 4 Above in red color, the semiclassical energies and in blue color the quantum ones. Below the fidelity between the quantum solutions with B 1 and B 2 . We add a probe qubit with the interaction ǫσ ( p ) ( σ ( 1 ) + σ ( 2 ) ) . Thus one has two effective Hamiltonians one with z z z B 1 = B 0 + ǫ , the other with B 2 = B 0 − ǫ . At the right, we consider a small magnetic field B x . F F 1.0 1.00000 0.99995 0.8 0.99990 0.6 0.99985 0.4 0.99980 0.2 0.99975 0.99970 B 0 B 0 � 3 � 2 � 1 1 2 3 � 3 � 2 � 1 1 2 3 Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Fidelity For two pure states, ρ 1 = | χ �� χ | and ρ 2 = | φ �� φ | , the fidelity is defined by F ( | χ �� χ | , | φ �� φ | ) = | � χ | φ � | 2 , the transition probability from one state to another. Its geometric interpretation is the closeness of states. For one mixed state ρ 2 , one has F ( | χ �� χ | ,ρ 2 ) = � χ | ρ 2 | χ � , that denotes the probability to be a pure state. For mixed states the fidelity should satisfy the properties: 0 ≤ ≤ 1 (1) F ( ρ 1 ,ρ 2 ) F ( ρ 1 ,ρ 2 ) = F ( ρ 2 ,ρ 1 ) (2) F ( Uρ 1 ,Uρ 2 ) = F ( ρ 1 ,ρ 2 ) (3) Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Fidelity Uhlmann-Jozsa proved that � ” � 2 “q √ ρ 1 ρ 2 √ ρ 1 F ( ρ 1 ,ρ 2 ) = Tr , satisfies the previous properties. Another definition satisfying the same properties was given by Mendonca et al, i.e., q q 1 − Tr ( ρ 2 1 − Tr ( ρ 2 F ( ρ 1 ,ρ 2 ) = Tr ( ρ 1 ρ 2 ) + 1 ) 2 ) . The fidelity has a fundamental role in communication theory because measures the accuracy of a transmission. Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Fidelity and Fidelity Susceptibility The fidelity ( P. Zanardi and N. Paunkovic, Phys. Rev. E 74 (2006)) can be used to determine when the ground state of a quantum system presents a sudden change as function of a control parameter. If we denote that parameter by λ one has F ( λ,λ + δλ ) = | � ψ ( λ ) | ψ ( λ + δλ ) � | 2 . Taylor series expansion of the fidelity ˛ ˛ d 2 F F ( λ c ,λ c + δλ ) = F ( λ c ,λ c ) + δλ dF + ( δλ ) 2 1 ˛ ˛ + · · · , ˛ ˛ dλ 2 dλ ˛ 2 ! ˛ ˛ ˛ λ = λ c λ = λ c the first derivative is zero because the fidelity is a minimum and the fidelity susceptibility is defined by (W. You et al Phys. Rev. E 76 (2007)) χ F = 21 − F ( λ c ,λ c + δλ ) . ( δλ ) 2 It is dependent of the Hamiltonian term that causes the phase transition. Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
Entanglement Suppose Alice and Bob are trying to create n copies of a particular bipartite state | Φ � , such that Alice hold the part A and Bob the part B. They are not allowed any quantum communication between them. However they have a large collection of shared singlet pairs | Ψ − � . How many singlet pairs must they use up in order to create n copies of | Φ � ? The answer is they need to create roughly nS vN ( Φ ) , the von Neumann entropy. Examples, the so called Bell states � � 1 | Φ ± � | + , + � ± | − , − � , = √ 2 � � 1 | Ψ ± � | + , − � ± | − , + � √ . = 2 which have maximum linear and von Neumann entropies. Guy Paic Fest (December 1, 2012) Quantum information approach Puebla, M´ exico
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