B [j&@sddlmZmZddlmZddlmZmZmZm Z m Z m Z m Z ddl mZddlmZddlmZddlmZddlmZdd lmZmZmZdd lmZmZGd d d eZd dZddZ dS))print_functiondivision)product)TupleAddMulMatrixlogexpandRational)Tr) prettyForm)Dagger)HermitianOperator) represent) numpy_ndarrayscipy_sparse_matrixto_numpy) TensorProducttensor_product_simpcseZdZdZefddZddZddZdd Zd d Z d d Z ddZ ddZ ddZ ddZddZddZddZZS)DensityaDensity operator for representing mixed states. TODO: Density operator support for Qubits Parameters ========== values : tuples/lists Each tuple/list should be of form (state, prob) or [state,prob] Examples ======== Create a density operator with 2 states represented by Kets. >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d 'Density'((|0>, 0.5),(|1>, 0.5)) cs@tt||}x*|D]"}t|tr0t|dkstdqW|S)Nz?Each argument should be of form [state,prob] or ( state, prob ))superr _eval_args isinstancerlen ValueError)clsargsarg) __class__>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states() (|0>, |1>) cSsg|] }|dqS)rr!).0rr!r!r" Bsz"Density.states..)rr)selfr!r!r"states5s zDensity.statescCstdd|jDS)a#Return list of all probabilities. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs() (0.5, 0.5) cSsg|] }|dqS)r!)r#rr!r!r"r$Qsz!Density.probs..)rr)r%r!r!r"probsDs z Density.probscCs|j|d}|S)atReturn specific state by index. Parameters ========== index : index of state to be returned Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states()[1] |1> r)r)r%indexstater!r!r" get_stateSszDensity.get_statecCs|j|d}|S)aReturn probability of specific state by index. Parameters =========== index : index of states whose probability is returned. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs()[1] 0.500000000000000 r')r)r%r)probr!r!r"get_probhszDensity.get_probcsfdd|jD}t|S)aop will operate on each individual state. Parameters ========== op : Operator Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.apply_op(A) 'Density'((A*|0>, 0.5),(A*|1>, 0.5)) csg|]\}}||fqSr!r!)r#r*r,)opr!r"r$sz$Density.apply_op..)rr)r%r.Znew_argsr!)r.r"apply_op}szDensity.apply_opc Ksg}xr|jD]h\}}|}t|tr^xLt|jddD]"}||||d|dq6Wq |||||q Wt|S)aExpand the density operator into an outer product format. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.doit() 0.5*|0><0| + 0.5*|1><1| r)repeatrr')rr rrrappend_generate_outer_prod)r%ZhintsZtermsr*r,rr!r!r"doits z Density.doitcCs|\}}|\}}t|dks0t|dkr8tdt|dtrxt|dkrxt|dkrxt|dt|d}nt|tt|}t|t||S)NrzHAtleast one-pair of Non-commutative instance required for outer product.r')Zargs_cncrrrrrrr)r%Zarg1Zarg2Zc_part1Znc_part1Zc_part2Znc_part2r.r!r!r"r2s    zDensity._generate_outer_prodcKst|f|S)N)rr3)r%Zoptionsr!r!r" _representszDensity._representcGs|jd|S)N\rho)r5)Z_print)r%printerrr!r!r"_print_operator_name_latexsz"Density._print_operator_name_latexcGs ttdS)Nuρ)r Zunichr)r%r6rr!r!r"_print_operator_name_prettysz#Density._print_operator_name_prettycKs|dg}t||S)Nindices)getr r3)r%kwargsr9r!r!r" _eval_traces zDensity._eval_tracecCst|S)zl Compute the entropy of a density matrix. Refer to density.entropy() method for examples. )entropy)r%r!r!r"r=szDensity.entropy)__name__ __module__ __qualname____doc__ classmethodrr&r(r+r-r/r3r2r4r7r8r<r= __classcell__r!r!)r r"rsrcCst|trt|}t|tr$t|}t|trR|}tt dd|D St|t rddl }|j |}| ||| StddS)aNCompute the entropy of a matrix/density object. This computes -Tr(density*ln(density)) using the eigenvalue decomposition of density, which is given as either a Density instance or a matrix (numpy.ndarray, sympy.Matrix or scipy.sparse). Parameters ========== density : density matrix of type Density, sympy matrix, scipy.sparse or numpy.ndarray Examples ======== >>> from sympy.physics.quantum.density import Density, entropy >>> from sympy.physics.quantum.represent import represent >>> from sympy.physics.quantum.matrixutils import scipy_sparse_matrix >>> from sympy.physics.quantum.spin import JzKet, Jz >>> from sympy import S, log >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> d = Density((up,0.5),(down,0.5)) >>> entropy(d) log(2)/2 css|]}|t|VqdS)N)r )r#er!r!r" szentropy..rNz4numpy.ndarray, scipy.sparse or sympy matrix expected)rrrrrrZ eigenvalskeysr sumrZnumpyZlinalgeigvalsr r)ZdensityrHZnpr!r!r"r=s      r=cCst|trt|n|}t|tr(t|n|}t|tr@t|tsXtdt|t|f|j|jkrr|jrrtd|tdd}t |||tdd S)a Computes the fidelity [1]_ between two quantum states The arguments provided to this function should be a square matrix or a Density object. If it is a square matrix, it is assumed to be diagonalizable. Parameters ========== state1, state2 : a density matrix or Matrix Examples ======== >>> from sympy import S, sqrt >>> from sympy.physics.quantum.dagger import Dagger >>> from sympy.physics.quantum.spin import JzKet >>> from sympy.physics.quantum.density import Density, fidelity >>> from sympy.physics.quantum.represent import represent >>> >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> amp = 1/sqrt(2) >>> updown = (amp * up) + (amp * down) >>> >>> # represent turns Kets into matrices >>> up_dm = represent(up * Dagger(up)) >>> down_dm = represent(down * Dagger(down)) >>> updown_dm = represent(updown * Dagger(updown)) >>> >>> fidelity(up_dm, up_dm) 1 >>> fidelity(up_dm, down_dm) #orthogonal states 0 >>> fidelity(up_dm, updown_dm).evalf().round(3) 0.707 References ========== .. [1] http://en.wikipedia.org/wiki/Fidelity_of_quantum_states zfstate1 and state2 must be of type Density or Matrix received type=%s for state1 and type=%s for state2z]The dimensions of both args should be equal and the matrix obtained should be a square matrixr'r) rrrrrtypeshapeZ is_squarer r r3)Zstate1Zstate2Z sqrt_state1r!r!r"fidelitys,  rKN)!Z __future__rr itertoolsrZsympyrrrrr r r Zsympy.core.tracer Z sympy.printing.pretty.stringpictr Zsympy.physics.quantum.daggerrZsympy.physics.quantum.operatorrZsympy.physics.quantum.representrZ!sympy.physics.quantum.matrixutilsrrrZ#sympy.physics.quantum.tensorproductrrrr=rKr!r!r!r"s $     L.