U Zʹj&@sddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZdd lmZdd lmZdd lmZdd lmZdd lmZmZmZddlmZmZddl m!Z!GdddeZ"ddZ#ddZ$dS))product)Add)Tuple)expand)Mul)Slog)MutableDenseMatrix prettyForm)Dagger)HermitianOperator) represent) numpy_ndarrayscipy_sparse_matrixto_numpy) TensorProducttensor_product_simp)TrcseZdZdZefddZddZddZdd Zd d Z d d Z ddZ ddZ ddZ ddZddZddZddZZS)Densitya Density operator for representing mixed states. TODO: Density operator support for Qubits Parameters ========== values : tuples/lists Each tuple/list should be of form (state, prob) or [state,prob] Examples ======== Create a density operator with 2 states represented by Kets. >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d Density((|0>, 0.5),(|1>, 0.5)) cs8t|}|D]"}t|tr*t|dkstdq|S)Nz?Each argument should be of form [state,prob] or ( state, prob ))super _eval_args isinstancerlen ValueError)clsargsarg __class__\/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/physics/quantum/density.pyr*s   zDensity._eval_argscCstdd|jDS)aReturn list of all states. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states() (|0>, |1>) cSsg|] }|dqS)rr".0rr"r"r# Dsz"Density.states..rrselfr"r"r#states7s zDensity.statescCstdd|jDS)a#Return list of all probabilities. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs() (0.5, 0.5) cSsg|] }|dqS)r"r$r"r"r#r&Ssz!Density.probs..r'r(r"r"r#probsFs 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> rr)r)indexstater"r"r# get_stateUszDensity.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_probjszDensity.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/r1opr"r#r&sz$Density.apply_op..)rr)r)r4new_argsr"r3r#apply_opszDensity.apply_opc Ksxg}|jD]d\}}|}t|trXt|jddD]"}||||d|dq2q |||||q t|S)aExpand the density operator into an outer product format. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.doit() 0.5*|0><0| + 0.5*|1><1| r)repeatrr+)rrrrrappend_generate_outer_prod)r)hintsZtermsr/r1rr"r"r#doits  z Density.doitcCs|\}}|\}}t|dks0t|dkr8tdt|dtrxt|dkrxt|dkrxt|dt|d}nt|tt|}t|t||S)NrzHAtleast one-pair of Non-commutative instance required for outer product.r+)Zargs_cncrrrrrr r)r)Zarg1Zarg2Zc_part1Znc_part1Zc_part2Znc_part2r4r"r"r#r9s   zDensity._generate_outer_prodcKst|f|SN)rr;)r)optionsr"r"r# _representszDensity._representcGsdS)Nz\rhor"r)printerrr"r"r#_print_operator_name_latexsz"Density._print_operator_name_latexcGstdS)Nuρr r?r"r"r#_print_operator_name_prettysz#Density._print_operator_name_prettycKs|dg}t||S)Nindices)getrr;)r)kwargsrCr"r"r# _eval_traces zDensity._eval_tracecCst|S)zl Compute the entropy of a density matrix. Refer to density.entropy() method for examples. )entropyr(r"r"r#rGszDensity.entropy)__name__ __module__ __qualname____doc__ classmethodrr*r,r0r2r6r;r9r>rArBrFrG __classcell__r"r"r r#rs rcCst|trt|}t|tr$t|}t|trR|}tt dd|D St|t rddl }|j |}| ||| StddS)aCompute 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.spin import JzKet >>> from sympy import S >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> d = Density((up,S(1)/2),(down,S(1)/2)) >>> entropy(d) log(2)/2 css|]}|t|VqdSr<r)r%er"r"r# szentropy..rNz4numpy.ndarray, scipy.sparse or SymPy matrix expected)rrrrrMatrixZ eigenvalskeysrsumrnumpyZlinalgeigvalsr r)ZdensityrTnpr"r"r#rGs      rGcCst|trt|n|}t|tr(t|n|}t|tr@t|tsXtdt|t|f|j|jkrr|jrrtd|tj }t |||tj  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 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] https://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 matrix) rrrrPrtypeshapeZ is_squarerZHalfrr;)Zstate1Zstate2Z sqrt_state1r"r"r#fidelitys, rXN)% itertoolsrZsympy.core.addrZsympy.core.containersrZsympy.core.functionrZsympy.core.mulrZsympy.core.singletonrZ&sympy.functions.elementary.exponentialr Zsympy.matrices.denser rPZ sympy.printing.pretty.stringpictr Zsympy.physics.quantum.daggerr Zsympy.physics.quantum.operatorrZsympy.physics.quantum.representrZ!sympy.physics.quantum.matrixutilsrrrZ#sympy.physics.quantum.tensorproductrrZsympy.physics.quantum.tracerrrGrXr"r"r"r#s$             H,