\hJ%SSKJr SSKJr SSKJr SSKJr SSKJ r SSK J r SSK J r SSKJr SS KJr SS KJr SS KJr SS KJr SS KJrJrJr SSKJr "SS\5rSr Sr!g))product)Add)Tuple)expand)Mul)Slog)MutableDenseMatrix prettyForm)Dagger)HermitianOperator) represent) numpy_ndarrayscipy_sparse_matrixto_numpy)Trc~^\rSrSrSr\U4Sj5rSrSrSr Sr Sr S r S r S rS rS rSrSrSrU=r$)DensityaDensity operator for representing mixed states. TODO: Density operator support for Qubits Parameters ========== values : tuples/lists Each tuple/list should be of form (state, prob) or [state,prob] Examples ======== Create a density operator with 2 states represented by Kets. >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d Density((|0>, 0.5),(|1>, 0.5)) c>[TU]U5nUH2n[U[5(a[ U5S:XaM)[ S5e U$)Nz?Each argument should be of form [state,prob] or ( state, prob ))super _eval_args isinstancerlen ValueError)clsargsarg __class__s U/var/www/auris/envauris/lib/python3.13/site-packages/sympy/physics/quantum/density.pyrDensity._eval_args)sNw!$'CsE**s3x1} "788  cV[URVs/sHoSPM sn6$s snf)zReturn 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>) rrr selfr!s r#statesDensity.states6'3#1v3443&cV[URVs/sHoSPM sn6$s snf)zReturn list of all probabilities. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs() (0.5, 0.5) r'r(s r#probs Density.probsEr,r-c*URUSnU$)aReturn 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)indexstates r# get_stateDensity.get_stateTs$ % # r%c*URUSnU$)aJReturn probability of specific state by index. Parameters =========== index : index of states whose probability is returned. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs()[1] 0.500000000000000 r/r3)r)r4probs r#get_probDensity.get_probis$yy" r%cfURVVs/sH up#X-U4PM nnn[U6$s snnf)a{op will operate on each individual state. Parameters ========== op : Operator Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.apply_op(A) Density((A*|0>, 0.5),(A*|1>, 0.5)) )r r)r)opr5r9new_argss r#apply_opDensity.apply_op~s6(;?))D)%RXt$)D!!Es-c Z/nURHup4UR5n[U[5(aF[ URSS9H,nUR X@R USUS5-5 M. MpUR X@R X35-5 M [U6$)aUExpand the density operator into an outer product format. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.doit() 0.5*|0><0| + 0.5*|1><1| r)repeatrr/)r rrrrappend_generate_outer_prod)r)hintstermsr5r9r!s r#doit Density.doits !YYMULLNE5#&&"5::a8CLL&?&?A@CA'H"HI9 T";";E"IIJ'E{r%cUR5up4UR5upV[U5S:Xd[U5S:Xa [S5e[U6[ [U65-n[U6[U6-U-$)NrzHAtleast one-pair of Non-commutative instance required for outer product.)args_cncrrrr)r)arg1arg2c_part1nc_part1c_part2nc_part2r=s r#rDDensity._generate_outer_prodsx MMO MMO MQ #h-1"434 4(^F3>2 2G}S']*R//r%c 6[UR540UD6$N)rrG)r)optionss r# _representDensity._represents000r%cg)Nz\rhor)printerr s r#_print_operator_name_latex"Density._print_operator_name_latexsr%c[S5$)Nuρr rYs r#_print_operator_name_pretty#Density._print_operator_name_prettys677r%c vURS/5n[UR5U5R5$)Nindices)getrrG)r)kwargsras r# _eval_traceDensity._eval_traces.**Y+$))+w',,..r%c[U5$)z[Compute the entropy of a density matrix. Refer to density.entropy() method for examples. )entropy)r)s r#rgDensity.entropys t}r%rX)__name__ __module__ __qualname____firstlineno____doc__ classmethodrr*r0r6r:r?rGrDrUr[r^rdrg__static_attributes__ __classcell__)r"s@r#rrs],   5 5**".80 18/r%rc[U[5(a [U5n[U[5(a [ U5n[U[ 5(a:UR 5R5n[[SU55*5$[U[5(aBSSK nURRU5nURXRU5-5*$[S5e)a{Compute the entropy of a matrix/density object. This computes -Tr(density*ln(density)) using the eigenvalue decomposition of density, which is given as either a Density instance or a matrix (numpy.ndarray, sympy.Matrix or scipy.sparse). Parameters ========== density : density matrix of type Density, SymPy matrix, scipy.sparse or numpy.ndarray Examples ======== >>> from sympy.physics.quantum.density import Density, entropy >>> from sympy.physics.quantum.spin import JzKet >>> from sympy import S >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> d = Density((up,S(1)/2),(down,S(1)/2)) >>> entropy(d) log(2)/2 c3<# UHo[U5-v M g7frSr ).0es r# entropy..s5WSV8WsrNz4numpy.ndarray, scipy.sparse or SymPy matrix expected)rrrrrMatrix eigenvalskeysrsumrnumpylinalgeigvalsr r)densityr}nps r#rgrgs4'7##G$'.//7#'6""##%**,s5W55566 G] + +))##G,wvvg./// BD Dr%c[U[5(a [U5OUn[U[5(a [U5OUn[U[5(a[U[5(d&[ S[ U5<S[ U5<S35eUR UR :waUR(a [ S5eU[R-n[X!-U-[R-5R5$)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 zBstate1 and state2 must be of type Density or Matrix received type=z for state1 and type=z for state2z]The dimensions of both args should be equal and the matrix obtained should be a square matrix) rrrrwrtypeshape is_squarerHalfrrG)state1state2 sqrt_state1s r#fidelityrsX#-VW"="=Yv 6F",VW"="=Yv 6F ff % %Z-G-Gv,V 67 7||v||#(8(8EF F!&&.K {!+-6 7 < < >>r%N)" itertoolsrsympy.core.addrsympy.core.containersrsympy.core.functionrsympy.core.mulrsympy.core.singletonr&sympy.functions.elementary.exponentialr sympy.matrices.denser rw sympy.printing.pretty.stringpictr sympy.physics.quantum.daggerrsympy.physics.quantum.operatorrsympy.physics.quantum.representr!sympy.physics.quantum.matrixutilsrrrsympy.physics.quantum.tracerrrgrrXr%r#rsN'&"6=7/<5ZZ*AAH)DX9?r%