Цi&SSKJr SSKJr SSKJr SSKJr SSKJ r SSK J r SSK J r SSKJr SS KJr SS KJr SS KJr SS KJr SS KJrJrJr SSKJrJr SSK J!r! "SS\5r"Sr#Sr$g))product)Add)Tuple)expand)Mul)Slog)MutableDenseMatrix prettyForm)Dagger)HermitianOperator) represent) numpy_ndarrayscipy_sparse_matrixto_numpy) TensorProducttensor_product_simp)Trc~^\rSrSrSr\U4Sj5rSrSrSr Sr Sr S r S r S rS rS rSrSrSrU=r$)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$)Nz?Each argument should be of form [state,prob] or ( state, prob ))super _eval_args isinstancerlen ValueError)clsargsarg __class__s \/var/www/html/ai-image-ml/venv/lib/python3.13/site-packages/sympy/physics/quantum/density.pyrDensity._eval_args*sNw!$'CsE**s3x1} "788  cV[URVs/sHoSPM sn6$s snf)zReturn list of all states. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states() (|0>, |1>) rrr"selfr#s r%statesDensity.states7'3#1v3443&cV[URVs/sHoSPM sn6$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.probsFr.r/c*URUSnU$)aReturn specific state by index. Parameters ========== index : index of state to be returned Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states()[1] |1> rr")r+indexstates r% get_stateDensity.get_stateUs$ % # r'c*URUSnU$)aJReturn probability of specific state by index. Parameters =========== index : index of states whose probability is returned. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs()[1] 0.500000000000000 r1r5)r+r6probs r%get_probDensity.get_probjs$yy" r'cfURVVs/sH up#X-U4PM nnn[U6$s snnf)a{op will operate on each individual state. Parameters ========== op : Operator Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.apply_op(A) Density((A*|0>, 0.5),(A*|1>, 0.5)) )r"r)r+opr7r;new_argss r%apply_opDensity.apply_ops6(;?))D)%RXt$)D!!Es-c Z/nURHup4UR5n[U[5(aF[ URSS9H,nUR X@R USUS5-5 M. MpUR X@R X35-5 M [U6$)aUExpand the density operator into an outer product format. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.doit() 0.5*|0><0| + 0.5*|1><1| r)repeatrr1)r"rrrrappend_generate_outer_prod)r+hintstermsr7r;r#s r%doit Density.doits !YYMULLNE5#&&"5::a8CLL&?&?A@CA'H"HI9 T";";E"IIJ'E{r'cUR5up4UR5upV[U5S:Xd[U5S:Xa [S5e[US[5(a<[U5S:Xa-[U5S:Xa[ US[ US5-5nO[U6[ [U65-n[U6[U6-U-$)NrzHAtleast one-pair of Non-commutative instance required for outer product.r1)args_cncrr rrrrr)r+arg1arg2c_part1nc_part1c_part2nc_part2r?s r%rFDensity._generate_outer_prods MMO MMO MQ #h-1"434 4 x{M 2 2s8}7IMQ&$Xa[ 1D%DEBhsH~ 66BG}S']*R//r'c 6[UR540UD6$N)rrI)r+optionss r% _representDensity._represents000r'cg)Nz\rhor+printerr"s r%_print_operator_name_latex"Density._print_operator_name_latexsr'c[S5$)Nuρr r[s r%_print_operator_name_pretty#Density._print_operator_name_prettys677r'c vURS/5n[UR5U5R5$)Nindices)getrrI)r+kwargsrcs 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%riDensity.entropys t}r'rZ)__name__ __module__ __qualname____firstlineno____doc__ classmethodrr,r2r8r<rArIrFrWr]r`rfri__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*$[S5e)a{Compute the entropy of a matrix/density object. This computes -Tr(density*ln(density)) using the eigenvalue decomposition of density, which is given as either a Density instance or a matrix (numpy.ndarray, sympy.Matrix or scipy.sparse). Parameters ========== density : density matrix of type Density, SymPy matrix, scipy.sparse or numpy.ndarray Examples ======== >>> from sympy.physics.quantum.density import Density, entropy >>> from sympy.physics.quantum.spin import JzKet >>> from sympy import S >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> d = Density((up,S(1)/2),(down,S(1)/2)) >>> entropy(d) log(2)/2 c3<# UHo[U5-v M g7frUr ).0es r% entropy..s5WSV8WsrNz4numpy.ndarray, scipy.sparse or SymPy matrix expected)rrrrrMatrix eigenvalskeysrsumrnumpylinalgeigvalsr r )densityrnps r%riris4'7##G$'.//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) rrrryr typeshape is_squarerHalfrrI)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 ry sympy.printing.pretty.stringpictr sympy.physics.quantum.daggerrsympy.physics.quantum.operatorrsympy.physics.quantum.representr!sympy.physics.quantum.matrixutilsrrr#sympy.physics.quantum.tensorproductrrsympy.physics.quantum.tracerrrirrZr'r%rsQ'&"6=7/<5ZZR*DDN)DX9?r'