+ iJ%^RIHt^RIHt^RIHt^RIHt^RIH t ^RI H t ^RI H t ^RIHt^R IHt^R IHt^R IHt^R IHt^R IHtHtHt^RIHt!RR]4tRt Rt!R#))product)Add)Tuple)expand)Mul)Slog)MutableDenseMatrix prettyForm)Dagger)HermitianOperator) represent) numpy_ndarrayscipy_sparse_matrixto_numpy)Trcaa]tRt^toRt]V3Rl4tRtRtRt Rt Rt Rt R t R tR tR tR tRtRtVtV;t#)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<\SV`V4pVF4p\V\4'd\ V4^8XdK+\ R4h V#)z?Each argument should be of form [state,prob] or ( state, prob ))super _eval_args isinstancerlen ValueError)clsargsarg __class__s&& }/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/physics/quantum/density.pyrDensity._eval_args)sNw!$'CsE**s3x1} "788  c ^\VPUu.uF q^,NK up!#uupi)zReturn list of all states. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states() (|0>, |1>) rrselfr s& r"statesDensity.states6)3#1vv3443*c ^\VPUu.uF q^,NK up!#uupi)zReturn list of all probabilities. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs() (0.5, 0.5) r&r's& r"probs Density.probsEr+r,c :VPV,^,pV#)aReturn specific state by index. Parameters ========== index : index of state to be returned Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states()[1] |1> r)r(indexstates&& r" get_stateDensity.get_stateTs$ % # r$c :VPV,^,pV#)aJReturn probability of specific state by index. Parameters =========== index : index of states whose probability is returned. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs()[1] 0.500000000000000 r1)r(r2probs&& r"get_probDensity.get_probis$yy" r$c nVPUUu.uFwr#W,V3NK ppp\V!#uuppi)a{op will operate on each individual state. Parameters ========== op : Operator Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.apply_op(A) Density((A*|0>, 0.5),(A*|1>, 0.5)) )rr)r(opr3r7new_argss&& r"apply_opDensity.apply_op~s6(;?))D)%RXt$)D!!Es1c .pVPFwr4VP4p\V\4'dV\ VP^R7F8pVP W@P V^,V^,4,4K: KVP W@P W34,4K \V!#)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| )repeat)rrrrrappend_generate_outer_prod)r(hintstermsr3r7r s&, r"doit Density.doits !YYMULLNE5#&&"5::a8CLL&?&?A@CA'H"HI9 T";";E"IIJ'E{r$cVP4wr4VP4wrV\V4^8Xg\V4^8Xd \R4h\V!\ \V!4,p\V!\V!,V,#)rzHAtleast one-pair of Non-commutative instance required for outer product.)args_cncrrrr)r(arg1arg2c_part1nc_part1c_part2nc_part2r;s&&& r"rBDensity._generate_outer_prodsx MMO MMO MQ #h-1"434 4(^F3>2 2G}S']*R//r$c 6\VP43/VB#N)rrE)r(optionss&,r" _representDensity._represents000r$cR#)z\rhor(printerrs&&*r"_print_operator_name_latex"Density._print_operator_name_latexsr$c\R4#)uρr rWs&&*r"_print_operator_name_pretty#Density._print_operator_name_prettys677r$c vVPR.4p\VP4V4P4#)indices)getrrE)r(kwargsr_s&, r" _eval_traceDensity._eval_traces.**Y+$))+w',,..r$c \V4#)z[Compute the entropy of a density matrix. Refer to density.entropy() method for examples. )entropy)r(s&r"reDensity.entropys t}r$rV)__name__ __module__ __qualname____firstlineno____doc__ classmethodrr)r.r4r8r=rErBrSrYr\rbre__static_attributes____classdictcell__ __classcell__)r! __classdict__s@@r"rrs_,   5 5**".80 18/r$rc\V\4'd \V4p\V\4'd \ V4p\V\ 4'd;VP 4P4p\\RV44)4#\V\4'dG^RI pVPPV4pVPWPV4,4)#\R4h)a{Compute the entropy of a matrix/density object. This computes -Tr(density*ln(density)) using the eigenvalue decomposition of density, which is given as either a Density instance or a matrix (numpy.ndarray, sympy.Matrix or scipy.sparse). Parameters ========== density : density matrix of type Density, SymPy matrix, scipy.sparse or numpy.ndarray Examples ======== >>> from sympy.physics.quantum.density import Density, entropy >>> from sympy.physics.quantum.spin import JzKet >>> from sympy import S >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> d = Density((up,S(1)/2),(down,S(1)/2)) >>> entropy(d) log(2)/2 c3D"TFq\V4,xK R#5irQr ).0es& r" entropy..s5WSV88Ws Nz4numpy.ndarray, scipy.sparse or SymPy matrix expected)rrrrrMatrix eigenvalskeysrsumrnumpylinalgeigvalsr r)densityr}nps& r"reres4'7##G$'.//7#'6""##%**,s5W55566 G] + +))##G,wvvg./// BD Dr$cF\V\4'd \V4MTp\V\4'd \V4MTp\V\4'd\V\4'g'\ R\ V4: R\ V4: R24hVP VP 8wdVP'd \ R4hV\P,p\W!,V,\P,4P4#)a#Computes the fidelity [1]_ between two quantum states The arguments provided to this function should be a square matrix or a Density object. If it is a square matrix, it is assumed to be diagonalizable. Parameters ========== state1, state2 : a density matrix or Matrix Examples ======== >>> from sympy import S, sqrt >>> from sympy.physics.quantum.dagger import Dagger >>> from sympy.physics.quantum.spin import JzKet >>> from sympy.physics.quantum.density import fidelity >>> from sympy.physics.quantum.represent import represent >>> >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> amp = 1/sqrt(2) >>> updown = (amp*up) + (amp*down) >>> >>> # represent turns Kets into matrices >>> up_dm = represent(up*Dagger(up)) >>> down_dm = represent(down*Dagger(down)) >>> updown_dm = represent(updown*Dagger(updown)) >>> >>> fidelity(up_dm, up_dm) 1 >>> fidelity(up_dm, down_dm) #orthogonal states 0 >>> fidelity(up_dm, updown_dm).evalf().round(3) 0.707 References ========== .. [1] https://en.wikipedia.org/wiki/Fidelity_of_quantum_states zBstate1 and state2 must be of type Density or Matrix received type=z for state1 and type=z for state2z]The dimensions of both args should be equal and the matrix obtained should be a square matrix) rrrrwrtypeshape is_squarerHalfrrE)state1state2 sqrt_state1s&& r"fidelityrsX#-VW"="=Yv 6F",VW"="=Yv 6F ff % %Z-G-Gv,V 67 7||v||#(8(8(8EF F!&&.K {!+-6 7 < < >>r$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.tracerrrerrVr$r"rsN'&"6=7/<5ZZ*AAH)DX9?r$