sg&ddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZdd lmZdd lmZdd lmZdd lmZdd lmZmZmZddlmZmZddl m!Z!GddeZ"dZ#dZ$y))product)Add)Tuple)expand)Mul)Slog)MutableDenseMatrix prettyForm)Dagger)HermitianOperator) represent) numpy_ndarrayscipy_sparse_matrixto_numpy) TensorProducttensor_product_simp)TrcteZdZdZefdZdZdZdZdZ dZ dZ d Z d Z d Zd Zd ZdZxZS)Densitya Density operator for representing mixed states. TODO: Density operator support for Qubits Parameters ========== values : tuples/lists Each tuple/list should be of form (state, prob) or [state,prob] Examples ======== Create a density operator with 2 states represented by Kets. >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d Density((|0>, 0.5),(|1>, 0.5)) ct||}|D]+}t|trt |dk(r"t d|S)Nz?Each argument should be of form [state,prob] or ( state, prob ))super _eval_args isinstancerlen ValueError)clsargsarg __class__s P/var/www/html/venv/lib/python3.12/site-packages/sympy/physics/quantum/density.pyrzDensity._eval_args*sPw!$' 8CsE*s3x1} "788 8  cRt|jDcgc]}|d c}Scc}w)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>) rrr!selfr"s r$stateszDensity.states7%3#s1v3443 $cRt|jDcgc]}|d c}Scc}w)a#Return list of all probabilities. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs() (0.5, 0.5) r'r(s r$probsz Density.probsFr+r,c*|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)indexstates r$ get_statezDensity.get_stateUs$ % # r%c*|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.r1)r)r2probs r$get_probzDensity.get_probjs$yy" r%cd|jDcgc] \}}||z|f}}}t|Scc}}w)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)) )r!r)r)opr3r6new_argss r$apply_opzDensity.apply_ops9(;?))D%RXt$DD!!Es,c Tg}|jD]\}}|j}t|trGt |jdD],}|j ||j |d|dz.m|j ||j ||zt|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.)r!rrrrappend_generate_outer_prod)r)hintstermsr3r6r"s r$doitz Density.doits !YY KMUDLLNE5#&"5::a8ICLLd&?&?A@CA'H"HII T$";";E5"IIJ KE{r%c|j\}}|j\}}t|dk(st|dk(r tdt|dtr:t|dk(r,t|dk(rt |dt |dz}nt|t t|z}t|t|z|zS)NrzHAtleast one-pair of Non-commutative instance required for outer product.r.)args_cncrrrrrrr)r)arg1arg2c_part1nc_part1c_part2nc_part2r9s r$r?zDensity._generate_outer_prods MMO MMO MQ #h-1"434 4 x{M 2s8}7IMQ&$Xa[ 1D%DEBhsH~ 66BG}S']*R//r%c 6t|jfi|SN)rrB)r)optionss r$ _representzDensity._represents000r%cy)Nz\rhor)printerr!s r$_print_operator_name_latexz"Density._print_operator_name_latexsr%ctdS)Nuρr rQs r$_print_operator_name_prettyz#Density._print_operator_name_prettys677r%c v|jdg}t|j|jS)Nindices)getrrB)r)kwargsrWs r$ _eval_tracezDensity._eval_traces.**Y+$))+w',,..r%ct|S)zl Compute the entropy of a density matrix. Refer to density.entropy() method for examples. )entropy)r)s r$r\zDensity.entropys t}r%)__name__ __module__ __qualname____doc__ classmethodrr*r/r4r7r;rBr?rNrSrUrZr\ __classcell__)r#s@r$rrsX,   5 5**".80&18/r%rct|tr t|}t|tr t |}t|t r:|j j}ttd|D St|trCddl }|jj|}|j||j|z Std)aCompute the entropy of a matrix/density object. This computes -Tr(density*ln(density)) using the eigenvalue decomposition of density, which is given as either a Density instance or a matrix (numpy.ndarray, sympy.Matrix or scipy.sparse). Parameters ========== density : density matrix of type Density, SymPy matrix, scipy.sparse or numpy.ndarray Examples ======== >>> from sympy.physics.quantum.density import Density, entropy >>> from sympy.physics.quantum.spin import JzKet >>> from sympy import S >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> d = Density((up,S(1)/2),(down,S(1)/2)) >>> entropy(d) log(2)/2 c38K|]}|t|zywrLr ).0es r$ zentropy..s51SV85srNz4numpy.ndarray, scipy.sparse or SymPy matrix expected)rrrrrMatrix eigenvalskeysrsumrnumpylinalgeigvalsr r)densityrnnps r$r\r\s4'7#G$'./7#'6"##%**,s5W55566 G] +))##G,wrvvg./// BD Dr%ct|tr t|n|}t|tr t|n|}t|trt|ts$t dt |dt |d|j |j k7r|jr t d|tjz}t||z|ztjzjS)a Computes the fidelity [1]_ between two quantum states The arguments provided to this function should be a square matrix or a Density object. If it is a square matrix, it is assumed to be diagonalizable. Parameters ========== state1, state2 : a density matrix or Matrix Examples ======== >>> from sympy import S, sqrt >>> from sympy.physics.quantum.dagger import Dagger >>> from sympy.physics.quantum.spin import JzKet >>> from sympy.physics.quantum.density import fidelity >>> from sympy.physics.quantum.represent import represent >>> >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> amp = 1/sqrt(2) >>> updown = (amp*up) + (amp*down) >>> >>> # represent turns Kets into matrices >>> up_dm = represent(up*Dagger(up)) >>> down_dm = represent(down*Dagger(down)) >>> updown_dm = represent(updown*Dagger(updown)) >>> >>> fidelity(up_dm, up_dm) 1 >>> fidelity(up_dm, down_dm) #orthogonal states 0 >>> fidelity(up_dm, updown_dm).evalf().round(3) 0.707 References ========== .. [1] https://en.wikipedia.org/wiki/Fidelity_of_quantum_states zBstate1 and state2 must be of type Density or Matrix received type=z for state1 and type=z for state2z]The dimensions of both args should be equal and the matrix obtained should be a square matrix) rrrrhrtypeshape is_squarerHalfrrB)state1state2 sqrt_state1s r$fidelityrysX#-VW"=Yv 6F",VW"=Yv 6F ff %Z-Gv,V 67 7||v||#(8(8EF F!&&.K {6!+-6 7 < < >>r%N)% itertoolsrsympy.core.addrsympy.core.containersrsympy.core.functionrsympy.core.mulrsympy.core.singletonr&sympy.functions.elementary.exponentialr sympy.matrices.denser rh sympy.printing.pretty.stringpictr sympy.physics.quantum.daggerrsympy.physics.quantum.operatorrsympy.physics.quantum.representr!sympy.physics.quantum.matrixutilsrrr#sympy.physics.quantum.tensorproductrrsympy.physics.quantum.tracerrr\ryrPr%r$rsQ'&"6=7/<5ZZR*DDN)DX9?r%