OtjJ%ddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZdd lmZdd lmZdd lmZdd lmZdd lmZmZmZddlmZGddeZdZ dZ!dS))product)Add)Tuple)expand)Mul)Slog)MutableDenseMatrix prettyForm)Dagger)HermitianOperator) represent) numpy_ndarrayscipy_sparse_matrixto_numpy)TrczeZdZdZefdZdZdZdZdZ 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]9}t|trt |dkst d:|S)Nz?Each argument should be of form [state,prob] or ( state, prob ))super _eval_args isinstancerlen ValueError)clsargsarg __class__s [/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/physics/quantum/density.pyrzDensity._eval_args)snww!!$'' 8 8CsE** 8s3xx1}} "78880= c2td|jDS)aReturn list of all states. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states() (|0>, |1>) cg|] }|d S)r.0r s r" z"Density.states..C333#s1v333r#rrselfs r"stateszDensity.states6 3333344r#c2td|jDS)a#Return list of all probabilities. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs() (0.5, 0.5) cg|] }|d S)r&r's r"r)z!Density.probs..Rr*r#r+r,s r"probsz Density.probsEr/r#c,|j|d}|S)atReturn specific state by index. Parameters ========== index : index of state to be returned Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states()[1] |1> rr)r-indexstates r" get_statezDensity.get_stateTs$ % # 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 r2r5)r-r6probs r"get_probzDensity.get_probis$y" r#c<fd|jD}t|S)aop will operate on each individual state. Parameters ========== op : Operator Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.apply_op(A) Density((A*|0>, 0.5),(A*|1>, 0.5)) c$g|] \}}|z|f Sr&r&)r(r7r:ops r"r)z$Density.apply_op..s&DDD%RXt$DDDr#)rr)r-r>new_argss ` r"apply_opzDensity.apply_op~s,(EDDD$)DDD!!r#c g}|jD]\}}|}t|trRt |jdD]:}||||d|dz;|||||zt|S)aExpand the density operator into an outer product format. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> from sympy.physics.quantum.operator import Operator >>> A = Operator('A') >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.doit() 0.5*|0><0| + 0.5*|1><1| r)repeatrr2)rrrrrappend_generate_outer_prod)r-hintstermsr7r:r s r"doitz Density.doits !Y K KMUDLLNNE5#&& K"5:a888IICLLd&?&?A@CA'H'H"HIIIII T$";";E5"I"IIJJJJE{r#c4|\}}|\}}t|dkst|dkrtdt|t t|z}t|t|z|zS)NrzHAtleast one-pair of Non-commutative instance required for outer product.)args_cncrrrr)r-arg1arg2c_part1nc_part1c_part2nc_part2r>s r"rDzDensity._generate_outer_prods MMOO MMOO MMQ  #h--1"4"4344 4(^F3>22 2G}S']*R//r#c @t|fi|SN)rrG)r-optionss r" _representzDensity._represents 00000r#cdS)Nz\rhor&r-printerrs r"_print_operator_name_latexz"Density._print_operator_name_latexswr#c tdS)Nuρr rUs r"_print_operator_name_prettyz#Density._print_operator_name_prettys6777r#c |dg}t||S)Nindices)getrrG)r-kwargsr[s r" _eval_tracezDensity._eval_traces9**Y++$))++w'',,...r#c t|S)zl Compute the entropy of a density matrix. Refer to density.entropy() method for examples. )entropyr,s r"r`zDensity.entropys t}}r#)__name__ __module__ __qualname____doc__ classmethodrr.r3r8r;r@rGrDrSrWrYr^r` __classcell__)r!s@r"rrs,    [  5 5 5 5 5 5**""".8000 111888///r#rc2t|trt|}t|trt |}t|t rM|}ttd|D St|trJddl }|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 c3:K|]}|t|zVdSrQr )r(es r" zentropy..s,551SVV8555555r#rNz4numpy.ndarray, scipy.sparse or SymPy matrix expected)rrrrrMatrix eigenvalskeysrsumrnumpylinalgeigvalsr r)densityrqnps r"r`r`s 4'7##%G$$'.//$7##'6"" D##%%**,,s55W555555666 G] + +D)##G,,wrvvg.//// BDD Dr#c t|trt|n|}t|trt|n|}t|trt|ts0t dt |dt |d|j|jkr|jrt d|tj z}t||z|ztj z 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 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) rrrrkrtypeshape is_squarerHalfrrG)state1state2 sqrt_state1s r"fidelityr|sX#-VW"="= IYv   6F",VW"="= IYv   6F ff % %7Z-G-G7jv,,,,V 677 7|v|##(8#EFF F!&.K {6!+-6 7 7 < < > >>r#N)" itertoolsrsympy.core.addrsympy.core.containersrsympy.core.functionrsympy.core.mulrsympy.core.singletonr&sympy.functions.elementary.exponentialr sympy.matrices.denser rk sympy.printing.pretty.stringpictr sympy.physics.quantum.daggerrsympy.physics.quantum.operatorrsympy.physics.quantum.representr!sympy.physics.quantum.matrixutilsrrrsympy.physics.quantum.tracerrr`r|r&r#r"rsm''''''&&&&&&""""""666666======777777//////<<<<<<555555ZZZZZZZZZZ******AAAAAAAAH)D)D)DX9?9?9?9?9?r#