^j&ddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZdd lmZdd lmZdd lmZdd lmZdd lmZmZmZddlmZmZddl m!Z!GddeZ"dZ#dZ$dS))product)Add)Tuple)expand)Mul)Slog)MutableDenseMatrix prettyForm)Dagger)HermitianOperator) represent) numpy_ndarrayscipy_sparse_matrixto_numpy) TensorProducttensor_product_simp)TrczeZdZdZefdZdZdZdZdZ dZ dZ d Z d Z d Zd Zd ZdZxZS)Densitya Density operator for representing mixed states. TODO: Density operator support for Qubits Parameters ========== values : tuples/lists Each tuple/list should be of form (state, prob) or [state,prob] Examples ======== Create a density operator with 2 states represented by Kets. >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d Density((|0>, 0.5),(|1>, 0.5)) ct|}|D]9}t|trt |dkst d:|S)Nz?Each argument should be of form [state,prob] or ( state, prob ))super _eval_args isinstancerlen ValueError)clsargsarg __class__s NC:\Projects\Historical\venv\Lib\site-packages\sympy/physics/quantum/density.pyrzDensity._eval_args*snww!!$'' 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..D333#s1v333r%rr!selfs r$stateszDensity.states7 3333344r%c2td|jDS)a#Return list of all probabilities. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.probs() (0.5, 0.5) cg|] }|d S)r(r)s r$r+z!Density.probs..Sr,r%r-r.s r$probsz Density.probsFr1r%c,|j|d}|S)atReturn specific state by index. Parameters ========== index : index of state to be returned Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states()[1] |1> rr!)r/indexstates r$ 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 r4r7)r/r8probs r$get_probzDensity.get_probjs$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$g|] \}}|z|f Sr(r()r*r9r<ops r$r+z$Density.apply_op..s&DDD%RXt$DDDr%)r!r)r/r@new_argss ` r$apply_opzDensity.apply_ops,(EDDD$)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)repeatrr4)r!rrrrappend_generate_outer_prod)r/hintstermsr9r<r"s r$doitz Density.doits !Y K KMUDLLNNE5#&& K"5:a888IICLLd&?&?A@CA'H'H"HIIIII T$";";E5"I"IIJJJJE{r%c|\}}|\}}t|dkst|dkrtdt|dtrRt|dkr?t|dkr,t |dt |dz}n t|t t|z}t|t|z|zS)NrzHAtleast one-pair of Non-commutative instance required for outer product.r4)args_cncrrrrrrr)r/arg1arg2c_part1nc_part1c_part2nc_part2r@s r$rFzDensity._generate_outer_prods MMOO MMOO MMQ  #h--1"4"4344 4 x{M 2 2 7s8}}7I7IMMQ&&$Xa[ 1D1D%DEEBBhsH~ 6 66BG}S']*R//r%c @t|fi|SN)rrI)r/optionss r$ _representzDensity._represents 00000r%cdS)Nz\rhor(r/printerr!s r$_print_operator_name_latexz"Density._print_operator_name_latexswr%c tdS)Nuρr rWs r$_print_operator_name_prettyz#Density._print_operator_name_prettys6777r%c |dg}t||S)Nindices)getrrI)r/kwargsr]s r$ _eval_tracezDensity._eval_traces9**Y++$))++w'',,...r%c t|S)zl Compute the entropy of a density matrix. Refer to density.entropy() method for examples. )entropyr.s r$rbzDensity.entropys t}}r%)__name__ __module__ __qualname____doc__ classmethodrr0r5r:r=rBrIrFrUrYr[r`rb __classcell__)r#s@r$rrs,    [  5 5 5 5 5 5**""".8000&111888///r%rc2t|trt|}t|trt |}t|t rM|}ttd|D St|trJddl }|j |}| |||z Std)aCompute the entropy of a matrix/density object. This computes -Tr(density*ln(density)) using the eigenvalue decomposition of density, which is given as either a Density instance or a matrix (numpy.ndarray, sympy.Matrix or scipy.sparse). Parameters ========== density : density matrix of type Density, SymPy matrix, scipy.sparse or numpy.ndarray Examples ======== >>> from sympy.physics.quantum.density import Density, entropy >>> from sympy.physics.quantum.spin import JzKet >>> from sympy import S >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> d = Density((up,S(1)/2),(down,S(1)/2)) >>> entropy(d) log(2)/2 c3:K|]}|t|zVdSrSr )r*es r$ zentropy..s,551SVV8555555r%rNz4numpy.ndarray, scipy.sparse or SymPy matrix expected)rrrrrMatrix eigenvalskeysrsumrnumpylinalgeigvalsr r)densityrsnps r$rbrbs 4'7##%G$$'.//$7##'6"" D##%%**,,s55W555555666 G] + +D)##G,,wrvvg.//// BDD Dr%c t|trt|n|}t|trt|n|}t|trt|ts0t dt |dt |d|j|jkr|jrt d|tj z}t||z|ztj z S)a Computes the fidelity [1]_ between two quantum states The arguments provided to this function should be a square matrix or a Density object. If it is a square matrix, it is assumed to be diagonalizable. Parameters ========== state1, state2 : a density matrix or Matrix Examples ======== >>> from sympy import S, sqrt >>> from sympy.physics.quantum.dagger import Dagger >>> from sympy.physics.quantum.spin import JzKet >>> from sympy.physics.quantum.density import fidelity >>> from sympy.physics.quantum.represent import represent >>> >>> up = JzKet(S(1)/2,S(1)/2) >>> down = JzKet(S(1)/2,-S(1)/2) >>> amp = 1/sqrt(2) >>> updown = (amp*up) + (amp*down) >>> >>> # represent turns Kets into matrices >>> up_dm = represent(up*Dagger(up)) >>> down_dm = represent(down*Dagger(down)) >>> updown_dm = represent(updown*Dagger(updown)) >>> >>> fidelity(up_dm, up_dm) 1 >>> fidelity(up_dm, down_dm) #orthogonal states 0 >>> fidelity(up_dm, updown_dm).evalf().round(3) 0.707 References ========== .. [1] https://en.wikipedia.org/wiki/Fidelity_of_quantum_states zBstate1 and state2 must be of type Density or Matrix received type=z for state1 and type=z for state2z]The dimensions of both args should be equal and the matrix obtained should be a square matrix) rrrrmrtypeshape is_squarerHalfrrI)state1state2 sqrt_state1s r$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 rm sympy.printing.pretty.stringpictr sympy.physics.quantum.daggerrsympy.physics.quantum.operatorrsympy.physics.quantum.representr!sympy.physics.quantum.matrixutilsrrr#sympy.physics.quantum.tensorproductrrsympy.physics.quantum.tracerrrbr~r(r%r$rs''''''&&&&&&""""""666666======777777//////<<<<<<555555ZZZZZZZZZZRRRRRRRR******DDDDDDDDN)D)D)DX9?9?9?9?9?r%