ó ¡¼™\c@ sddlmZmZddlmZddlmZmZmZm Z m Z m Z m Z ddl mZddlmZddlmZddlmZddlmZdd lmZmZmZdd lmZmZd efd „ƒYZd „Zd„Z dS(iÿÿÿÿ(tprint_functiontdivision(tproduct(tTupletAddtMultMatrixtlogtexpandtRational(tTr(t prettyForm(tDagger(tHermitianOperator(t represent(t numpy_ndarraytscipy_sparse_matrixtto_numpy(t TensorProductttensor_product_simptDensitycB s‰eZdZed„ƒZd„Zd„Zd„Zd„Zd„Z d„Z d„Z d „Z d „Z d „Zd „Zd „ZRS(sDensity 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)) cC s]tt|ƒj|ƒ}x>|D]6}t|tƒoCt|ƒdkstdƒ‚qqW|S(Nis?Each argument should be of form [state,prob] or ( state, prob )(tsuperRt _eval_argst isinstanceRtlent ValueError(tclstargstarg((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyR's  cC s$tg|jD]}|d^q ŒS(sReturn list of all states. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states() (|0>, |1>) i(RR(tselfR((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pytstates5s cC s$tg|jD]}|d^q ŒS(s#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) i(RR(RR((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pytprobsDs cC s|j|d}|S(stReturn 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> i(R(Rtindextstate((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyt get_stateSscC s|j|d}|S(s¢Return 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 i(R(RR tprob((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pytget_probhscC s6g|jD]\}}|||f^q }t|ŒS(såop 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(RtopR!R#tnew_args((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pytapply_op}s,cK sªg}x—|jD]Œ\}}|jƒ}t|tƒrxbt|jddƒD]+}|j||j|d|dƒƒqMWq|j||j||ƒƒqWt|ŒS(s¥Expand 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| trepeatiii(RRRRRtappendt_generate_outer_prod(RthintsttermsR!R#R((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pytdoit”s  # cC sã|jƒ\}}|jƒ\}}t|ƒdksHt|ƒdkrWtdƒ‚nt|dtƒr¯t|ƒdkr¯t|ƒdkr¯t|dt|dƒƒ}nt|Œtt|Œƒ}t|Œt|Œ|S(NisHAtleast one-pair of Non-commutative instance required for outer product.i(targs_cncRRRRRR R(Rtarg1targ2tc_part1tnc_part1tc_part2tnc_part2R%((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyR*±s$!cK st|jƒ|S(N(RR-(Rtoptions((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyt _representÅscG s|jd|ŒS(Ns\rho(t_print(RtprinterR((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyt_print_operator_name_latexÈscG sttdƒƒS(Ns\N{GREEK SMALL LETTER RHO}(R tunichr(RR8R((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyt_print_operator_name_prettyËscK s+|jdgƒ}t|jƒ|ƒjƒS(Ntindices(tgetR R-(RtkwargsR<((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyt _eval_traceÎscC s t|ƒS(sl Compute the entropy of a density matrix. Refer to density.entropy() method for examples. (tentropy(R((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyR@Òs(t__name__t __module__t__doc__t classmethodRRRR"R$R'R-R*R6R9R;R?R@(((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyRs           cC sÐt|tƒrt|ƒ}nt|tƒr<t|ƒ}nt|tƒrx|jƒjƒ}tt d„|Dƒƒ ƒSt|t ƒrÀddl }|j j |ƒ}|j ||j|ƒƒ Stdƒ‚dS(sTCompute 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.represent import represent >>> from sympy.physics.quantum.matrixutils import scipy_sparse_matrix >>> from sympy.physics.quantum.spin import JzKet, Jz >>> from sympy import S, log >>> 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 cs s|]}|t|ƒVqdS(N(R(t.0te((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pys þsiÿÿÿÿNs4numpy.ndarray, scipy.sparse or sympy matrix expected(RRRRRRt eigenvalstkeysRtsumRtnumpytlinalgteigvalsRR(tdensityRLtnp((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyR@Ús cC sét|tƒrt|ƒn|}t|tƒr<t|ƒn|}t|tƒ sbt|tƒ r‡tdt|ƒt|ƒfƒ‚n|j|jkr±|jr±tdƒ‚n|tddƒ}t |||tddƒƒj ƒS(s» 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 Density, 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 sfstate1 and state2 must be of type Density or Matrix received type=%s for state1 and type=%s for state2s]The dimensions of both args should be equal and the matrix obtained should be a square matrixii( RRRRRttypetshapet is_squareR R R-(tstate1tstate2t sqrt_state1((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pytfidelitys,!!N(!t __future__RRt itertoolsRtsympyRRRRRRR tsympy.core.traceR t sympy.printing.pretty.stringpictR tsympy.physics.quantum.daggerR tsympy.physics.quantum.operatorR tsympy.physics.quantum.representRt!sympy.physics.quantum.matrixutilsRRRt#sympy.physics.quantum.tensorproductRRRR@RU(((s<lib/python2.7/site-packages/sympy/physics/quantum/density.pyts4Ë .