ó ¡¼™\c@sddZddlmZmZmZmZmZmZddlm Z m Z m Z ddlm Z ddl mZddlmZddd d d d d dgZde fd„ƒYZdefd„ƒYZdefd„ƒYZd efd„ƒYZd efd„ƒYZd efd„ƒYZd e fd„ƒYZd e fd„ƒYZd„Zd„ZdS(sPauli operators and statesiÿÿÿÿ(tItMultAddtPowtexptInteger(tOperatortKettBra(t ComplexSpace(tMatrix(tKroneckerDeltatSigmaXtSigmaYtSigmaZt SigmaMinust SigmaPlust SigmaZKett SigmaZBratqsimplify_paulit SigmaOpBasecBsMeZdZed„ƒZed„ƒZed„ƒZd„Zd„Z RS(s Pauli sigma operator, base classcCs |jdS(Ni(targs(tself((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pytnamescCst|jdƒtk S(Ni(tboolRtFalse(R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pytuse_namescCstfS(N(R(R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt default_argsscOstj|||ŽS(N(Rt__new__(tclsRthints((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRscKs tdƒS(Ni(R(RtotherR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_commutator_BosonOp!s( t__name__t __module__t__doc__tpropertyRRt classmethodRRR (((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRs  cBsqeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z RS( s¦Pauli sigma x operator Parameters ========== name : str An optional string that labels the operator. Pauli operators with different names commute. Examples ======== >>> from sympy.physics.quantum import represent >>> from sympy.physics.quantum.pauli import SigmaX >>> sx = SigmaX() >>> sx SigmaX() >>> represent(sx) Matrix([ [0, 1], [1, 0]]) cOstj|||ŽS(N(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR=scKs5|j|jkrtdƒSdtt|jƒSdS(Nii(RRRR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_commutator_SigmaY@s cKs5|j|jkrtdƒSdtt|jƒSdS(Niiþÿÿÿ(RRRR (RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_commutator_SigmaZFs cKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR LscKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_anticommutator_SigmaYOscKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_anticommutator_SigmaZRscCs|S(N((R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt _eval_adjointUscGs"|jrdt|jƒSdSdS(Ns{\sigma_x^{(%s)}}s {\sigma_x}(RtstrR(RtprinterR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_print_contents_latexXs cGsdS(NsSigmaX()((RR,R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_print_contents^scCs6|jr2|jr2t|jƒjt|ƒdƒSdS(Ni(t is_Integert is_positiveR Rt__pow__tint(Rte((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt _eval_powerascKsR|jddƒ}|dkr:tddgddggƒStd|dƒ‚dS(NtformattsympyiisRepresentation in format s not implemented.(tgetR tNotImplementedError(RtoptionsR5((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_represent_default_basises  (R!R"R#RR&R'R R(R)R*R-R.R4R:(((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR %s          cBsheZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z RS( s¨Pauli sigma y operator Parameters ========== name : str An optional string that labels the operator. Pauli operators with different names commute. Examples ======== >>> from sympy.physics.quantum import represent >>> from sympy.physics.quantum.pauli import SigmaY >>> sy = SigmaY() >>> sy SigmaY() >>> represent(sy) Matrix([ [0, -I], [I, 0]]) cOstj||ŒS(N(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR†scKs5|j|jkrtdƒSdtt|jƒSdS(Nii(RRRR (RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR'‰s cKs5|j|jkrtdƒSdtt|jƒSdS(Niiþÿÿÿ(RRRR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_commutator_SigmaXs cKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_anticommutator_SigmaX•scKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR)˜scCs|S(N((R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR*›scGs"|jrdt|jƒSdSdS(Ns{\sigma_y^{(%s)}}s {\sigma_y}(RR+R(RR,R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR-žs cGsdS(NsSigmaY()((RR,R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR.¤scCs6|jr2|jr2t|jƒjt|ƒdƒSdS(Ni(R/R0R RR1R2(RR3((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR4§scKsS|jddƒ}|dkr;tdt gtdggƒStd|dƒ‚dS(NR5R6isRepresentation in format s not implemented.(R7R RR8(RR9R5((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR:«s  ( R!R"R#RR'R;R<R)R*R-R.R4R:(((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR ns         cBsheZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z RS( s­Pauli sigma z operator Parameters ========== name : str An optional string that labels the operator. Pauli operators with different names commute. Examples ======== >>> from sympy.physics.quantum import represent >>> from sympy.physics.quantum.pauli import SigmaZ >>> sz = SigmaZ() >>> sz ** 3 SigmaZ() >>> represent(sz) Matrix([ [1, 0], [0, -1]]) cOstj||ŒS(N(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRÌscKs5|j|jkrtdƒSdtt|jƒSdS(Nii(RRRR (RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR;Ïs cKs5|j|jkrtdƒSdtt|jƒSdS(Niiþÿÿÿ(RRRR (RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR&Õs cKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR<ÛscKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR(ÞscCs|S(N((R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR*áscGs"|jrdt|jƒSdSdS(Ns{\sigma_z^{(%s)}}s {\sigma_z}(RR+R(RR,R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR-äs cGsdS(NsSigmaZ()((RR,R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR.êscCs6|jr2|jr2t|jƒjt|ƒdƒSdS(Ni(R/R0RRR1R2(RR3((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR4íscKsR|jddƒ}|dkr:tddgddggƒStd|dƒ‚dS(NR5R6iiiÿÿÿÿsRepresentation in format s not implemented.(R7R R8(RR9R5((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR:ñs  ( R!R"R#RR;R&R<R(R*R-R.R4R:(((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR´s         cBsŒeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z d „Zd „Zd„ZRS(sáPauli sigma minus operator Parameters ========== name : str An optional string that labels the operator. Pauli operators with different names commute. Examples ======== >>> from sympy.physics.quantum import represent, Dagger >>> from sympy.physics.quantum.pauli import SigmaMinus >>> sm = SigmaMinus() >>> sm SigmaMinus() >>> Dagger(sm) SigmaPlus() >>> represent(sm) Matrix([ [0, 0], [1, 0]]) cOstj||ŒS(N(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRscKs.|j|jkrtdƒSt|jƒ SdS(Ni(RRR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR;s cKs1|j|jkrtdƒStt|jƒSdS(Ni(RRRR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR&s cKsd|S(Ni((RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR'#scKs t|jƒS(N(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_commutator_SigmaMinus&scKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR))scKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR<,scKst tdƒS(Ni(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR(/scKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_anticommutator_SigmaPlus2scCs t|jƒS(N(RR(R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR*5scCs |jr|jrtdƒSdS(Ni(R/R0R(RR3((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR48scGs"|jrdt|jƒSdSdS(Ns{\sigma_-^{(%s)}}s {\sigma_-}(RR+R(RR,R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR-<s cGsdS(Ns SigmaMinus()((RR,R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR.BscKsR|jddƒ}|dkr:tddgddggƒStd|dƒ‚dS(NR5R6iisRepresentation in format s not implemented.(R7R R8(RR9R5((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR:Es  (R!R"R#RR;R&R'R=R)R<R(R>R*R4R-R.R:(((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRús             cBs•eZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z d „Zd „Zd„Zd„ZRS(sÞPauli sigma plus operator Parameters ========== name : str An optional string that labels the operator. Pauli operators with different names commute. Examples ======== >>> from sympy.physics.quantum import represent, Dagger >>> from sympy.physics.quantum.pauli import SigmaPlus >>> sp = SigmaPlus() >>> sp SigmaPlus() >>> Dagger(sp) SigmaMinus() >>> represent(sp) Matrix([ [0, 1], [0, 0]]) cOstj||ŒS(N(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRhscKs-|j|jkrtdƒSt|jƒSdS(Ni(RRR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR;ks cKs1|j|jkrtdƒStt|jƒSdS(Ni(RRRR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR&qs cKs(|j|jkrtdƒSd|SdS(Niiþÿÿÿ(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR'ws cKs t|jƒS(N(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR=}scKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR)€scKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR<ƒscKsttdƒS(Ni(RR(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR(†scKs tdƒS(Ni(R(RRR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_anticommutator_SigmaMinus‰scCs t|jƒS(N(RR(R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR*ŒscCs||S(N((RR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt _eval_mulscCs |jr|jrtdƒSdS(Ni(R/R0R(RR3((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR4’scGs"|jrdt|jƒSdSdS(Ns{\sigma_+^{(%s)}}s {\sigma_+}(RR+R(RR,R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR-–s cGsdS(Ns SigmaPlus()((RR,R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR.œscKsR|jddƒ}|dkr:tddgddggƒStd|dƒ‚dS(NR5R6iisRepresentation in format s not implemented.(R7R R8(RR9R5((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR:Ÿs  (R!R"R#RR;R&R'R=R)R<R(R?R*R@R4R-R.R:(((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRNs               cBsƒeZdZd„Zed„ƒZed„ƒZed„ƒZd„Z d„Z d„Z d„Z d „Z d „Zd „ZRS( s‚Ket for a two-level system quantum system. Parameters ========== n : Number The state number (0 or 1). cCs+|dkrtdƒ‚ntj||ƒS(Niisn must be 0 or 1(ii(t ValueErrorRR(Rtn((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR³s cCs |jdS(Ni(tlabel(R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRB¸scCstS(N(R(R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt dual_class¼scCs tdƒS(Ni(R (RRC((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_hilbert_spaceÀscKst|j|jƒS(N(R RB(RtbraR((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_eval_innerproduct_SigmaZBraÄscKs%|jdkr|Stdƒ|SdS(Niiÿÿÿÿ(RBR(RtopR9((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_apply_operator_SigmaZÇscKs#|jdkrtdƒStdƒS(Nii(RBR(RRHR9((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_apply_operator_SigmaXÍscKs,|jdkrttdƒSt tdƒS(Nii(RBRR(RRHR9((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_apply_operator_SigmaYÐscKs'|jdkrtdƒStdƒSdS(Nii(RBRR(RRHR9((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_apply_operator_SigmaMinusÓs cKs'|jdkrtdƒStdƒSdS(Ni(RBRR(RRHR9((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_apply_operator_SigmaPlusÙs cKsq|jddƒ}|dkrY|jdkrCtdgdggƒStdgdggƒStd|dƒ‚dS(NR5R6iisRepresentation in format s not implemented.(R7RBR R8(RR9R5((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR:ßs  ;(R!R"R#RR$RBR%RDRERGRIRJRKRLRMR:(((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyR¨s        cBs5eZdZd„Zed„ƒZed„ƒZRS(s{Bra for a two-level quantum system. Parameters ========== n : Number The state number (0 or 1). cCs+|dkrtdƒ‚ntj||ƒS(Niisn must be 0 or 1(ii(RARR(RRB((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRós cCs |jdS(Ni(RC(R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRBøscCstS(N(R(R((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRDüs(R!R"R#RR$RBR%RD(((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRès  cCsFt|tƒot|tƒs+t||ƒS|j|jkrl|j|jkr\t||ƒSt||ƒSnÖt|tƒr4t|tƒr”tdƒSt|tƒr´tt|jƒSt|tƒrÕt t|jƒSt|t ƒrtdƒdt|jƒdSt|t ƒrBtdƒdt|jƒdSnt|tƒrýt|tƒrdt t|jƒSt|tƒr}tdƒSt|tƒrtt|jƒSt|t ƒrÌt tdƒt|jƒdSt|t ƒrBttdƒt|jƒdSnEt|tƒr¢t|tƒr,tt|jƒSt|tƒrMt t|jƒSt|tƒrftdƒSt|t ƒrƒt |jƒ St|t ƒrBt |jƒSn t|t ƒrpt|tƒrÛtdƒt|jƒdSt|tƒr t tdƒt|jƒdSt|tƒr&t |jƒSt|t ƒr?tdƒSt|t ƒrBtdƒdt|jƒdSnÒt|t ƒr:t|tƒr©tdƒt|jƒdSt|tƒr×ttdƒt|jƒdSt|tƒrôt |jƒ St|t ƒrtdƒt|jƒdSt|t ƒrBtdƒSn||SdS(sO Internal helper function for simplifying products of Pauli operators. iiiN( t isinstanceRRRR RR RRRR(tatb((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyt_qsimplify_pauli_products|   "  "    " c CsGt|tƒr|St|tttfƒrNt|ƒ}|d„|jDƒŒSt|tƒrC|jƒ\}}g}x·|r.|j dƒ}xŽt |ƒrt|t ƒrt|dt ƒr|j |dj kr|j dƒ}t ||ƒ}|jƒ\}} t| Œ}||}qW|j|ƒqxWt|Œt|ŒS|S(sõ Simplify an expression that includes products of pauli operators. Parameters ========== e : expression An expression that contains products of Pauli operators that is to be simplified. Examples ======== >>> from sympy.physics.quantum.pauli import SigmaX, SigmaY >>> from sympy.physics.quantum.pauli import qsimplify_pauli >>> sx, sy = SigmaX(), SigmaY() >>> sx * sy SigmaX()*SigmaY() >>> qsimplify_pauli(sx * sy) I*SigmaZ() css|]}t|ƒVqdS(N(R(t.0targ((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pys …si(RNRRRRttypeRRtargs_cnctpoptlenRRRQtappend( R3tttctnctnc_stcurrtxtytc1tnc1((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyRjs,   N(R#R6RRRRRRtsympy.physics.quantumRRRR tsympy.matricesR t(sympy.functions.special.tensor_functionsR t__all__RR R RRRRRRQR(((s:lib/python2.7/site-packages/sympy/physics/quantum/pauli.pyts". IFFTZ@ i