o jg&@sddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z ddlmZdd lmZdd lmZdd lmZdd lmZdd lmZmZmZddlmZmZddl m!Z!GdddeZ"ddZ#ddZ$dS))product)Add)Tuple)expand)Mul)Slog)MutableDenseMatrix prettyForm)Dagger)HermitianOperator) represent) numpy_ndarrayscipy_sparse_matrixto_numpy) TensorProducttensor_product_simp)TrcseZdZdZefddZddZddZdd Zd d Z d d Z ddZ ddZ ddZ ddZddZddZddZZS)Densitya Density operator for representing mixed states. TODO: Density operator support for Qubits Parameters ========== values : tuples/lists Each tuple/list should be of form (state, prob) or [state,prob] Examples ======== Create a density operator with 2 states represented by Kets. >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d Density((|0>, 0.5),(|1>, 0.5)) cs8t|}|D]}t|trt|dkstdq|S)Nz?Each argument should be of form [state,prob] or ( state, prob ))super _eval_args isinstancerlen ValueError)clsargsarg __class__U/var/www/html/zoom/venv/lib/python3.10/site-packages/sympy/physics/quantum/density.pyr*s zDensity._eval_argscCtdd|jDS)aReturn list of all states. Examples ======== >>> from sympy.physics.quantum.state import Ket >>> from sympy.physics.quantum.density import Density >>> d = Density([Ket(0), 0.5], [Ket(1),0.5]) >>> d.states() (|0>, |1>) cSg|]}|dqS)rr".0rr"r"r# Dz"Density.states..rrselfr"r"r#states7 zDensity.statescCr$)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) cSr%)r"r&r"r"r#r(Sr)z!Density.probs..r*r+r"r"r#probsFr.z Density.probscC|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,indexstater"r"r# get_stateUzDensity.get_statecCr1)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/r2)r,r3probr"r"r#get_probjr6zDensity.get_probcsfdd|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)) csg|] \}}||fqSr"r")r'r4r7opr"r#r(sz$Density.apply_op..)rr)r,r:new_argsr"r9r#apply_opszDensity.apply_opc Ksxg}|jD]2\}}|}t|tr,t|jddD]}||||d|dqq|||||qt|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/)rrrrrappend_generate_outer_prod)r,hintstermsr4r7rr"r"r#doits  z Density.doitcCs|\}}|\}}t|dkst|dkrtdt|dtr>> 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 css|] }|t|VqdSrJr)r'er"r"r# szentropy..rNz4numpy.ndarray, scipy.sparse or SymPy matrix expected)rrrrrMatrix eigenvalskeysrsumrnumpylinalgeigvalsr r)densityrdnpr"r"r#rUs      rUcCst|tr t|n|}t|trt|n|}t|tr t|ts,tdt|t|f|j|jkr9|jr9td|tj }t |||tj  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 zfstate1 and state2 must be of type Density or Matrix received type=%s for state1 and type=%s for state2z]The dimensions of both args should be equal and the matrix obtained should be a square matrix) rrrr^rtypeshape is_squarerHalfrrB)state1state2 sqrt_state1r"r"r#fidelitys, rnN)% itertoolsrsympy.core.addrsympy.core.containersrsympy.core.functionrsympy.core.mulrsympy.core.singletonr&sympy.functions.elementary.exponentialr sympy.matrices.denser r^ sympy.printing.pretty.stringpictr sympy.physics.quantum.daggerr sympy.physics.quantum.operatorrsympy.physics.quantum.representr!sympy.physics.quantum.matrixutilsrrr#sympy.physics.quantum.tensorproductrrsympy.physics.quantum.tracerrrUrnr"r"r"r#s&            H ,