1、1,Introduction to Quantum Information Processing CS 467 / CS 667 Phys 467 / Phys 767 C&O 481 / C&O 681,Richard Cleve DC 3524 clevecs.uwaterloo.caCourse web site at: http:/www.cs.uwaterloo.ca/cleve/courses/cs467,Lecture 11 (2005),2,Contents,Continuation of density matrix formalismTaxonomy of various
2、normal matricesBloch sphere for qubitsGeneral quantum operations,3,Continuation of density matrix formalismTaxonomy of various normal matricesBloch sphere for qubitsGeneral quantum operations,4,Recap: density matrices (I),The density matrix of the mixed state (1, p1), (2, p2), ,(d, pd) is:,1. & 2. 0
3、 + 1 and 0 1 both have,Examples (from previous lecture):,5,Recap: density matrices (II),7. The first qubit of 01 10,Examples (continued):,has:,.? (later),6,Recap: density matrices (III),Applying U to yields U U,Measuring state with respect to the basis 1, 2,., d,yields: k th outcome with probability
4、 k kand causes the state to collapse to k k,Quantum operations in terms of density matrices:,Since these are expressible in terms of density matrices alone (independent of any specific probabilistic mixtures), states with identical density matrices are operationally indistinguishable,7,Characterizin
5、g density matrices,Three properties of :Tr = 1 (Tr M = M11 + M22 + . + Mdd ) = (i.e. is Hermitian) 0, for all states ,Moreover, for any matrix satisfying the above properties, there exists a probabilistic mixture whose density matrix is ,Exercise: show this,8,Continuation of density matrix formalism
6、Taxonomy of various normal matricesBloch sphere for qubitsGeneral quantum operations,9,Normal matrices,Definition: A matrix M is normal if MM = MM,Theorem: M is normal iff there exists a unitary U such that M = UDU, where D is diagonal (i.e. unitarily diagonalizable),Examples of abnormal matrices:,i
7、s not even diagonalizable,is diagonalizable, but not unitarily,10,Unitary and Hermitian matrices,with respect to some orthonormal basis,Normal:,Unitary: MM = I which implies |k |2 = 1, for all k,Hermitian: M = M which implies k R, for all k,Question: which matrices are both unitary and Hermitian?,An
8、swer: reflections (k +1,1, for all k),11,Positive semidefinite,Positive semidefinite: Hermitian and k 0, for all k,Theorem: M is positive semidefinite iff M is Hermitian and, for all , M 0,(Positive definite: k 0, for all k),12,Projectors and density matrices,Projector: Hermitian and M 2 = M, which
9、implies that M is positive semidefinite and k 0,1, for all k,Density matrix: positive semidefinite and Tr M = 1, so,Question: which matrices are both projectors and density matrices?,Answer: rank-one projectors (k = 1 if k = k0 and k = 0 if k k0 ),13,Taxonomy of normal matrices,14,Continuation of de
10、nsity matrix formalismTaxonomy of various normal matricesBloch sphere for qubitsGeneral quantum operations,15,Bloch sphere for qubits (I),Consider the set of all 2x2 density matrices ,Note that the coefficient of I is , since X, Y, Z have trace zero,They have a nice representation in terms of the Pa
11、uli matrices:,Note that these matricescombined with Iform a basis for the vector space of all 2x2 matrices,We will express density matrices in this basis,16,Bloch sphere for qubits (II),We will express,First consider the case of pure states , where, without loss of generality, = cos()0 + e2isin()1 (
12、, R),Therefore cz = cos(2), cx = cos(2)sin(2), cy = sin(2)sin(2),These are polar coordinates of a unit vector (cx , cy , cz) R3,17,Bloch sphere for qubits (III),Pure states are on the surface, and mixed states are inside (being weighted averages of pure states),Note that orthogonal corresponds to an
13、tipodal here,18,Continuation of density matrix formalismTaxonomy of various normal matricesBloch sphere for qubitsGeneral quantum operations,19,General quantum operations (I),Example 1 (unitary op): applying U to yields U U,General quantum operations (a.k.a. “completely positive trace preserving map
14、s”, “admissible operations” ): Let A1, A2 , , Am be matrices satisfying,20,General quantum operations (II),Example 2 (decoherence): let A0 = 00 and A1 = 11,This quantum op maps to 0000 + 1111,Corresponds to measuring “without looking at the outcome”,For = 0 + 1,21,General quantum operations (III),Ex
15、ample 3 (trine state “measurent”):,Let 0 = 0, 1 = 1/20 + 3/21, 2 = 1/20 3/21,Then,The probability that state k results in “outcome” Ak is 2/3, and this can be adapted to actually yield the value of k with this success probability,Define A0 = 2/300A1= 2/311 A2= 2/322,22,General quantum operations (IV),Example 4 (discarding the second of two qubits): Let A0 = I0 and A1 = I1,State becomes ,State becomes,Note 1: its the same density matrix as for (0, ), (1, ),Note 2: the operation is the partial trace Tr2 ,23,THE END,
