Richard Cleve DC 3524 cleve@cs.uwaterloo Course web site at:

1. Introduction to Quantum Information ProcessingCS 467 / CS 667Phys 467 / Phys 767C&O 481 / C&O 681 Richard Cleve DC 3524 cleve@cs.uwaterloo.ca Course web site at: http://www.cs.uwaterloo.ca/~cleve/courses/cs467 Lecture 11 (2005)

2. Contents • Continuation of density matrix formalism • Taxonomy of various normal matrices • Bloch sphere for qubits • General quantum operations

3. Continuation of density matrix formalism • Taxonomy of various normal matrices • Bloch sphere for qubits • General quantum operations

4. 6. 0 with prob. ¼ 1 with prob. ¼ 0 +1 with prob. ¼ 0 − 1 with prob. ¼ 3. 0 with prob. ½ 1 with prob. ½ 4. 0 +1 with prob. ½ 0 −1 with prob. ½ Recap: density matrices (I) The density matrix of the mixed state ((ψ1, p1), (ψ2, p2), …,(ψd, pd)) is: Examples (from previous lecture): 1. & 2. 0 +1 and −0 −1 both have

5. 5. 0 with prob. ½ 0 +1 with prob. ½ Recap: density matrices (II) Examples (continued): has: ...?(later) 7. The first qubit of 01 −10

6. Recap: density matrices (III) Quantum operations in terms of density matrices: • Applying U to yieldsUU† • Measuring state  with respect to the basis1, 2,..., d, • yields: kthoutcome with probability kk • —and causes thestate to collapse tokk 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. Characterizing density matrices • Three properties of: • Tr= 1 (TrM = 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 • Taxonomy of various normal matrices • Bloch sphere for qubits • General quantum operations

9. eigenvectors: Normal matrices Definition: A matrix M is normal if M†M = MM† Theorem:M is normal iff there exists a unitary U such that M = U†DU, where D is diagonal (i.e. unitarily diagonalizable) Examples of abnormal matrices: is not even diagonalizable is diagonalizable, but not unitarily

10. Unitary and Hermitian matrices Normal: with respect to some orthonormal basis Unitary:M†M = I which implies |k |2= 1, for all k Hermitian:M = M† which implies k  R, for all k Question: which matrices areboth unitary and Hermitian? Answer: 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 M2 = M, which implies that M is positive semidefinite and k  {0,1}, for all k Density matrix: positive semidefinite and TrM=1, so Question: which matrices areboth projectors and density matrices? Answer: rank-one projectors (k = 1 if k = k0 and k = 0 ifk k0)

13. normal unitary Hermitian reflection positive semidefinite projector density matrix rank one projector Taxonomy of normal matrices

14. Continuation of density matrix formalism • Taxonomy of various normal matrices • Bloch sphere for qubits • General quantum operations

15. Bloch sphere for qubits (I) Consider the set of all 2x2 density matrices  They have a nice representation in terms of the Pauli matrices: Note that these matrices—combined with I—form a basis for the vector space of all 2x2 matrices We will express density matrices  in this basis Note that the coefficient of I is ½, since X, Y, Z have trace zero

16. Bloch sphere for qubits (II) We will express First consider the case of pure states , where, without loss of generality,  =cos()0 +e2isin()1 (,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. 0 + = 0 +1 –i – = 0 – 1 +i = 0 +i1 + – –i = 0 –i1 +i 1 Bloch sphere for qubits (III) Note that orthogonal corresponds to antipodal here Pure states are on the surface, and mixed states are inside (being weighted averages of pure states)

18. Continuation of density matrix formalism • Taxonomy of various normal matrices • Bloch sphere for qubits • General quantum operations

19. General quantum operations(a.k.a.“completely positive trace preserving maps”,“admissible operations”): LetA1, A2 , …, Am be matrices satisfying Then the mapping is a general quantum op General quantum operations (I) Example 1 (unitary op): applying U to yieldsUU†

20. After looking at the outcome,  becomes 00 with prob. ||2 11 with prob. ||2 General quantum operations (II) Example 2 (decoherence):letA0 =00andA1 =11 This quantum op maps  to 0000 + 1111 For ψ =0 +1, Corresponds to measuring  “without looking at the outcome”

21. DefineA0 =2/300 A1=2/311 A2=2/322 General quantum operations (III) Example 3 (trine state“measurent”): Let 0 = 0, 1 =1/20 + 3/21, 2 =1/20 3/21 Then The probability that state k results in “outcome” Ak is 2/3, and this can be adapted to actually yield the value of kwith this success probability

22. General quantum operations (IV) Example 4 (discarding the second of two qubits): LetA0 =I0andA1 =I1 State becomes  State becomes Note 1: it’s the same density matrix as for ((0, ½), (1, ½)) Note 2: the operation is the partial trace Tr2 

23. THE END