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

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

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

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 3. 0 with prob. ½ 1 with prob. ½ 4. 0 + 1 with prob. ½ 0 − 1 with prob. ½ 6. 0 with prob. ¼ 1 with prob. ¼ 0 + 1 with prob. ¼ 0 − 1 with prob. ¼

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

Recap: density matrices (III) Quantum operations in terms of density matrices: Applying U to  yields U U† Measuring state  with respect to the basis 1, 2,..., d, yields: k th outcome with probability k k —and causes the state 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

Characterizing 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

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

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: eigenvectors: is not even diagonalizable is diagonalizable, but not unitarily

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 are both unitary and Hermitian? Answer: reflections (k  {+1,1}, for all k)

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)

Projectors and density matrices Projector: Hermitian and M 2 = M, which 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 )

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

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

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

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

Bloch sphere for qubits (III) + 0 1 – +i –i +i = 0 + i1 –i = 0 – i1 – = 0 – 1 + = 0 +1 Note that orthogonal corresponds to antipodal here Pure states are on the surface, and mixed states are inside (being weighted averages of pure states)

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

General quantum operations (I) General quantum operations (a.k.a. “completely positive trace preserving maps”, “admissible operations” ): Let A1, A2 , …, Am be matrices satisfying Then the mapping is a general quantum op Example 1 (unitary op): applying U to  yields U U†

General quantum operations (II) Example 2 (decoherence): let A0 = 00 and A1 = 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” After looking at the outcome,  becomes 00 with prob. ||2 11 with prob. ||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 Define A0 = 2/300 A1= 2/311 A2= 2/322 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

General quantum operations (IV) Example 4 (discarding the second of two qubits): Let A0 = I0 and A1 = 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 

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