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1 Introduction to Quantum Information Processing CS 467 / CS 667 Phys 467 / Phys 767 C&O 481 / C&O 681 Richard Cleve DC 3524 Course web site at: Lecture 11 (2005)

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

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

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4 Recap: density matrices (I) The density matrix of the mixed state ( ( ψ 1 , p 1 ), ( ψ 2 , p 2 ), …, ( ψ d , p d ) ) is: 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 Examples (from previous lecture):

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

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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 k k —and 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

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7 Characterizing density matrices Three properties of : Tr = 1 ( Tr M = M 11 + M M dd ) = † ( 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

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

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9 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 eigenvectors:

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10 Unitary and Hermitian matrices with respect to some orthonormal basis Normal: 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 )

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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 )

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12 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 = k 0 and k = 0 if k k 0 )

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13 Taxonomy of normal matrices normal unitaryHermitian reflection positive semidefinite projector density matrix rank one projector

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

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15 Bloch sphere for qubits (I) Consider the set of all 2x2 density matrices Note that the coefficient of I is ½, since X, Y, Z have trace zero 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

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16 Bloch sphere for qubits (II) We will express First consider the case of pure states , where, without loss of generality, = cos ( ) 0 + e 2i sin ( ) 1 ( , R ) Therefore c z = cos (2 ), c x = cos (2 ) sin (2 ), c y = sin (2 ) sin (2 ) These are polar coordinates of a unit vector ( c x, c y, c z ) R 3

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17 Bloch sphere for qubits (III) ++ 00 11 –– +i+i –i–i +i = 0 + i1+i = 0 + i1 –i = 0 – i1–i = 0 – i1 – = 0 – 1– = 0 – 1 + = 0 +1+ = 0 +1 Pure states are on the surface, and mixed states are inside (being weighted averages of pure states) Note that orthogonal corresponds to antipodal here

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

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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 maps”, “admissible operations” ): Let A 1, A 2, …, A m be matrices satisfying Then the mappingis a general quantum op

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20 General quantum operations (II) Example 2 (decoherence): let A 0 = 0 0 and A 1 = 1 1 This quantum op maps to 0 0 0 0 + 1 1 1 1 Corresponds to measuring “ without looking at the outcome” For ψ = 0 + 1 , After looking at the outcome, becomes 0 0 with prob. | | 2 1 1 with prob. | | 2

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21 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” A k is 2/3, and this can be adapted to actually yield the value of k with this success probability Define A 0 = 2/3 0 0 A 1 = 2/3 1 1 A 2 = 2/3 2 2

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22 General quantum operations (IV) Example 4 (discarding the second of two qubits): Let A 0 = I 0 and A 1 = 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 Tr 2

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