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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.

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Presentation on theme: "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."— Presentation transcript:

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 Course web site at: Lecture 11 (2005)

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

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

4 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):

5 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)

6 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

7 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

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

9 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:

10 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 )

11 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 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 )

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

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

15 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

16 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

17 17 Bloch sphere for qubits (III) ++ 00 11 –– +i+i –i–i +i = 0 + i1+i = 0 + i1 –i = 0 – i1–i = 0 – i1 – = 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

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

19 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

20 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

21 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 

22 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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