Download presentation

Published byJustus Savage Modified over 2 years ago

1
**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 Course web site at:

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
**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

5
**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

6
**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

7
**Characterizing density matrices**

Three properties of : Tr = 1 (Tr M = M11 + M 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
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

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 are both 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 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 )

13
**Taxonomy of normal matrices**

unitary Hermitian reflection positive semidefinite projector density matrix rank one

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 + e2isin()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
**Bloch sphere for qubits (III)**

+ 0 1 – +i –i +i = 0 + i1 –i = 0 – i1 – = 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)

18
**Continuation of density matrix formalism**

Taxonomy of various normal matrices Bloch sphere for qubits General quantum operations

19
**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†

20
**General quantum operations (II)**

Example 2 (decoherence): let A0 = 00 and A1 = 11 This quantum op maps to 0000 + 1111 For ψ = 0 + 1, Corresponds to measuring “without looking at the outcome” After looking at the outcome, becomes 00 with prob. ||2 11 with prob. ||2

21
**General quantum operations (III)**

Example 3 (trine state “measurent”): Let 0 = 0, 1 = 1/20 + 3/21, 2 = 1/20 3/21 Define A0 = 2/300 A1= 2/311 A2= 2/322 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

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: it’s the same density matrix as for ((0, ½), (1, ½)) Note 2: the operation is the partial trace Tr2

23
THE END

Similar presentations

OK

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.

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.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on charge-coupled device images Ppt on principles of peace building review Ppt on education system in india during british rule Ppt on 555 timer application Ppt on time management download Ppt on articles for class 2 Cardiovascular system anatomy and physiology ppt on cells Ppt on seven segment display pin Ppt on articles of association of a company Ppt on tunnel diode characteristics