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This introduction to quantum information processing covers density matrices, normal matrices, the Bloch sphere for qubits, and general quantum operations. It discusses the taxonomy of normal matrices, properties of density matrices, and quantum operations in terms of density matrices. The lecture explores eigenvectors, unitary and Hermitian matrices, positive semidefinite matrices, projectors and density matrices, and the Bloch sphere representation for qubits. It also delves into general quantum operations and provides examples encompassing various matrix operations.
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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)
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
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. 0 with prob. ½ 0 +1 with prob. ½ Recap: density matrices (II) Examples (continued): has: ...?(later) 7. The first qubit of 01 −10
Recap: density matrices (III) Quantum operations in terms of density matrices: • Applying U to yieldsUU† • Measuring state with respect to the basis1, 2,..., d, • yields: kthoutcome with probability kk • —and causes thestate to collapse tokk 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 (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
Continuation of density matrix formalism • Taxonomy of various normal matrices • Bloch sphere for qubits • General quantum operations
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
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)
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 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)
normal unitary Hermitian reflection positive semidefinite projector density matrix rank one projector Taxonomy of normal matrices
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 +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
0 + = 0 +1 –i – = 0 – 1 +i = 0 +i1 + – –i = 0 –i1 +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)
Continuation of density matrix formalism • Taxonomy of various normal matrices • Bloch sphere for qubits • General quantum operations
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 yieldsUU†
After looking at the outcome, becomes 00 with prob. ||2 11 with prob. ||2 General quantum operations (II) Example 2 (decoherence):letA0 =00andA1 =11 This quantum op maps to 0000 + 1111 For ψ =0 +1, Corresponds to measuring “without looking at the outcome”
DefineA0 =2/300 A1=2/311 A2=2/322 General quantum operations (III) Example 3 (trine state“measurent”): Let 0 = 0, 1 =1/20 + 3/21, 2 =1/20 3/21 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
General quantum operations (IV) Example 4 (discarding the second of two qubits): LetA0 =I0andA1 =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
