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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:
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Contents Continuation of density matrix formalism
Taxonomy of various normal matrices Bloch sphere for qubits General quantum operations
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Continuation of density matrix formalism
Taxonomy of various normal matrices Bloch sphere for qubits General quantum operations
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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
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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
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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
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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
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Continuation of density matrix formalism
Taxonomy of various normal matrices Bloch sphere for qubits General quantum operations
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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
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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)
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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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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 )
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Taxonomy of normal matrices
unitary Hermitian reflection positive semidefinite projector density matrix rank one
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Continuation of density matrix formalism
Taxonomy of various normal matrices Bloch sphere for qubits General quantum operations
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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
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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
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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)
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Continuation of density matrix formalism
Taxonomy of various normal matrices Bloch sphere for qubits General quantum operations
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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†
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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
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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
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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
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THE END
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