27may_2_slides.pdf

Measurements and Time EvolutionMatthias Christandl Lecture 1

1Montag, 27. Mai 13

Measurements

measurement of A

A =X

i

aiPiprojector on to eigenspace

to ith eigenvalue

Labelling with eigenvalues often convenient, but not necessary

projective measurement

set of orthogonal projectors that sum to identity

{Pi}, Pi = P i , P2i = Pi,

X

i

Pi = id

independent of eigenvalue

Is this the most general measurement?

i

i :=PiPiprob[i]

prob[i] = trPi

2Montag, 27. Mai 13

POVMs

projective measurement

POVMpositive operator-valued measure

set of positive-semidefinite operators that sum to identity

i

{Qi}, Qi 0,X

i

Qi = id

i :=Pi(A |00|B)Pi

prob[i]Qi = 0|BPi|0B

prob[i] = trPi(A |00|B)= trAQiA

A

|00|B

|Qi| = |A0|BPi|A|0B 0

X

i

Qi =X

i

0|BPi|0B

= 0|B(X

i

Pi)|0B

= 0|B idAB |0B = idA3Montag, 27. Mai 13

POVMs: Examples

POVM i

{Qi}, Qi 0,X

i

Qi = id

Aprob[i] = trAQiA

Example 1: Mixture of two projective measurements

with 50% probability measure in z-direction with 50% probability measure in x-direction

Example 2: Tetrahedron

Qi =1

2|ii| =

1

2

1

2(id + ai )

a0/1 =

r2

3(1, 0, 1

2), a2/3 =

r2

3(0,1, 1

2)

Q0 =1

2|0ih0|, Q1 =

1

2|1ih1|, Q2 =

1

2|+ih+|, Q3 =

1

2|ih|

4Montag, 27. Mai 13

Time Evolution

time evolution

| eiHt|

with time-independent Hamiltonian for a fixed amount of time

without loss of generality (discretization)

U

Example: Qubit rotation

UtUt

Ut = eite ~2

unit vector

UtUt =

1

2(id + Ut(r )Ut ) =

1

2(id + (Rtr) )

Wunderformel

R(e, t)rotations in the Bloch sphere5Montag, 27. Mai 13

Rotations in the Bloch sphereUt = e

ite ~2 UtUt =

1

2(id + Ut(r )Ut ) =

1

2(id + (Rtr) )

Example: magnetic field in x-direction, qubit in z-directionqubit rotates around x-axis

trzt1

-1

2 4 6 8 10 12

-0.5

0.5

visibility

Example: Hadamard transform

H =1p2

1 11 1

=

x + zp2

= iei(1p2,0, 1p

2) ~2

H|0 = 12(|0+ |1) = |+

H|1 = 12(|0 |1) = |

6Montag, 27. Mai 13

Physical Operations as CPTP Maps

8Montag, 27. Mai 13

CPTP maps

U

A

|00|B tr

A0

B0

A0

trB0U(A |00|B)U

completely positive trace-preserving map

tr(A) = trA

(A) 0, for all A 0

A A0

idC(AC) 0, for all AC 0for all C

Stinespring: Every CPTP map is of this form!

implies: every state evolution is unitary

9Montag, 27. Mai 13

Operator-Sum Representation

U

A

|00|B tr

A0

B0

A0

(A) = trB0U(A |00|B)U =X

i

i|B0U |0BA0|BU |iB0

=X

i

EiAEi

Kraus operators:matrices, mapping A into A

10Montag, 27. Mai 13

CPTP maps: Examples

A A0

Depolarising channel

Bit flip channel

() = (1 p)+ p121 = (1 3

4p)+

1

4p(XX + Y Y + ZZ)

() = (1 p)+ pXX

() = (1 p)+ pZZPhase flip channel

Amplitude damping channel() = E0E

0 + E1E

1

E0 =

1 00p1

E1 =

0p

0 0

Nielsen-Chuang, CUP 2001

Kraus operator

Kraus operator


Measurements as CPTP mapsfor simplicity, we only show projective measurements

measurement i

i :=PiPiprob[i]

prob[i] = trPi

() =X

i

pi|ii| i

=X

i

|ii| PiPi

Example: z-axis

() = (tr|00|)|00|+ (tr|11|)|11| =

p0 00 p1


Entangled with Environment

() = p1|00|+ p1|11|measurement

U|00|B trB

= |++| |+ = 12(|0+ |1)

1

2(|00 + |11)

1

2(|00|+ |11|)

decoherence is entanglement with

environment

U = |0000|+ |1110|+ |0101|+ |1011|


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