Michael A. Nielsen

University of Queensland

Quantum Noise

Goals: 1. To introduce a tool – the density matrix – that is

used to describe noise in quantum systems, and to give some examples.

Density matrices

Generalization of the quantum state used to describe noisy quantum systems.

Terminology: “Density matrix” = “Density operator”

Ensemble

Fundamental point of view

Quantum subsystem,j jp

What we’re going to do in this lecture, and why we’re doing it

Most of the lecture will be spent understanding the density matrix.

Unfortunately, that means we’ve got to master arather complex formalism.

It might seem a little strange, since the density matrix is never essential for calculations – it’s a mathematical tool, introduced for convenience.

The density matrix seems to be a very deep abstraction – once you’ve mastered the formalism, it becomes far easier to understand many other things, including quantum noise, quantum error-correction, quantum entanglement, and quantum communication.

Why bother with it?

I. Ensemble point of view

I magine that a quantum system is in the state withprobability .

j

jp

We do a measurement described by projectors .kP

Probability of outcome Pr | state j jk

k k p j j jk

k

P p trj j j k

k

p P

where is th density matre .ixj j jj

p tr kP

all mecomple asuremtely determi ent statistnes ics.

Probability of outcome k

Qubit examples

Suppose 0 with probability 1. 1 1 0

Then 0 0 1 0 .0 0 0

0 0 0Then 1 1 0 1 .

1 0 1

Suppose 1 with probability 1.

0 1Suppose with probability 1.

2

i

1 10 1 0 1 1 1Then 1 .

12 22 2

ii ii

i i

Qubit example

Suppose 0 with probability , and 1 withprobability 1 .

pp

Then 0 0 1 1 1p p

1 0 0 0 0

1 .0 0 0 1 0 1

pp p

p

Measurement in the 0 , 1 basis yields

0 1Pr 0 tr 0 0 1 0

0 1 0

p

p

.p Similarly, Pr 1 1 .p

Why work with density matrices?

Answer: Simplicity!

0 with probability 0.1

1 with probability 0.1

The quantum state is:

0 1 with probability 0.15

2

0 1 with probability 0.15

2

0 1 with probability 0.25

2

i

0 1 with probability 0.25

2

i

12

12

0

0

Dynamics and the density matrix

Suppose we have a quantum system in the state withprobability .

j

jp

The quantum system undergoes a dynamics described by the unitary matrix .U

The quantum system is now in the state with probability .

j

j

Up

initial density mThe is atrix .j j jjp

† fi nal density matrThe is x ' .i j j jjpU U

† .j j jjU pU U †' .U U

Single-qubit examples

Suppose 0 with probability , and 1 withprobability 1 .

pp

0Then .

0 1

p

p

Suppose an gate is applied. Then 'X X X 1 0

.0

p

p

12Suppose 0 and 1 with equal probabilities .

Then .2I “Completely mixed state”

†

Suppose any unitary gate is applied.

Then ' = .2 2

UI I

U U

How the density matrix changes during a measurement

Suppose a measurement described byprojectors is perf ormed on an ensemble giving rise to the density matrix . I f the measurement gives result

show that the corresponding post-me

kP

k

Worked Exercise:

'

asurement densitymatrix is

.tr

k kk

k k

P P

P P

Characterizing the density matrix

What class of matrices correspond to possible density matrices?

Suppose is a density matrix.j j jjp

Then tr( ) trj j jjp jj

p 1

For any vector ,

j j jj

a

a a p a a 2

j jjp a 0

Given that tr =1 and is a positive matrix,show that there is some set of states and probabilities

such that = .j

j j j jjp p

Exercise:

tr =1 and is a positive matrix. Summary:

Summary of the ensemble point of view

tr =1, and is a positive matrix.Conversely, given any matrix satisf ying these properties,there exists a set of states and probabilities

Character

such that = .

ization:

j j

j j jj

p

p

The density matrix f or a system in state

with probabili

Defi niti

ty

on:

is .j

j j j jj

p p

'

A measurement described by projectors gives result with probability tr ,and the post-

measurement density matrix is

Mea

.t

surement:

r

k

k

k kk

k k

Pk P

P P

P P

†Dynamics : ' .U U

A simple example of quantum noise

X

With probability p the not gate is applied.

With probability 1-p the not gate fails, and nothing happens.

j j jjp 1j j j j j jj

p pX X p p 1pX X p

If we were to work with state vectors instead of densitymatrices, doing a series of noisy quantum gates wouldquickly result in an incredibly complex ensemble of states.

How good a not gate is this?

XHow "good" a not gate is this, f or a particular input ?

1E pX X p

The usual way two states and are compared is to compute the , or overlap:

fi delit)

y( , .

a b

F a b a b

To compare with we compfi delity

ute the,

( , ) .

j j jja p

F a a a

The fidelity measures how similarthe states are, ranging from 0 (totallydissimilar), up to 1 (the same).

We compare the ideal output,, to the actual output.X

A quantum operation

Fidelity measures for two mixed states are a surprisingly complex topic!

How good a not gate is this?

XHow "good" a not gate is this, f or a particular input ?

1E pX X p We compare the ideal output,

, to the actual output.X

The fi delity of the gate is thus ( , )F X E XE X

1p p X X

21p p X

The fi delity ranges between , f or 0 , and 1, f or 0 1 / 2 .

p

II. Subsystem point of viewAlice Bob

klklk l

jP Pr( ) tr jj P I * trmn jklklmn

P I k l m n *

mn jklklmnl n P I k m *

jkl mlklml P k

* tr jkl mlklmP k l

Atr jP

*Awhere is

known as th reduced densitymatr

e of system .x Ai

kl mlklmk l

II. Subsystem point of viewAlice Bob

klklk l

jP

A reduced density matri is the f or sys .x tem A

A*

A B

Pr( ) tr , wheretr

j

kl mlklm

j Pk l

A

All the statistics f or measurements on system A canbe recovered f rom .

How to calculate: a method, and an example

*B

Show that this new defi nition agrees with the old, that is, tExercise:

r when.

kl mlklm

klkl

k lk l

1 2 1 2 1 2 1 2

An alternative, more convenient defi nition f or the partialtrace is to defi ne: tr trB a a b b a a b b

2 1 1 2b b a aThen extend the defi nition linearly to arbitrary matrices.

A B

I f the system is in the state thExa entr

mple: a ba a b b b b a a a a

The example of a Bell state

00 11

2

A B

Suppose = . Then the reduced densitymatrix f or the fi rst system is given by:

Exam

tr

ple:

B B B Btr 00 00 tr 00 11 tr 11 00 tr 11 11

2

0 0 +1 1

2

2I

12

12

From , it'sjust like having the state 0with probability , and the state 1 wi

Alice's point of view

th probability .

Under dynamics and measurement, the densitymatrix behaves just as it does in the ensemblepoint of view.

III. The density matrix as fundamental object

'

A measurement described by projectors gives result with probability tr ,and the post-

measurement density matrix is

Pos

.t

tulate 3:

r

k

k

k kk

k k

Pk P

P P

P P

† The dynamics of a closed quantum system are

described by Postulate 2

' .:

U U

Postulate 1: A quantum system is described by a positivematrix (the density matrix), with unit trace, acting ona complex inner product space known as state space.

We take the tensor product to fi nd the state space of a composite system. The state of one componentis f ound by taking the partial trace over the remainder of

Postulate

the s

4:

ystem.

A system in state with probability has density matrix.

j j

j jj

pp

Alice Bob

Why teleportation doesn’t allow FTL communication

Alice Bob

01 01

Why teleportation doesn’t allow FTL communication

Alice Bob

00 11The initial state f or the protocol is

2

B 2

Bob's initial reduced density matrix is just the reduceddensity matrix f or a Bell state, .I

Why teleportation doesn’t allow FTL communication

1 2 3 4

2

B B Z B X B XZ

Alice Bob

01 11 4

12 4

13 4

14 4

with probability ;

with probability ;

with probability ; and

with probability .

B

B Z

B X

B XZ

Why teleportation doesn’t allow FTL communication

Alice Bob

A 1 1'B

Bob's fi nal reduced density matrix is thus

tr ...

4

B B

4Z Z X X XZ ZX

2 2 2 2* * * *

2 2 2 2* * * *

4

2I

Why teleportation doesn’t allow FTL communication

Alice Bob

Bob’s reduced density matrix after Alice’s measurementis the same as it was before, so the statistics of anymeasurement Bob can do on his system will be thesame after Alice’s measurement as before!

Why teleportation doesn’t allow FTL communication

Fidelity measures for quantum gates

It should have a simple, clear, unambiguous operationalinterpretation.

Research problem: Find a measure quantifying how well a noisy quantum gate works that has the following properties:

It should have a clear meaning in an experimentalcontext, and be relatively easy to measure in a stablefashion.It should have “nice” mathematical properties thatfacilitate understanding processes like quantumerror-correction.

Candidates abound, but nobody has clearly obtained asynthesis of all these properties. It’d be good to do so!