Michael A. Nielsen

University of Queensland

Quantum Mechanics II: Examples

Goals: 1. To apply the principles introduced in the last

lecture to some illustrative examples: superdense coding, and quantum teleportation.

2. Revised form of postulates 2 (dynamics) and 3 (measurement).

3. Introduce more elements of the Dirac notation.4. Discuss the philosophy underlying quantum

information science.

Alice Bob

ab

Superdense coding

ab

Theorist’s impressionof a measuring device

Alice Bob

ab

Superdense coding

ab

Alice Bob

ab

Superdense coding

00 11

2

00 :Apply I 00 11 00 11

2 2

ab 01:Apply Z 00 11 00 11

2 2

10 :Apply X 00 11 10 01

2 2

11:Apply XZ 00 11 10 01

2 2

0 1 ; 1 0X X

0 0 ; 1 1Z Z

1 ebit + 1 qubit of communication 2 bits of classical communication

Superdense coding can be viewed as a statement about the interchangeability of physical resources.

01 10

2

Could Alice and Bob still communicatetwo bits using the superdense coding protocol if the initialstate shared by Alice and Bob w

Worked exerci

as ?

se:

Revised measurement postulate

12

I f we measure in an orthonormal basis ,..., , then we obtain the resulR

t wiecall post

th probabiulat

litye

:

(

3

) .d

j

e e j

P j e

The measurement disturbs the system, leaving it in a state determined by the outcome.

je

1

I magine we measure a quantum system, , in the orthonormal basis Pr

,o

.ble

. .m

:

.,Ad

Ae e

Suppose system is part of a larger system, consisting oftwo components, and .

AA B

How should we describe the eff ect of the measurement onthe larger system?

Revised measurement postulate

12

I f we measure in an orthonormal basis ,..., , then we obtain the resulR

t wiecall post

th probabiulat

litye

:

(

3

) .d

j

e e j

P j e

The measurement disturbs the system, leaving it in a state determined by the outcome.

je

1

1

The replaces the orthogonal states ,...revised postulatecomplete se

,with a of orthogonat subspacel ,...,s .

d

m

e eV V

1 2 ... .mV V V V

1 2 3 1 2 3sp , , sp , spe e e e e e

1 2 3Example: e e e

1

A general measurement can be thought of as asking theq which of the subspaces ,..., are we in?uestion " "mV V

Revised measurement postulate

12

I f we measure in an orthonormal basis ,..., , then we obtain the resulR

t wiecall post

th probabiulat

litye

:

(

3

) .d

j

e e j

P j e

The measurement disturbs the system, leaving it in a state determined by the outcome.

je

1 1

Mathematically, it is convenient to describe the subspaces,..., in terms of their corresp projectors,onding ,..., .mm P PV V

1 2

1 2 3 1 2

Example: The projector onto sp , acts as

P e eP e e e e e

I n general, the projector onto a subspace acts as the identity on that subspace, and annihilates everythingorthogonal to .

P V

V

Revised measurement postulate

1Let ,..., be a set of projectors onto a complete set oforthogonal subspaces of state space.

mP P

This set of projectors defi nes a measurement.

I f we measure then we get outcome with probability Pr( ) .j

jj P

; j j jk jkjP I P P P

The measurement unavoidably disturbs the system, leaving

it in the post-measurement state j

j

P

P

Example: A two-outcome measurement on a qutrit

A general state of a may be writtenqutri 1t 0 2 .

1

2

projects onto sp 0 , 1 ; and

projects onto sp 2 .

P

P

221Pr(1)

0

P

' 11 22

1

0 1=

P

P

22Pr(2) P

' 22

2

2= ~ 2

P

P

Example: Measuring the first of two qubits

1 2

Suppose we want to perf orm a measurement in the basis, f or the fi rst of two qubits. e e

00 01 10 11

0

I f the state of two qubits is 00 01 10 11then measuring the fi rst qubit in the computational basisgives the result 0 with probabili

Example

ty Pr 0

:

P I

00 01 10 11 00 0100 01 10 11 00 01

2 2

00 01

1

1

2

2

The rule is to fi rst f orm the corresponding projectors , onto the state space of that qubit, and then to

tensor them with the identity on the second qubit, obtaining and .P I P I

P P

Example: Measuring the first two of three qubits

1 2 3 4

Suppose we have three qubits in the state .e a e b e c e d

1 2 3 4

1

Measuring the fi rst two qubits in the basis , , ,gives the result 1 with probability Pr(1)

e e e e

P I

1= e a 2=

1 2 3 4, , , is an orthonormal basis f or the statespace of the fi rst two qubits.e e e e

, , , are normalized states of the third qubit.a b c d

1Post-measurement state is .e a

TeleportationAlice Bob

TeleportationAlice Bob

01 01

TeleportationAlice Bob

00 11

2

0 1

00 110 1

2

000 011 100 111

2

00 11 00 111 100

2 2 2 2

01 10 01 101 101

2 2 2 2

01 10 01 101 110

2 2 2 2

00 11 00 111 111

2 2 2 2

00 1110 1

2 2

00 1110 1

2 2

01 1011 0

2 2

01 1011 0

2 2

TeleportationAlice Bob

01 0 1Z

0 1X

0 1ZX

0 1I

1 ebit + 2 classical bits of communication 1 qubit of communication

Teleportation can be viewed as a statement about theinterchangeability of physical resources.

1 ebit + 1 qubit of communication 2 bits of classical communication

Compare with superdense coding:

1 qubit of communication = 2 bits of communication (Mod 1 ebit)

The fundamental question of information science

1. Given a physical resource – energy, time, bits, space, entanglement; and

2. Given an information processing task – data compression, information transmission, teleportation; and

3. Given a criterion for success;

We ask the question:How much of 1 do I need to achieve 2, while satisfying 3?

“How to write a quant-ph”

Pursuing this question in the quantum case has led to, and presumably will continue to lead to, interesting new information processing capabilities.

Are there any fundamental scientific questions that can be addressed by this program?

What fundamental problems are addressed by quantum information science?

You Your challenger

Knowing the rules Understanding the game

Knowing the rules of quantum mechanics

Understanding quantum mechanics

What high-level principles are implied by quantum mechanics?

Superfluidity, like the fractional quantum Hall effect, is an emergent phenomenon – a low-energy collective effect of huge numbers of particles that cannot be deduced from the microscopic equations of motion in a rigorous way and that disappears completely when the system is taken apart (Anderson, 1972)”

“I give my class of extremely bright graduate students, who have mastered quantum mechanics but are otherwise unsuspecting and innocent, a take-home exam in which they are asked to deduce superfluidity from first principles. There is no doubt a special place in hell being reserved for me at this very moment for this mean trick, for the task is impossible.

Robert B. Laughlin, 1998 Nobel Lecture

Quantum processes

teleportation

communication

cryptography

theory of entanglement

Shor’s algorithm

quantum error-correction

Complexity

quantum phase transitions

Quantum information science as an approachto the study of complex quantum systems

A few quanta of miscellanea

The “outer product” notation

The trace operation

Historical digression on measurement

Quantum dynamics in continuous time: an alternative form of the second postulate

The spectral theorem – diagonalizing Hermitian matrices

Outer product notation

Let and be vectors.

Defi ne a linear operation (matrix) by

Example: 1 0 0 1 1 1

*I fCon

, and thnection to matrices:

en .j j kj ja a j b b j a b k b a

1* * *

2 1 2But 1 .k

a

a b b b a

1* * *

2 1 2 3Thus .

a

a b a b b b

Outer product notation

1 0 0 1 0

0Example:

0

1 1 0 11

Example:

1 0

0 1Example: Z

0 0 1 1

1 0

0 0

0 0

0 1

1 0 1 0 1 0 1

0Example:

0 0

0 0 0

1 0 1 01

Example:1 0

0 1 0 1 1 0

1Exa le:

0mp X

Find an outer product representatioExercise: n f or .Y

Outer product notation

1 Suppose ,..., is an orthonormal basis f or state

space. Prove tha

Exerc

t .

ise:

d

j jj

e e

I e e

One of the advantages of the outer product notationis that it provides a convenient tool with which to describe projectors, and thus quantum measurements.

1 2

1 2 3 1 2

The projector onto sp , acts as Recall: P e e

P e e e e e

1 1 2 2 1 2 3 1 2

This gives us a simple explicit f ormula f or , sincePe e e e e e e e e

1

More generally, the projector onto a subspace spanned byorthonormal vectors ,..., is given by .

m

j jj

e eP e e

† Prove that Exer .ci : =se a b b a

The spectral theorem

†

†1

1

Suppose is a Hermitian matrix, . Then is ,

diag , , ,where is unitary, and

Theorem:diagonal

, , are the eigenvalue

iz

s

ab e

.

l

of d

d

A A AA

A U UU A

1But diag , , = .jd jj j

Thus , where is the

eigenvector of , .j j j j jj

j j j

A e e e U j

A A e e

, where is the projector onto the eigenspace of .

k k kk

k

A P PA

Examples of the spectral theorem

1 0 0 0 1 1

0 1Example: Z

0 1 0 1 has eigenvectors , with

1 0 2corr

Exam

esponding eigenvlaues 1.

ple: X

1 11 1- = 1 1 - 1 -1

1 -12 2

1 1 1 11 1=

1 1 1 12 2

0 1=

1 0

Historical digression: the measurementpostulate formulated in terms of “observables”

A complete set of projectors onto orthogonalsubspaces. Outcome occurs with probability Pr( ) .The corresponding post-measurement state is

.

Our f orm: j

j

j

j

Pj

j P

P

P

A measurement is described by an ,a Hermitian operator , with spectral decomposiOld f orm: o

tion

bservable

.j jj

MM P

The possible measurement outcomes correspond to theeigenvalues , and the outcome occurs with probability Pr( ) .The corresponding post-measurement state is

.

j j

j

j

j

j P

P

P

An example of observables in action

Suppose we "measE uxample: re ".Z

has spectral decomposition 0 0 - 1 1, sothis is just like measuring in the computational basis,and calling the outcomes "1" and "-1", respectively, f or0 and 1.

Z Z

Find the spectral decomposition of .Show that measuring corresponds to measuringthe parity of two qubits, with the result +1 correspondingto even parity, and the result

Exercis

-1 correspon

:

i

e

d

Z ZZ Z

ng to oddparity.

Suppose we measure the observable f or astate which is an eigenstate of that observable. Showthat, with certainty, the outcome of the measurement isthe corresponding eigenvalue

Exerci

of the ob

se: M

servable.

The trace operation

tr j jjA A

0 1 1 0

tr 0; trExamples 21 0

: .0 1

X X I I

trj jj

AB AB

Cyclicity proper try: r .t =tAB BA

jk kjjkA B kj jkjk

B A kkkBA tr BA

Prove that Exer trcise b: =a .a b

An alternative form of postulate 2

Postulate 2: The evolution of a closed quantum system

is described by a unitary transf ormation.

' U

But quantum dynamics occurs in continuous time!

The evolution of a closed quantum system is described by:

where is a constant Hermitian matrix known a

Schroedinger's equation

Hamiltonians the

of the system.

di H

dtH

The eigenvectors of are known as the of the system, and the corresponding eigenvalues are kno

energy eigenstates wn

as the energies.

H

Example: has energy eigenstates 0 1 / 2 and

0 1 / 2, with corresponding energies H X

An alternate form of postulate 2

Connection to old formof postulate 2

The solution of Schroedinger's ( )e exp( ) (0quation is )t iHt

exp( )U iHt ' U