QUBITS IN PHASE SPACE - INAOE - P · 2004. 12. 14. · “Quantum computers in phase space”: C. Miquel, J.P.P & M. Saraceno PRA 64, 032609 (2002) • HOWEVER: We can still define

QUBITS IN PHASE SPACE

JUAN PABLO PAZ

Department of Physics, University of Buenos Aires (ARGENTINA)

Los Alamos National Laboratory (USA)

Quantum Optics II

Cozumel, December 2004

J.P. Paz, A. Roncaglia & M. Saraceno, “Qubits in phase space: Wigner function approach toquantum error correction and the mean king problem”, quant-ph/0410137

• Continuous Wigner functions

Discrete Wigner functions (some problems and solutions)

• Discrete Wigner functions for systems of qubits (Wootterset al, quant-ph/0401135): a new approach.

• Applications: Wigner function of Stabilizer States (Bell states). Aphase space solution to “The mean king problem”.

QUBITS IN PHASE SPACE: OUTLINE

PHASE SPACE DISTRIBUTIONS: WIGNER FUNCTIONS

A WAY TO VISUALIZE A QUANTUM STATE USING “INTUITIVE” TOOLS(CLASSICAL LIKE)

Gaussian peaks

(classical)

Oscillations! Wigner functioncan be negative due toquantum interference.

CONTINUOUS WIGNER FUNCTIONS

• PROPERTIES:

W(x,p) is real

Integral along lines give all marginal distributions:

€

dx dpW1(x, p)W2(x, p) =12πh

Tr(ρ1∫ ρ2)Use it to compute inner products as:

€

dx dpW (x, p) = Probability(aX + bP = c)∫

€

ax + bp = c

• CONTINUOUS:

€

W (x, p) =dy2πh

eipy / h∫ < x +y2| ρ | x − y

2>

• LINES IN PHASE SPACE • STATES IN HILBERT SPACE

CONTINUOUS WIGNER FUNCTIONS

DISCRETE WIGNER FUNCTIONS

€

U n = n +1,mod(N) V k = k +1,mod(N)

• A SYSTEM WITH A FINITE DIMENSIONAL (N) HILBERT SPACE (AN ATTEMPT)

€

{ n , n =1,K,N}{ k , k =1,K,N}

Position basis (arbitrary)

Momentum basis

€

k =1N

exp(i2π nk /N)n=1

N

∑ n

• LINES IN PHASE SPACE • STATES IN HILBERT SPACE?

• PHASE SPACE DISPLACEMENT OPERATORS

• PHASE SPACE = N x N GRID

• DEFINE ‘LINES’ IN THE GRID AS SETS (q,p) SUCH THAT aq+bp=c (mod N arithmetic)

• WARNING: PROBLEMS ARISE!

U. Leonhardt, PRA 53, 2998 (1996), C. Miquel, J.P.P & M. Saraceno PRA 64, 032609 (2002)

0 1 2 3 4 5 6 7

0

1

3

4

2

5

6

7

q+p=0N=8


q & p are integers, arithmetic mod N

0 1 2 3 4 5 6 7

0

1

3

4

2

5

6

7

q+p=0N=8

0 1 2 3 4 5 6 7

0

1

3

4

2

5

6

7

2q+2p=0


Lines may have moremore than N pointsq & p are integers, arithmetic mod N

0 1 2 3 4 5 6 7

0

1

3

4

2

5

6

7

q+p=0N=8

0 1 2 3 4 5 6 7

0

1

3

4

2

5

6

7

2q+2p=0


Lines may have moremore than N pointsq & p are integers, arithmetic mod N

WHY??: {1,2,…,N} is NOT a FINITE FIELD: 2 x 4 = 0 (mod 8)

(it is a field if N is a prime number GF(N))

• HOWEVER: We can still define a discrete Wigner function with all the ‘good’ properties inan 2N x 2N grid (BUT lines & states are no longer naturally connected)

• SAME PROPERTIES THAN IN THE CONTINUOUS CASE:

W(q,p) is real

Sum over lines is always possitive!

€

Nq,p=1

2N

∑ W1(q, p)W2(q, p) = Tr(ρ1 ρ2)

€

W (q, p)q,p=1

2N

∑ = ϕ ρ ϕ , D(b,a)ϕ = exp(i2πc /N)ϕ

€

ax − bp = c


“Quantum computers in phase space”: C. Miquel, J.P.P & M. Saraceno PRA 64, 032609 (2002)

• HOWEVER: We can still define a discrete Wigner function with all the ‘good’ properties inan 2N x 2N grid (BUT lines & states are no longer naturally connected)

Computationalstate

positiveos

cilla

tory

• SAME PROPERTIES THAN IN THE CONTINUOUS CASE:

W(q,p) is real

Sum over lines is always possitive!

€

Nq,p=1

2N

∑ W1(q, p)W2(q, p) = Tr(ρ1 ρ2)

€

W (q, p)q,p=1

2N

∑ = ϕ ρ ϕ , D(b,a)ϕ = exp(i2πc /N)ϕ

€

ax − bp = c


“Quantum computers in phase space”: C. Miquel, J.P.P & M. Saraceno PRA 64, 032609 (2002)

USE WIGNER FUNCTIONS TO REPRESENT THE STATE AND THE

EVOLUTION OF QUANTUM COMPUTERS

see C. Miquel, J.P.P & M. Saraceno, Phys Rev A 65 (2002), 062309

Phase spacerepresentation ofGrover’salgorithm

See paper for interesting (useful?) analogiesbetween quantum algorithms and quantummaps.

Also: C. Lopez & J.P. Paz, PRA 68 052305(2004): use of W.F. in quantum walks;


DISCRETE WIGNER FUNCTIONS: A NEW APPROACH

0 1

0

1

ω

ω2

ω3

ω4

ω5

ω6

ω ω2 ω3 ω4 ω5 ω6

All the lines contain N points

K. Gibbons, D. Hoffman and W.Wootters, quant-ph/0401155 an approach that works when

€

N = pn

Consider (qubits). Phase space is an NxN grid. Position and momentumcoordinates must be considered elements of the Finite Field

€

N = 2n

€

GF(2n )

What is ?

€

GF(2n ) Set of polynomials with binarycoefficients . Usual sum between polynomials. Usualproduct between polynomials modulo (an n-th degreeirreducible polynomial)

€

2n

€

π (x) = a0 + a1x +K+ an−1xn−1

€

ai = 0,1

€

πG (x)

All non-zero elements in areexpressed in terms of a “generating”element

€

GF(2n )

€

GF(2n ) = 0,1,ω,K,ωN−2{ }

€

q + p = 0




€

N = pn


€

N = 2n

€

GF(2n )

What is ?

€


€

2n

€

π (x) = a0 + a1x +K+ an−1xn−1

€

ai = 0,1

€

πG (x)


€

GF(2n )

€

GF(2n ) = 0,1,ω,K,ωN−2{ }

€

q +ω p = 0

0 1

0

1

ω

ω2

ω3

ω4

ω5

ω6

ω ω2 ω3 ω4 ω5 ω6




€

N = pn


€

N = 2n

€

GF(2n )

What is ?

€


€

2n

€

π (x) = a0 + a1x +K+ an−1xn−1

€

ai = 0,1

€

πG (x)


€

GF(2n )

€

GF(2n ) = 0,1,ω,K,ωN−2{ }

€

q +ω 2p = 0

0 1

0

1

ω

ω2

ω3

ω4

ω5

ω6

ω ω2 ω3 ω4 ω5 ω6


Phase space recovers usual geometric featues

When axis are labeled with elements of the finite field GF(N) and lines aredefined with linear equations of the form aq+bp=c, then:

Lines which are not parallel intersect in only one point

There are exactly N(N+1) lines

These lines are grouped in N+1 sets of parallel lines (striations)

Striations for N=4, two qubits.

€

q = c

€

p = c

€

q + p = c

€

ωq + p = c

€

ω 2 q + p = c


Then one can associate every line with a pure state!

Association between lines and states should be such that:

States associated with parallel lines are orthogonal (overlap rule). Then astriation is associated with a bases of Hilbert space.

States associated with non-parallel lines have an overlap 1/N. Then basesassociated with different striations must be “mutually unbiased” (MUB).

Basis 1

Basis 2

Basis 3

Basis 4

Basis 5


Phase space translation operators can also be naturally defined

0 1 ω ω2

0

1

ω

ω2

00 01 10 11

00

01

10

11

Eigenstates of Z1 and Z2

Eigenstates of X1 and X2

€

00



0 1 ω ω2

0

1

ω

ω2

00 01 10 11

00

01

10

11



€

X1 00 = 01



0 1 ω ω2

0

1

ω

ω2

00 01 10 11

00

01

10

11



€

X2 00 = 10



0 1 ω ω2

0

1

ω

ω2

00 01 10 11

00

01

10

11



€

X1X2 00 = 11



0 1 ω ω2

0

1

ω

ω2

00 01 10 11

00

01

10

11



€

00 x



0 1 ω ω2

0

1

ω

ω2

00 01 10 11

00

01

10

11



€

Z1 00 x = 01 x



0 1 ω ω2

0

1

ω

ω2

00 01 10 11

00

01

10

11



€

Z2 00 x = 10 x



0 1 ω ω2

0

1

ω

ω2

00 01 10 11

00

01

10

11



€

Z1Z2 00 x = 11 x

Horizontal translations:

Vertical translations:

€

⊗X qi

€

⊗Z pi

General translations: composition of vertical plus horizontal

€

T(q, p) = ⊗X qi Z pi

CAN THIS BE USEFUL?

Yes: problems where states with symmetries under T(q,p) play a special roleYes: problems where states with symmetries under T(q,p) play a special role

Bell states are eigenstates of σx⊗ σx and σz⊗ σz

All Bell states can be obtained from one by translating one of them

00 01 10 11

00

01

10

11

State iseigenstate of

σz⊗ σz


σx⊗ σx

Stabilizer states (error correction, etc)

Bell states are an example of stabilizer states

Wigner function for a Bell state

CAN THIS BE USEFUL?

Yes: problems where states with symmetries under T(q,p) play a special roleYes: problems where states with symmetries under T(q,p) play a special role

Bell states are eigenstates of σx⊗ σx and σz⊗ σz

All Bell states can be obtained from one by translating one of them

00 01 10 11

00

01

10

11


σz⊗ σz


σx⊗ σx

Stabilizer states (error correction, etc)

Bell states are an example of stabilizer states

Wigner function for a Bell state

00 01 10 11

00

01

10

11€

1200 + 11( )

€

= 0

€

=14

CAN THIS BE USEFUL?

00 01 10 11

00

01

10

11€

1200 + 11( )

Wigner function for all Bell states

00 01 10 11

00

01

10

11

€

1201 + 10( )

€

X1

CAN THIS BE USEFUL?

00 01 10 11

00

01

10

11€

1200 + 11( )

Wigner function for all Bell states

00 01 10 11

00

01

10

11

€

1201 + 10( )

€

X1

00 01 10 11

00

01

10

11

€

1201 − 10( )

€

X1Z1

00 01 10 11

00

01

10

11

€

1200 − 11( )

€

Z1

THE MEAN KING PROBLEM The problem:

o A physicist prepares a spin 1/2 particle and gives it to a mean King.

o The King measures the spin along ONE of the cartesian (x,y,z) components.

o He gives the particle back to the physicist who can do whatever he wants with it..

o Finally, the King tells the physicist in which direction he measured the spin (x,y or z).

o Then the physicist MUST tell the result of the measurement performed by the king..

or face death...

o Untold part: The Queen helped the physicist by telling him “entangle!”







or face death...


Various mithological versions: KINGPHYSICIST







or face death...


Various mithological versions: KINGPHYSICIST

SPIN 1/2 PARTICLE!

THE MEAN KING PROBLEM

xσ

zσ

yσ

Received states according to

King’s measurement

The problem: A physicist prepares a spin 1/2 particle and gives it to a mean King. The King measuresthe spin along ONE of the cartesian (x,y,z) components. He gives the particle back to the physicist who cando whatever he wants with it.. Finally, the King tells the physicist in which direction he measured the spin (x,yor z) and the physicist MUST tell the result of the measurement. OR ELSE…

Solution: The physicist should use entanglement! He will prepare a Bell state, and then...

Solution in terms of the Discrete Wigner function :

Initial state:

€

1200 + 11( )

€

X

€

Z

€

Y

SOLUTION?

THE MEAN KING PROBLEM: PHASE SPACE SOLUTIONThe King will prepare for us

one of six “line” statesFind a state such that its Wigner adds up

to zero along three lines

THE MEAN KING PROBLEM: PHASE SPACE SOLUTIONThe King will prepare for us

one of six “line” statesFind a state such that its Wigner adds up

to zero along three lines

If the physicist detectssuch state he can rule

out one option fromeach line!

THE MEAN KING PROBLEM: PHASE SPACE SOLUTIONMoreover, the state and its three rotateddescendents must be orthogonal (theymust form a basis of the Hilbert space)

€

116

€

716

€

−116

Solution

€

Z1Z2

€

X1X2

€

Y1Y2

THE MEAN KING PROBLEM: PHASE SPACE SOLUTIONThe solution is to measure an observable which is diagonal in the basis

J.P. Paz, A. Roncaglia & M. Saraceno, “Qubits in phase space: Wigner function approach toquantum error correction and the mean king problem”, quant-ph/0410137

• Various definitions of discrete Wigner functions are available (itcan always be done using a 2Nx2N phase space grid but in thiscase the association between lines and pure states is lost).

• Using position and momentum coordinates as elements of thefinite field GF(N) (Wootters suggestion) one can have a phasespace consisting of an NxN grid with a “nice” geometric structure.

• Parallel lines map onto orthogonal states. A striation with parallellines maps onto a bases of Hilbert space. Different striations maponto Mutually Unbiased Bases.

• Useful? It may be a powerful tool (maybe…) for problems wherestabilizer states (eigenstates of operators which are tensorproducts of Pauli operators) play an important role. Errorcorrection, etc.

Summary:

CONTINUOUS AND DISCRETE WIGNER FUNCTIONS

• A SYSTEM WITH A FINITE DIMENSIONAL (N) HILBERT SPACE

€

{ n , n =1,K,N}{ k , k =1,K,N}

Position basis (arbitrary)

Momentum basis

€

k =1N

exp(i2π nk /N)n=1

N

∑ n

• PHASE SPACE DISPLACEMENT OPERATORS ARE EASY TO CONSTRUCT

€

U n = n +1,mod(N)V k = k +1,mod(N)

Position shift

Momentum shift

€

D(q, p) =UqV −p exp(iπ qp /N)

• PHASE SPACE POINT OPERATORS ARE NOT EASY TO CONSTRUCT

€

A(q, p) =1N 2

n=1

N

∑ exp(i2π (np + mp) /N) D(n,m)n=1

N

∑Naive attempt

Does not have all the right properties (for all values of N)

Discrete Wigner function: How does it look like?

Quantum states in phase space

Gaussian wavepacketComputational state

positive oscillatory oscillatory

osci

llato

ry

OSCILLATIONS DUE TOPERIODIC BOUNDARYCONDITIONS

Corrects phase errors: Z1 , Z2 , Z3

The stabilizer is defined by: M1=X1 X2

M2=X2 X3

Encoded states

m1=+1

m2=+1

m1= -1

m2=+1

m1= -1

m2= -1

m1=+1

m2= -1

Z1

Z2

Z3

W(q,p) of theencoded states mustbe invariant underthese translations

Translation of W(q,p) byZi gives an orthogonalrepresentation

3-qubit quantum error correcting code

Wigner representation of logical states

These states have eigenvalues m1=m2=+1

Eigenstates of ZN=Z1Z2Z3 (which commutes with the stabilizer)

Wigner representation invariantunder the translations

M1=X1 X2

M2=X2 X3

ZN=Z1Z2Z3

000 001 010 100 101 111 011 110

000

001

110

111

100

011

101

010

These symmetriesreduce our grid of 64

independent values to 8real parameters

0L

Wigner representation of logical states

These states have eigenvalues m1=m2=+1

Eigenstates of ZN=Z1Z2Z3 (which commutes with the stabilizer)

000 001 010 100 101 111 011 110

000

001

110

111

100

011

101

010

1LBy applying X1 we obtainWigner representation invariant

under the translations

M1=X1 X2

M2=X2 X3

ZN=Z1Z2Z3

These symmetriesreduce our grid of 64

independent values to 8real parameters

Documents

QUBITS IN PHASE SPACE - INAOE - P · 2004. 12. 14. · “Quantum computers in phase space”: C. Miquel, J.P.P & M. Saraceno PRA 64, 032609 (2002) • HOWEVER: We can still define