L2. The geometry of quantum statesOscar Rosas-Ortiz

Departamento de Fı́sica

Cinvestav

México

(CADI 2012, ITESM Edo Méx.)

L2. The geometry of quantum states – p. 1

Introduction

L2. The geometry of quantum states – p. 2

Introduction

L2. The geometry of quantum states – p. 3

Pure states

For a pure state the ket

|Φ〉 =∑

k

ck|φk〉

provides a complete description of the quantum system.

Simplest case:

|Φ〉 = a|0〉+ b|1〉, |a|2 + |b|2 = 1, |0〉 =(

1

0

), |1〉 =

(0

1

).

L2. The geometry of quantum states – p. 4

Simplest measuring devices define pure states

• Simplest measuring device: A filter checking for |ψ〉.

Πψ = |ψ〉〈ψ|.

• If |ϕ〉 is the initial vector then Πψ|ϕ〉 = Γ|ψ〉, with Γ := 〈ψ|ϕ〉.

The result of testing |ψ〉 on |ϕ〉 is:

⋆ Affirmative if |ϕ〉 ∝ |ψ〉⋆ Negative if |ϕ〉⊥|ψ〉⋆ Uncertain if |ϕ〉 = α|ψ〉+ |η〉, with |η〉⊥|ψ〉 (⇒ α = Γ).

|Γ|2 = |〈ψ|ϕ〉|2 ⇔ “transition probability”

P (ψ) = |Γ|2 = 〈ϕ|ψ〉〈ψ|ϕ〉 = 〈ϕ|Πψ|ϕ〉 ≡ 〈Πψ〉ϕ ≡ 〈Πϕ〉ψL2. The geometry of quantum states – p. 5

Simplest measuring devices define pure states

A “projective measurement” is described by an observable A, with

spectral decomposition (αk ∈ R)

A =∑

k

αkΠαk ⇒ A|αk〉 = αk|αk〉.

Upon measuring the arbitrary state |α〉, the probability P (αk) of gettingresult αk is given by

P (αk) = 〈Παk〉α(⇒ 〈A〉α =

∑

k

αkP (αk)

)

Given the outcome αk occurred, the state of the quantum system

immediately after the measurement is

Παk |α〉√P (αk)

.

L2. The geometry of quantum states – p. 6

Theory of filters

L2. The geometry of quantum states – p. 7

Stern-Gerlach (preparation of pure states)

Assume that Z has the eigenvalues +1 and −1 with eigenvectors |0〉and |1〉. The spectral decomposition of Z is given by

Z =

1∑

k=0

zkΠk = |0〉〈0| − |1〉〈1| =

1 0

0 −1

.

The measurement of Z on the state

|ψ〉 = |0〉+ |1〉√2

gives

+1 with probability 12

−1 with probability 12

L2. The geometry of quantum states – p. 8

Mixed state: insufficient information• Ignorance about which of the mutually exclusive pure states |ψk〉 hasbeen prepared.

• We can only ascribe probabilities Pk ≥ 0,∑

k

Pk = 1, to each |ψk〉.

• At t = t0 the set of kets |ψ1〉, |ψ2〉, . . ., with statistical weightsP1, P2, . . ., represent the quantum state.

• The average value

〈A〉 =∑

k

〈A〉kPk, Pk ≥ 0,∑

k

Pk = 1

can be managed as follows

〈A〉 =∑

k

Pk〈ψk|AI|ψk〉 =∑

k,ℓ

Pk〈ψk|A|ψℓ〉〈ψℓ|ψk〉 =∑

k,ℓ

Pk〈ψℓ|ΠkA|ψℓ〉

=∑

ℓ

〈ψℓ|

[(

∑

k

PkΠk

)

A

]

|ψℓ〉 ≡ Tr

[(

∑

k

PkΠk

)

A

]

L2. The geometry of quantum states – p. 9

Measurements on statistical mixtures

Then 〈A〉 = TrρA, with

ρ :=∑

k

PkΠk, Πk ≡ ρk := |ψk〉〈ψk|, Pk ≥ 0,∑

k

Pk = 1.

• For A = I we get

〈I〉 = Trρ =∑

k

PkTrΠk =∑

k

Pk

(∑

i

〈ψi|ρk|ψi〉)

=∑

k

Pk = 1.

• For any |ψ〉

〈ρ〉ψ = 〈ψ|(∑

k

PkΠk

)|ψ〉 =

∑

k

Pk|〈ψ|ψk〉|2 ≥ 0.

L2. The geometry of quantum states – p. 10

The density operator

• The density operator ρ is Hermitean (H), positive definite (+) and oftrace equal to unity (1).

• Conversely, any (H+1)-ρ̃ must have a spectral decomposition

ρ̃ =∑

k

λkΠk, λk ≥ 0,∑

k

λk = 1.

• Pure state = statistical mixture having |ψ〉 as its sole element.

ρ = Πψ = |ψ〉〈ψ| ⇒ ρ2 = ρ ⇒ Trρ2 = Trρ = 1.

• Mixed state = statistical mixture

ρ =∑

k

PkΠk ⇒ ρ2 =∑

k

P 2kΠk ⇒ Trρ2 =∑

k

P 2 ≤ Trρ

L2. The geometry of quantum states – p. 11

The density operator

Trρ2 = 1 + 2(|ρ12|2 − ρ11ρ22)

ρA =

1 0

0 0

, Trρ2A = 1, ρB =1

2

1 1

1 1

, Trρ2B = 1, ρC =

2 1

1 1

, Trρ2C =7

9

• Since ρ|ψk〉 = Pk|ψk〉, for the von Neumann entropy

S(ρ) = −Trρ log2 ρ ≡ −∑

k

Pk log2 Pk, 0 log2 0 ≡ 0

⋆ Pure state: S(ρ) = −1 · log2(1) = 0

⋆ Maximally mixed: Pk = 1N , so that S(ρ) = log2(N)

L2. The geometry of quantum states – p. 12

Bloch-Poincaré sphere

ρ =I + ~α · ~σ

2, ||~α|| ≤ 1

⋆ ρ is pure iff ||~α|| = 1

⋆ The maximally mixed state ρ = I2

is at origin.L2. The geometry of quantum states – p. 13

Convexity

• A set S is convex if for each pair of points x and y in S, the linesegment joining x and y is in S

αx+ βy ∈ S, α ≥ 0, β ≥ 0, α + β = 1

• Affine combination

y =

N∑

k=1

λkxk,

N∑

k=1

λk = 1, λk ∈ R

L2. The geometry of quantum states – p. 14

Convex Set

• A convex combination is an affine combination such that λk ≥ 0:

ρ :=∑

k

PkΠk,∑

k

Pk = 1, Pk ≥ 0

• THEO. A set S is convex iff every convex combination of points of Slies in S.

• The convex hull of a set S is the intersection of all the convex setswhich contain S.

• THEO. For any set S, the convex hull of S consists precisely of allconvex combinations of elements of S.

L2. The geometry of quantum states – p. 15

Convex polytope and simplex

• THEO. If S is a nonempty subset of Rn then every x in convS can beexpressed as a convex combination of n+ 1 of fewer points of S.

• The convex hull of a finite set of points is called a polytope. IfS = {x1, . . . , xk+1} and DimS = k, then convS is a k-dimensionalsimplex. The points xk are called vertices.

⋆ 0-simplex S0= a single point {x1}⋆ 1-simplex S1= a line segment [x1, x2]

⋆ 2-simplex S2= a triangle (joining x3 /∈ affS1)⋆ 3-simplex S3= a tetrahedron (joining x4 /∈ affS2)⋆ 4-simplex S4= a pentatope (joining x5 /∈ affS3)

L2. The geometry of quantum states – p. 16

Convex polytope and simplex

L2. The geometry of quantum states – p. 17

Entanglement

Entropy of entanglement for a 3D cross section of the 6D manifold ofpure states of two qubits. The hotter the color the more entangled thestate. (Bengtsson and Zyczkowski)

√2|ϕ+〉 = |00〉+ |11〉,

√2|ψ+〉 = |01〉+ |10〉

L2. The geometry of quantum states – p. 18

Discussion

1. How close are two quantum states?

2. The trace distance

D(ρ1, ρ2) =1

2Tr|ρ1 − ρ2|, |A| =

√A†A

is convex.

3. For instance, given two Bloch vectors

ρk =I + ~αk · ~σ

2, k = 1, 2

one notice that the distance between two qubit states is equal to

one half the ordinary Euclidean distance between them on the

Bloch sphere!

D(ρ1, ρ2) =||~α1 − ~α2||

2L2. The geometry of quantum states – p. 19

Discussion

4. The unbiased basis description is intrinsically convex

5. Geometric description of quantum phenomena is free ofinterpretation

6. Study of entanglement, quantum tomography, and quantumdecoherence among others.

L2. The geometry of quantum states – p. 20

Gracias!!

L2. The geometry of quantum states – p. 21

IntroductionIntroductionPure statessmall Simplest measuring devices define pure statessmall Simplest measuring devices define pure statesTheory of filterssmall Stern-Gerlach (preparation of pure states)Mixed state: insufficient informationMeasurements on statistical mixturesThe density operatorThe density operatorBloch-Poincar'e sphereConvexityConvex SetConvex polytope and simplexConvex polytope and simplexEntanglementDiscussionDiscussion