Download L2. The geometry of quantum states

L2. The geometry of quantum states
Oscar 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
|Φi =
X
k
ck |φk i
provides a complete description of the quantum system.
Simplest case:
|Φi = a|0i + b|1i,
2
2
|a| + |b| = 1,
|0i =
1
0
,
|1i =
0
1
.
L2. The geometry of quantum states – p. 4
Simplest measuring devices define pure states
• Simplest measuring device: A filter checking for |ψi.
Πψ = |ψihψ|.
• If |ϕi is the initial vector then Πψ |ϕi = Γ|ψi, with Γ := hψ|ϕi.
The result of testing |ψi on |ϕi is:
⋆ Affirmative if |ϕi ∝ |ψi
⋆ Negative if |ϕi⊥|ψi
⋆ Uncertain if |ϕi = α|ψi + |ηi, with |ηi⊥|ψi (⇒ α = Γ).
|Γ|2 = |hψ|ϕi|2
⇔
“transition probability”
P (ψ) = |Γ|2 = hϕ|ψihψ|ϕi = hϕ|Πψ |ϕi ≡ hΠψ iϕ ≡ hΠϕ iψ
L2. 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)
X
A=
αk Παk ⇒ A|αk i = αk |αk i.
k
Upon measuring the arbitrary state |αi, the probability P (αk ) of getting
result αk is given by
!
X
P (αk ) = hΠαk iα
⇒ hAiα =
αk P (αk )
k
Given the outcome αk occurred, the state of the quantum system
immediately after the measurement is
Παk |αi
p
.
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 |0i
and |1i. The spectral decomposition of Z is given by


1
X
1 0

.
Z=
zk Πk = |0ih0| − |1ih1| =
0 −1
k=0
The measurement of Z on the state
|0i + |1i
|ψi = √
2
gives


 +1 with probability

 −1 with probability
1
2
1
2
L2. The geometry of quantum states – p. 8
Mixed state: insufficient information
• Ignorance about which of the mutually exclusive pure states |ψk i has
been prepared.
X
• We can only ascribe probabilities Pk ≥ 0,
Pk = 1, to each |ψk i.
k
• At t = t0 the set of kets |ψ1 i, |ψ2 i, . . ., with statistical weights
P1 , P2 , . . ., represent the quantum state.
• The average value
hAi =
X
k
hAik Pk ,
X
Pk ≥ 0,
Pk = 1
k
can be managed as follows
hAi
=
X
Pk hψk |AI|ψk i =
k
X
=
hψℓ |
ℓ
X
Pk hψk |A|ψℓ ihψℓ |ψk i =
k,ℓ
"
X
k
Pk Π k
! #
A |ψℓ i ≡ Tr
X
Pk hψℓ |Πk A|ψℓ i
k,ℓ
"
X
k
Pk Π k
! #
A
L2. The geometry of quantum states – p. 9
Measurements on statistical mixtures
Then hAi = TrρA, with
X
ρ :=
Pk Πk ,
Πk ≡ ρk := |ψk ihψk |,
k
• For A = I we get
hIi = Trρ =
X
Pk TrΠk =
k
X
k
• For any |ψi
hρiψ = hψ|
Pk
X
k
Pk Πk
!
Pk ≥ 0,
!
X
hψi |ρk |ψi i
i
|ψi =
X
k
X
Pk = 1.
X
Pk = 1.
k
=
k
Pk |hψ|ψk i|2 ≥ 0.
L2. The geometry of quantum states – p. 10
The density operator
• The density operator ρ is Hermitean (H), positive definite (+) and of
trace equal to unity (1).
• Conversely, any (H+1)-e
ρ must have a spectral decomposition
X
X
ρe =
λk Πk ,
λk ≥ 0,
λk = 1.
k
k
• Pure state = statistical mixture having |ψi as its sole element.
ρ = Πψ = |ψihψ|
ρ2 = ρ
⇒
• Mixed state = statistical mixture
X
X
2
ρ=
Pk Πk ⇒ ρ =
Pk2 Πk
k
k
⇒
⇒
Trρ2 = Trρ = 1.
2
Trρ =
X
k
P 2 ≤ Trρ
L2. The geometry of quantum states – p. 11
The density operator
Trρ2 = 1 + 2(|ρ12 |2 − ρ11 ρ22 )
ρA

=
1
0


1
 , Trρ2A = 1, ρB = 1 
2
1
0
0
1
1


 , Trρ2B = 1, ρC = 
2
1
1

 , Trρ2C = 7
9
1
• Since ρ|ψk i = Pk |ψk i, for the von Neumann entropy
X
S(ρ) = −Trρ log2 ρ ≡ −
Pk log2 Pk , 0 log2 0 ≡ 0
k
⋆ Pure state: S(ρ) = −1 · log2 (1) = 0
⋆ Maximally mixed: Pk =
1
N,
so that S(ρ) = log2 (N )
L2. The geometry of quantum states – p. 12
Bloch-Poincaré sphere
I +α
~ · ~σ
ρ=
,
2
||~
α|| ≤ 1
⋆ ρ is pure iff ||~
α|| = 1
⋆ The maximally mixed state ρ =
I
2
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 line
segment joining x and y is in S
αx + βy ∈ S,
α ≥ 0, β ≥ 0,
α+β =1
• Affine combination
y=
N
X
k=1
λk x k ,
N
X
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 :=
X
k
Pk Πk ,
X
k
Pk = 1,
Pk ≥ 0
• THEO. A set S is convex iff every convex combination of points of S
lies in S.
• The convex hull of a set S is the intersection of all the convex sets
which contain S.
• THEO. For any set S, the convex hull of S consists precisely of all
convex 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 be
expressed 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. If
S = {x1 , . . . , xk+1 } and DimS = k, then convS is a k-dimensional
simplex. The points xk are called vertices.
⋆ 0-simplex S 0 = a single point {x1 }
⋆ 1-simplex S 1 = a line segment [x1 , x2 ]
⋆ 2-simplex S 2 = a triangle (joining x3 ∈
/ affS 1 )
⋆ 3-simplex S 3 = a tetrahedron (joining x4 ∈
/ affS 2 )
⋆ 4-simplex S 4 = a pentatope (joining x5 ∈
/ affS 3 )
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 of
pure states of two qubits. The hotter the color the more entangled the
state. (Bengtsson and Zyczkowski)
√ +
2|ϕ i = |00i + |11i,
√ +
2|ψ i = |01i + |10i
L2. The geometry of quantum states – p. 18
Discussion
1. How close are two quantum states?
2. The trace distance
1
D(ρ1 , ρ2 ) = Tr|ρ1 − ρ2 |,
2
√
|A| = A† A
is convex.
3. For instance, given two Bloch vectors
I +α
~ k · ~σ
,
ρk =
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!
||~
α1 − α
~ 2 ||
D(ρ1 , ρ2 ) =
2
L2. The geometry of quantum states – p. 19
Discussion
4. The unbiased basis description is intrinsically convex
5. Geometric description of quantum phenomena is free of
interpretation
6. Study of entanglement, quantum tomography, and quantum
decoherence among others.
L2. The geometry of quantum states – p. 20
Gracias!!
L2. The geometry of quantum states – p. 21