Download 4.Operator representations and double phase space

PG4
Substrato Clássico de Emaranhamento e Descoerência Quânticos
Professor
Alfredo Ozorio de Almeida
Ementa
Representações de operadores no espaço de fases
Apresentação do espaço de fases duplo
Traço parcial de estados emaranhados
Emaranhamento clássico
Equação mestra de Lindblad: descoerência, dissipação e difusão
Teoria semiclássica da equação mestra no espaço de fases duplo
1. Operator representations
in phase space
This means that any operator Â
can be considered as
a linear superposition of dyadic operators:
Aˆ   dq dq A(q , q ) q q
because
tr Aˆ q' q'   dq dq A(q , q ) tr q q q' q'
 A(q , q )  q Aˆ q
Which other sets of operators form good bases
for representing arbitrary operators?
Consider the set of unitary translation operators:
so that
or
similarly
translates the momenta:
or
Then the Baker-Hausdorf relation
allows us to define the phase space translation operators:
where
In quantum optics, they are known as displacement operators
and they are defined in terms of lowering and raising operators:
and
The classical translations form a group,
as do the corresponding quantum operators:
1
The cocycle is defined by the area
of the triangle
formed by the pair of vectors 1 and 2 .
2

This generalizes to
where
is the area of
the n-sided polygon:
1

n
The trace of a translation operator is
Suppose then that we can express any operator Â
as a superposition of translations:
Then
A( ) is the chord representation (the chord symbol)
for the operator Â
This is a first phase space representation for operators:
the space of all the translations in phase space.
It is easily verified that
So, the chord symbol is related to the position representation
by a symmetrized Fourier transform:
What is the result of a full Fourier transform
of the translation operators?
This is the operator for a reflection
through the phase space point x.
It is the (Fourier) conjugate of the translation operator,
just as p p is conjugate to q q
In terms of the dyadic position operators:
Again, a symmetrized Fourier transform.
We can verify that R̂x acts as the unitary operator
that reflects states about x = (p, q) directly.
Also, we have the group relations:
These complete the product rules
for the affine group of
translations and reflections
2
ˆ
R
Note that, in particular, x  1̂ ,
so the (degenerate) eigenvalues of R̂x
are +1 and -1.
Therefore, the reflection operators are both unitary and Hermitian.
Can they be used as a basis for the representation of operators?
The assumption that
leads to
This is the Weyl Representation:
A(x) is the Weyl symbol of the operator Â.
Are the reflection operators true observables?
The parity, +1, or -1, around the origin
is an observable wave property.
This is currently measured in quantum optics.
There, the natural basis are the even and odd states
of the Harmonic Oscillator.
For reflections around other centres, x,
translate the whole HO basis, just as the
translation of the ground state generates
coherent states.
The pair of representations based on translation chords
and reflection centres are Fourier conjugates,
just as are the position and the momentum representations.
The Weyl representation is related to the position representation
q Aˆ q by a symmetrized Fourier transform:
Some important properties:
The Weyl or chord symbols for products of operators
are not simple, but
In the case of the density operator, ̂ ,
it is convenient to normalize:
L
Defining the Wigner function and the chord function,
or the quantum characteristic function.
The expectation of any observable results from
the formulae for the trace of products:
But W(x) may take on negative values,
though
L
The chord function behaves like a characteristic function,
in that we obtain moments as derivatives:
The zero’th moment is just the normalization:
Projections of the Wigner function produce
true probability distributions:
Whereas, sections of the chord function are
true characteristic functions:
 dp W (p, q)  Pr(q)
With a little algebra we then find that
for density operators with even, or odd symmetry
about the origin,
Thus, all Wigner functions for symmetric density operators
attain their largest amplitude at the symmetry point,
but this is negative for odd symmetry.
More on reflection symmetry…
,
;
a comment on probabilities:
x, such that Pˆx ˆ  ˆ, whereas the invariance of the density operator with respect to Pˆx
Examples:
All cases related to HO with L=1 and unit mass
These are the only cases of positive Wigner functions
for pure states:
CLASSICAL?
two classical Gaussians with an interference pattern
centred on their midpoint. The spatial frequency
increases with the separation:
the same configuration is now reinterpreted:
both the classical Gaussians interfere around the origin,
while their cross-correlation generates the pair of peaks at
an odd cat
Is this a Wigner or a chord function?
Note that the amplitude is larger at the symmetry centre
than on the classical peaks.
…
(
)
The symmetry centre where the Wigner function is maximal
is nowhere near the classical energy curve,
This is a good parity basis for a pure state,
Wigner function for
an eigenstate of
Comment on the Husimi function:
This is most appropriate for quantum chaos,
because it highlights the classical region.
But this coarse-graining of the quantum interferences
is not an advantage for quantum information theory:
The opposite of the chord function.
These antithetical representations are both
intimately related to the translation operators,
since the Husimi function can be rewritten, for
:
Wigner
Husimi
All these are examples of pure states:
How can we know that a given Wigner function,
or chord function represents a pure state?
Let us study the effect of translations on the density operator,
The sensitivity of a state
to translations determines
its phase space correlations:
In particular,
which is the highest value attained by this correlation function,
The existence of two alternative expressions for
the phase space correlations of pure states leads to
Fourier Invariance:
Grosso modo, this remarkable property implies
that large scales of a pure state chord function
must be accompanied by very fine structure
of its oscillations.
This is also true of Wigner functions:
Large structures are accompanied by
“subplanckian structures”
that are eroded by decoherence.
2. Operator representations
in double phase space
Note similarity between dyadic basis,
and product state basis,
Then, natural to relate double Hilbert space
to double phase space:
The operator
corresponds to the
Lagrangian plane, Q = constant,
in double phase space. But adapt coordinates:
Justification in classical mechanics:
Canonical transformations,
x
A Lagrangian surface
in the double phase space:




Q  ( q , q )
Define:
C : x  x  (q , p )
P  (  p , p ) 
  (  ,   )

p  dq   p  dq  0
_
x
 P  dQ  0

If both surfaces are tori,
if L=1, a 2-D product torus,
but with
If each Lagrangian surface in single phase space corresponds to a state,
Just like product states: projects as a rectangle onto P, or Q.
i. Note that the Lagrangian surface is not a product.
ii. Note that projections onto P and Q may be singular.
S(Q) is the generating function of a canonical transformation:
For symplectic transformations (linear canonical),
S(Q) is quadratic and the semiclassical propagator is exact.
Legendre transforms create new generating functions:
Nontrivial change of coordinates in double phase space:
x  ( p, q ) 
x  x_
2
y  J   J ( x  x )
0  1
J 

1 0 
New Lagrangian coordinate planes
correspond to unitary operators:
y  0  I
  0  T
(identity operator)
(phase space translation)
Phase space translations form a group.
x  x
x  x  
Exact correspondence to quantum operators:
The transformation from horizontal to the vertical basis
is given by the full Fourier transform ( as with states):
The Reflection Operator
Now represent arbitrary operators in terms of
reflection centres or translation chords:
THE WEYL AND THE CHORD REPRESENTATION
Again, we use half the coordinates of double phase space,
inside a Lagrangian plane that is a phase space on its own.
(Balazs and Jennings + geometry)
In contrast, the Q space or the P space are not phase spaces,
but they are conjugate double phase space planes,
just as x and y.
Semiclassical form of representations of unitary operators
in terms of centres or chords is the same as for other
Lagrangian planes.
The Weyl representation is a superposition of
For a symplectic transformation there is only one branch
of y(x) and the semiclassical form is exact.
In general there may be caustics,
where the Lagrangian surface projects singularly
onto the y=0 plane. (the identity plane)
Turning on a Hamiltonian for a small time:
No caustics !
p
Recall WKB theory
for states in Hilbert space:
p (q )
qc

q   det
q
Action: S  p (q )
q
1
2
eiS ( q ) 
 is an invariant coordinate
within the surface
q
Caustics:

det

q
Change of coordinates (Maslov):

p   det
p
1
2
eiS ( p ) 
S
with
 q( p)
p
Quantum evolution is generated by the classical evolution
In the semiclassical approximation
Evolution of a vertical plane;
classical trajectories lead to the Hamilton-Jacobi equation:
S
S
( q, t )  H ( , t )
t
q
p
p
q
q
q




Dyadic operators,
also have a WKB approximation:
q 


 q



i  


 a (q ) a (q ) exp S (q )  S (q )



 A(Q)e

 
iS ( Q ) 
The apropriate Lagrangian surface in double phase space
is defined by
S
P(Q) 
Q
In particular, WKB for the density operator:
 

The Weyl representation of the Hamiltonian, or any smooth
observable, coincides with the corresponding classical function
within first order in Planck’s constant.
(Not so with the reflection operator)
Their chord representation is not smooth:
Metaplectic transformations:
(unitary transformations, corresponding to linear canonical
(symplectic) transformations)
Thus, both the centre and the chord representations
are invariant with respect to metaplectic transformations,
because
Evolução de operadores unitários
corresponde classicamente a visualizar
uma transformação canônica clássica, C ,
com coordenadas que evoluem continuamente:
Kt : x0  xt 
 C ' (t ) : x  x  Kt  C  K  t ( x )
Mecânica quântica:
Cˆ (t )  Kˆ t Cˆ Kˆ t
Propagação não-linear de uma reflexão:
visão no espaço de fases simples

 (t )
Propagação não-linear de uma reflexão:
visão no espaço de fases duplo
É a evolução de
um plano inicial
3. The partial trace
I2 A 
2
The different forms of the partial trace
depend on the Hilbert –Schmidt product
of each basis with the identity.
For the position basis:
Recall the matrix notation:
A(Q1 , (q, q))  q1 , (q2  q) A q1 , (q2  q)
PRODUCT STATES
Cross correlations imply entanglement
It is more usual to measure the entanglement by the purity
of just one of the partial traces:
Why is this the same measure as for subsystem-2 ?
Use Fourier invariance of quantum correlations:
Equality is not expected for the second moment of marginal distributions.
PROJECTION OF WIGNER FUNCTIONS
Quantum tomography is the converse of the property
that provides a probability density along any Lagrangian plane
as a projection of the Wigner function:
It is possible to reconstruct this representation
from the full set of probability densities.