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 can be considered as a linear superposition of dyadic operators: because

Which other sets of operators form good bases for representing arbitrary operators? so that or Consider the set of unitary translation operators: 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: The cocycle is defined by the area of the triangle formed by the pair of vectors and. This generalizes to where is the area of the n-sided polygon:

Suppose then that we can express any operator as a superposition of translations: The trace of a translation operator is Then

is the chord representation (the chord symbol) for the operator This is a first phase space representation for operators: the space of all the translations in phase space. So, the chord symbol is related to the position representation by a symmetrized Fourier transform: It is easily verified that

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 is conjugate to In terms of the dyadic position operators: Again, a symmetrized Fourier transform.

We can verify that 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

Note that, in particular,, so the (degenerate) eigenvalues of 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 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: LL The expectation of any observable results from the formulae for the trace of products: But W(x) may take on negative values, though Defining the Wigner function and the chord function, or the quantum characteristic function.

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:

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, whereas the invariance of the density operator with respect to

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

Is this a Wigner or a chord function? Note that the amplitude is larger at the symmetry centre than on the classical peaks. an odd cat

…

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 :

Canonical transformations, Define: Justification in classical mechanics: A Lagrangian surface in the double phase space:

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: New Lagrangian coordinate planes correspond to unitary operators: (identity operator) (phase space translation) Phase space translations form a group.

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 !

Action: is an invariant coordinate within the surface Recall WKB theory for states in Hilbert space:

Caustics: Change of coordinates (Maslov): with 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:

Dyadic operators, also have a WKB approximation: The apropriate Lagrangian surface in double phase space is defined by 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: Mecânica quântica:

Propagação não-linear de uma reflexão: visão no espaço de fases simples

É a evolução de um plano inicial Propagação não-linear de uma reflexão: visão no espaço de fases duplo

3. The partial trace

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:

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.