Defense Intelligence Reference Document Quantum Tomography Of Negative Energy States In The Vacuum

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directly observe quantum states, and so the true nature of an individual quantum
system is hidden. However, no principal obstacle exists to observing all complementary
aspects in a series of distinct experiments on identically prepared quantum objects.
In the sections that follow, we briefly review the several parts that comprise the
tomography machinery, and then put the whole picture together to understand what
the entire process is. No effort will be made for completeness because the subject of
quantum tomography takes up volumes of books. The reader will be referred to the
key literature of importance.
Wigner Functions
In classical optics the state of an electromagnetic oscillator is perfectly described by the
statistics of the classical amplitude a. The amplitude may be completely fixed (then
the field is coherent), or a may fluctuate (then the field is partially coherent or
incoherent). In classical optics as well as in classical mechanics, we can characterize
the statistics of the complex amplitude a or, equivalently, the statistics of the
component position q and momentum p by introducing a phase space distribution called
the Wigner function, W(q.p).Hi W(q,p) quantifies the probability of finding a particular
pair of q and p values in their simultaneous measurement. Knowing W(q,p) for a
particular quantum state that is under study, all statistical quantities of the
electromagnetic oscillator can be predicted by calculation. In this sense W(q,p)
describes the state in classical physics. The motivation for introducing the Wigner
function was the desire to find a quantum mechanical description similar to that in
classical statistical physics. However, in quantum mechanics Heisenberg's Uncertainty
Principle prevents one from observing position and momentum simultaneously and
precisely. In addition to this, we also cannot directly observe quantum states either.
Nevertheless, we are perfectly entitled to use the concept of quantum states as if they
were existing entities. We use their properties to predict the statistics of observations.
It is well known that the quantum mechanical wave function depends exclusively on
either the position or the momentum and contains nevertheless all the information
about the quantum system under study. However, E. Wigner showed that it is possible
to define a formal quantum mechanical analog to the classical distribution function. He
showed that we could use W(q,p) as a quantum phase space distribution exclusively to
calculate observables in a classical-like fashion. Wigner discovered that W(q,p) is a
real-valued function, but it is usually not just positive; it can also become negative.
This is a very nonclassical behavior for a probability distribution. It is for this reason
that W(q,p) came to be called a quasiprobability distribution.
W(q,p) has several properties and mathematical postulates, but it turns out that just
one postulate is sufficient for the purposes of quantum tomography [38]. Using this
postulate, it is assumed that W(q,p) behaves like a joint probability distribution for q
and p without ever mentioning any simultaneous observation of position and
momentum. The reduced, or marginal, distributions r: W(q, p)dp or r: W(q, p)dq
''' Recall in Sect. IIB-1 that the real and the imaginary parts of the complex amplitude r, can be regarded as the
position and the momentum of the electromagnetic oscillator.
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