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quantum

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state is a superposition of the possibilities:
A(x,y) x,y
Pxy | i
The description of the two particles is much larger than the description of one particle;
it is a function in twice the number of dimensions. This is also true in probability, when
the statistics of two random things are correlated. If two particles are uncorrelated, the
probability distribution for their joint position P(x,y) is a product of the probability of
finding one at one position and the other at the other position:
P(x,y) = P (x)P (y)
x y
In quantum mechanics, two particles can be in special states where the amplitudes of
their position are uncorrelated. For quantum amplitudes, the word entanglement replaces
the word correlation, but the analogy is exact. A disentangled wavefunction has the form:
A(x,y) = ψ (x)ψ (y)
x y
while an entangled wavefunction does not have this form. Like correlation in probability,
there are many more entangled states than disentangled ones. For instance, when two
particles which start out with an equal amplitude to be anywhere in a box have a strong
attraction and a way to dissipate energy, they can easily come together to make a bound
state. The bound state still has an equal probability to be anywhere, so that each particle is
equally likely to beeverywhere, but the two particles will become entangled so that wherever
one particle is, the other is too.
C. Entanglement
Quantum entanglement, also called the quantum non-local connection, is a property of a
quantum mechanical state of a system of two or more objects in which the quantum states of
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the constituting objects are linked together so that one object can no longer be adequately
described without full mention of its counterpart - even if the individual objects are spatially
separated in a spacelike manner. The property of entanglement was understood in the early
days of quantum theory, although not by that name. Quantum entanglement is at the
heart of the EPR paradox developed in 1935. This interconnection leads to non-classical
correlations between observable physical properties of remote systems, often referred to as
nonlocal correlations.
Quantum mechanics holds that observable, for example, spin are indeterminate until
such time as some physical intervention is made to measure the observable of the object in
question. In the singlet state of two spins it is equally likely that any given particle will be
observed to be spin-up as that it will be spin-down. Measuring any number of particles will
result in an unpredictable series of measures that will tend more and more closely to half up
and half down. However, if this experiment is done with entangled particles the results are
quite different. For example, when two members of an entangled pair are measured, their
spin measurement results will be correlated. Two (out of infinitely many) possibilities are
that the spins will be found to always have opposite spins (in the spin anti-correlated case),
or that they will always have the same spin (in the spin correlated case). Measuring one
member of the pair therefore tells you what spin the other member would have if it were
also measured. The distance between the two particles is irrelevant.
Theories involving ’hidden variables’ have been proposed in order to explain this result;
these hidden variables account for the spin of each particle, and are determined when the
entangled pair is created. It may appear then that the hidden variables must be in com-
munication no matter how far apart the particles are, that the hidden variable describing
one particle must be able to change instantly when the other is measured. If the hidden
variables stop interacting when they are far apart, the statistics of multiple measurements
must obey an inequality (called Bell’s inequality), which is, however, violated - both by
quantum mechanical theory and in experiments.
Whenpairs ofparticles aregeneratedby the decay of other particles, naturally or through
induced collision, these pairsmay be termed "entangled", in that such pairs oftennecessarily
have linked and opposite qualities, i.e. of spin or charge. The assumption that measurement
in effect "creates" the state of the measured quality goes back to the arguments of, among
others: Schrödinger, andEinstein, Podolsky, andRosen concerning Heisenberg’s uncertainty
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principle and its relation to observation (see also the Copenhagen interpretation). The
analysis of entangled particles by means of Bell’s theorem, can lead to an impression of
non-locality (that is, that there exists a connection between the members of such a pair that
defies both classical and relativistic concepts of space and time). This is reasonable if it is
assumed that each particle departs the scene of the pair’s creation in an ambiguous state
(as per a possible interpretation of Heisenberg). In such a case, for a given measurement
either outcome remains a possibility; only measurement itself would precipitate a distinct
value. On the other hand, if each particle departs the scene of its "entangled creation" with
properties that would unambiguously determine the value of the quality to be subsequently
measured, then a postulated instantaneous transmission of information across space and
time would not be required to account for the result. The Bohm interpretation postulates
that a guide wave exists connecting what are perceived as individual particles such that the
supposed hidden variables are actually the particles themselves existing as functions of that
wave.
Observation of wavefunction collapse can lead to the impression that measurements per-
formed on one system instantaneously influence other systems entangled with the measured
system, even when far apart. Yet another interpretation of this phenomenon is that quan-
tumentanglement doesnot necessarily enablethetransmission ofclassical informationfaster
than the speed of light because a classical information channel is required to complete the
process.
D. Hadamard Transform
The Hadamard transform (also known as the Walsh-Hadamard transform, Hadamard-
Rademacher-Walsh transform, Walsh transform, or Walsh-Fourier transform) is an ex-
ample of a generalized class of Fourier transforms. It is named for the French mathe-
matician Jacques Solomon Hadamard, the German-American mathematician Hans Adolph
Rademacher, and the American mathematician Joseph Leonard Walsh. It performs an
orthogonal, symmetric, involutional, linear operation on 2m real numbers (or complex num-
bers, although the Hadamard matrices themselves are purely real).
The Hadamard transform can be regarded as being built out of size-2 discrete Fourier
transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size 2 2
× ×···×
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2 2. It decomposes an arbitrary input vector into a superposition of Walsh functions.
×
The Hadamard transform Hm is a 2m 2m matrix, the Hadamard matrix (scaled by a
×
normalization factor), that transforms 2m real numbers xn into 2m real numbers Xk. The
Hadamard transformcan be defined in two ways: recursively, or by using the binary (base-2)
representation of the indices n and k.
Recursively, we define the 1 1 Hadamard transform H0 by the identity H0 = 1, and