Information Technology Reference
In-Depth Information
An Example: Consider a slightly more complex example of a source consisting
of a sequence of n photons polarized randomly, with equal probability of phase 0
or phase angle
y
(e.g., As a very simple example of a source with low von
Neumann entropy, consider N photons polarized randomly, equiprobably at 0
or 1.) In this case, the states are S
¼fj
a
0
i; j
a
1
ig;
where the first state
j
a
0
i¼j
0
i;
corresponds to phase 0, and the other state
j
a
1
i¼
cos
yj
0
iþ
sin
yj
1
i
corresponds to
phase angle
y
, and the probabilities are both p
ð
0
Þ¼
p
ð
1
Þ¼
1
2
. The density matrix
1
1
1
is
r ¼
2
j
a
0
ih
a
0
jþ
2
j
a
1
ih
a
1
j¼
2
ðj
0
ih
0
jþð
cos
yj
0
iþ
sin
yj
1
iÞð
cos
yh
0
jþ
sin
yh
1
jÞÞ¼
2
ðð
1
þ
cos
2
yÞj
0
ih
0
jþ
cos
y
sin
yj
0
ih
1
jþ
cos
y
sin
yj
1
ih
0
jþ
sin
2
yj
1
ih
1
jÞ
which has
1
2
2matrixform
2
1
þ
cos
2
y
over the basis vector
:
cos
y
sin
y
j
0
i
j
1
i
Then we can
find an appropriate
b
which gives a change of basis with new basis states
j
0
0
i¼j
0
i
and
cos
y
sin
y
sin
2
y
j
1
0
i¼
cos
bj
0
iþ
sin
bj
1
i;
providing
a diagonal density matrix
r
0
¼
over the basis vector
:
ð
1
þ
cos
2
yÞþ
cos
y
sin
y
tan
b
j
0
0
i
j
1
0
i
0
1
2
Although
cos
y
sin
yþ
sin
2
y
tan
b
0
this source has high Shannon entropy H
S
(p), it will have low von Neumann entropy
H
VN
(
r
) in the case of a small magnitude phase angle
y
However, note that the
entropies are the same in the special case where
y
=
p
/2, so the states
j
a
0
i¼j
0
i
,
j
a
1
i¼j
1
i
are orthogonal and the density matrix is simply the diagonal matrix
r ¼
over the
1
2
0
1
2
ðj
0
ih
0
jþj
1
ih
1
jÞ
which has a diagonal density matrix
r ¼
1
2
0
.
j
0
i
j
1
i
basis vector
For technical reasons, the unitary compression and decompression mappings
need to preserve the number of bits (some of which are ignored). An n-to-n
0
quantum compressor is a unitary transformation that maps n-qubit strings to
n-qubit strings; the first n
0
qubits that are output by the compressor are taken as
the compressed version of its input, and the remaining n
n
0
qubits are discarded.
An n
0
-to-n decompressor is a unitary transformation that maps n-qubit strings to
n-qubit strings; the first n
0
qubits input to the decompressor are the compressed
version of the uncompressed n qubits, and the remaining n
n
0
qubits are all 0.
The source to the compression scheme is assumed to be a sequence of n qubits
sampled independently from (S, p). The observed output is the result of first
compressing the input qubits, then decompressing them, and measuring the result
(over a basis containing the n inputs). The compression rate is n/n
0
and the
compression factor is n
0
/n. The fidelity of the compression scheme is the probability
that the observed output is equal to the original input (that is, the probability
that the original qubits are correctly recovered from the compressed qubits). The
goal here is a quantum compression with both a high fidelity and a high
compression rate.
Example (Continued): Consider again the example of a source consisting of a
sequence of n photons polarized randomly, with equal probability of phase 0 or
phase angle
y
.If
y
has small magnitude, then a quantum encoder can compress
these photons into an entangled state using just a few photons. Furthermore, a
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