QUANTUM COMPUTING - Bio-Inspired and Nanoscale Integrated Computing - page 73

unitary transformations and so concludes that I/O bandwidth cannot be sig-
nificantly reduced by such quantum methods for sensing. (Also see Holevo [97],
Fuchs and Caves [98] for proof that quantum methods cannot increase the
bandwidth for transmission of classical information.)

3.5. QUANTUM COMPRESSION

As noted above, quantum methods cannot increase the bandwidth for transmis-
sion of classical information; still, in certain cases, entangled states can be
compressed to fewer qubits. This quantum compression could have important
applications in practice, where the number of usable qubits is very limited.
Schumacher [99] considered compression and decompression of a noiseless
source of n quantum bits (qubits), each sampled independently from a given
mixed state quantum ensemble. For such a quantum source, the compression
factor obtainable by classical information theory is limited by the Shannon
entropy, which in general (except in the case where the quantum ensemble
has only orthogonal states) is less than the quantum compression factor given
by the von Neumann entropy. In particular, Schumacher [99] proved a quantum
noiseless coding theorem that states that the source's von Neumann entropy is the
number of qubits per source state which is necessary and sufficient to asympto-
tically (in the limit of large code-block size) encode the output of the source with
arbitrarily high fidelity. The quantum noiseless coding of Schumacher has
asymptotically optimal fidelity and size; the resulting compressed number of
qubits can be far fewer than in the classical case.

S HANNON E NTROPY AND THE L IMITATIONS OF C LASSICAL M ETHODS FOR
N OISELESS C OMPRESSION . Suppose n characters from a finite alphabet S are each
sampled independently over

some probability distribution p. In classical

information theory, the Shannon entropy of each character is H S ð p Þ¼

P a 2S p ð a Þ log p ð a Þ . A string of these n bits may be compressed without loss to
a bit string of mean length H S (p)n.

T HE VON N EUMANN ENTROPY AND Q UANTUM N OISELESS C OMPRESSION . Follow-
ing Schumacher [99], we assume there is a finite quantum state ensemble (S, p)
which is a mixed state consisting of a finite number of qubit states
S ¼fj a 0 i; ...; j a j S j 1 ig , where each j a i i2 S has probability p i . The compressor
is assumed to act on blocks of n qubits (so is a block compressor), and is assumed
to know this underlying ensemble (S, p). The density matrix of (S, p)isan j S jj S j

matrix r ¼ P j S j 1

i ¼ 0 p i j a i ih a i j:

The von Neumann entropy (see [99, 100]) correspond-
ing to (S, p)isH VN ðrÞ¼ Tr ðr log rÞ:

In general, the Shannon entropy H S (p)is
greater than or equal to the von Neumann entropy. These entropies are equal only
when the states in S are mutually orthogonal.