##### This material has been published in
IEEE
Trans. Information Theory
**46** (2000), 801-819,
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## Christian Krattenthaler and Paul Slater

# Asymptotic Redundancies for Universal
Quantum Coding

### (19 pages)

**Abstract.**
Clarke and Barron have recently shown that the Jeffreys' invariant
prior of Bayesian theory yields the common asymptotic
(minimax and maximin) redundancy of universal data compression
in a parametric setting. We seek a possible analogue
of this result for the two-level *quantum* systems.
We restrict our considerations to
prior
probability distributions belonging
to a certain
one-parameter family, *q*_{u}, *-\infty *< *u* < 1.
Within this setting, we
are able to compute exact redundancy formulas, for which we find
the asymptotic limits.
We compare our
quantum asymptotic redundancy formulas to those derived by
naively applying the (non-quantum) counterparts
of Clarke and Barron, and find certain common features.
Our results are based on formulas we obtain for the
eigenvalues and eigenvectors of
2^{n} x 2^{n} (Bayesian
density) matrices, *\zeta*_{n}(*u*).
These matrices are the weighted averages (with respect to *q*_{u})
of all possible tensor products of *n* identical 2 x 2 density matrices,
representing the two-level quantum systems.
We propose a form of *universal* coding
for the situation in which
the density matrix describing an ensemble of quantum signal states
is unknown. A sequence of *n* signals would be
projected onto the dominant eigenspaces of *\zeta*_{n}(*u*).

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