The formal concept of information is tied to physics, at least as far as applicability is a concern. There seems to be one issue, which, despite notable exceptions (e.g., [11999Zeilinger,Footnote 6] and [22000Muthukrishnan and C. R. Stroud]), has not yet been acknowledged widely: the principal irreducibility of quantum information in base n. Define a ``nit'' as a unit of information equal to the amount of information obtained by learning which of n equally likely events occurred. An n-state particle can be prepared in a single one of n possible states. Then, this particle carries one nit of information, namely to ``be in a single one from n different states.'' Subsequent measurements may confirm this statement. The most natural code basis for such a configuration is n, and not a binary one.
Classically, there is no preferred code basis whatsoever. Every classical state is postulated to be determined by a point in phase space. Formally, this amounts to an infinite amount of information in whatever base, since with probability one, all points are random; i.e., algorithmically incompressible [31994Calude]. Operationally, only a finite amount of classical information is accessible. Yet, the particular base in which this finite amount of classical information is represented is purely conventional. The same holds true for discrete classical systems, such as n modes of vibration on a string, where the restriction to these particular states is rather arbitrary.
The fundamental difference between classical and quantum information with respect to code bases could be illustrated by the following example. Physically, each nit could be represented by an n-level system. A single measurement collapses an n-state superposition and yields only one output, not log2 n outputs. In the nonentangled k particle case, the k mutually commuting observables could be some physical quantity (e.g., energy levels) associated with each particle. This sets the stage for the more general observables associated with ``entangled'' states. References [11999Zeilinger] and [42002Donath and Svozil] discuss examples with Bell states and Greenberger-Horne-Zeilinger states for the binary case, respectively.
In what follows, let us always consider a complete system of base states B associated with a unique ``context'' or ``communication frame'' F={F1,F2, ¼,Fk }, which corresponds to co-measurable observables with n outcomes. For n=2, their explicit form has been enumerated in [42002Donath and Svozil]. In this particular case, the F's can be identified with certain projection operators from the set of all possible mutually orthogonal ones, whose two eigenvalues can be identified with the two states. For three or more particles, this is no longer possible.
It should be emphasized that only the case of an entanglement between different particles but not within each particle is considered. If more than one observable could be associated with each particle, then these can become entangled as well, and then k n-ary observables will no longer be sufficient to describe k particles.
For a single n-state particle, the nit can be formalized as a state partition which is fine grained into n elements, one state per element. That is, if the set of states is represented by {1,¼,n}, then the nit is defined by {{1} ,¼,{n}}. Of course, any labeling would suffice, as long as the structure is preserved. It does not matter whether one calls the states, for instance, ``+,'' ``0'' and ``-'', or ``1,'' ``2'' and ``3'', resulting in a trit represented by {{+} ,{0} ,{-}} or {{1} ,{2} ,{3}} (here, the term ``trit'' stands for a nit with n=3). Thus, nits are defined modulo isomorphisms (i.e., one-to-one translations) of the state labels. To complete the setup of the single particle case, let us recall that any such state set would correspond to an orthonormal basis of n-dimensional Hilbert space.
Before proceeding to the most general case,
we shall consider the case of two particles
with three states per particle in all details.
We shall adopt an n-ary search strategy.
Assume that the first and the second particle
has three orthogonal states labeled by
a1,b1,c1
and
a2,b2,c2,
respectively.
Then nine product states can be formed and labeled from 1 to 9 in
lexicographic order; i.e.,
a1a2 º 1, ¼, c1c2 º 9.
Consider a set of two
comeasurable three-valued observables inducing two state partitions
of the set of states S={1,2,¼, 9} with three partition
elements with the properties
that (i) the set theoretic intersection of any two elements of the two
partitions is a single state, and (ii) the union of all these nine
intersections is just the set of state S.
As can be easily checked, an example for such state partitions are
| (1) |
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A Hilbert space representation of this setting can be obtained as
follows.
Define the states in S to be one-dimensional linear subspaces of
nine-dimensional Hilbert space; e.g.,
1 º (1,0,0,0,0,0,0,0,0),¼,9 º (0,0,0,0,0,0,0,0,1).
The trit operators are given by (the terms ``trit operator,'' ``observable,'' and the
corresponding state partition will be used synonymously)
| (2) |
If F2 = diag (d,e,f,d,e,f,d,e,f)
and a,b,c,d,e,f, are six different prime numbers,
then, due to the uniqueness of prime decompositions,
the two trit operators
can be combined to a single
``context'' operator
| (3) |
A generalization to k particles in n states per particle is straightforward. We obtain k partitions of the product states with n elements per partition in such a way that every single product state is obtained by the set theoretic intersection of k elements of all the different partitions.
Every single such partition can be interpreted as a nit. All such sets are generated by permuting the set of states, which amounts to nk! equivalent sets of state partitions. However, since they are mere one-to-one translations, they represent the same nits. This equivalence, however, does not concern the property of (non)entanglement, since the permutations take entangled states into nonentangled ones. We shall give an example below.
Again, the standard orthonormal basis of
nk-dimensional Hilbert space is identified with the set of states S={1,2,¼,nk}; i.e.,
(superscript ``T'' indicates transposition)
| (4) |
The nit operators are defined via diagonal matrices
which contain equal amounts nk-1 of mutually n different numbers
such as different primes q1,¼,qn; i.e.,
| (5) |
There exist nk! sets of nit operators, which are are obtained by forming a (nk ×nk)-matrix whose rows are the diagonal components of F1,¼,Fk from Eq. (5) and permuting all the columns. The resulting new operators F1¢,¼,Fk¢ are also nit operators.
All partitions discussed so far are equally weighted and well balanced, as all elements of them contain an equal number of states. In principle, one could also consider nonbalanced partitions. For example, one could take the partition [`F]1={{1},{2,3},{4,5,6,7,8,9}} instead of F1 in (1), represented the by trit diagonal operator diag (a,b,b,c,c,c,c,c,c). Yet any such attempt would result in a deviation from the optimal n-ary search strategy, and in an nonoptimal measurement procedures. Another, more principal, disadvantage would be the fact that such a state separation could not reflect the inevitable n-arity of the quantum choice.
In terms of partitions, entanglement occurs for diagonal
or antidiagonal arrangements of states
which do not add up to completed blocks.
Take, for example, the state partition scheme of Fig. 1,
which results in nonentangled states and state measurements.
A modified, entangled scheme can be established by just grouping the states
into diagonal and counterdiagonal groups as drawn in Fig.
2.
The corresponding trits are
| (6) |
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We can now introduce new 2×3 basis vectors grouped into the two bases
{a1¢,b1¢,c1¢}
and
{a2¢,b2¢,c2¢} by
| (7) |
In summary we have shown that, by adopting an n-ary search strategy, k particles (entangled or not) specify k nits in such a way that k mutually commuting measurements of independent observables with n outcomes are necessary and sufficient to determine the information. This finding is compatible to Zeilinger's foundational principle for quantum mechanics [11999Zeilinger]. In general, the main emphasis in the area of quantum computation has been in the area of binary decision problems. It is suggested that these investigations should be extended to decision problems with n alternatives (e.g., [51952Kleene,pp. 332-340]), for which quantum information theory seems to be extraordinarily well equipped.