Unitary Operators Over Quantum Systems with Several Levels

High-dimensional quantum states generalise multi-valued logics. The analogous of Pauli transformations acting on these quantum states determine a subgroup structure in U(n), which acts over the maximally entangled Bell states. These properties are suitable to produce superdense coding communication protocols.


Introduction
Qubits are realised as points in the unit sphere of the 2-dimensional complex Hilbert space. When considering k-truth values, or equivalently, quantum states with k levels or k observable values, any superposition of them is a point in the unit sphere of the k-dimensional complex Hilbert space, in this sense it represents a higher-dimensional quantum state [1].
We analyse several generalisations of Pauli transformations and maximally entangled orthonormal bases in the corresponding Hilbert spaces and the generated subgroups structures generated in the corresponding symmetry groups U(k) and SU(k). The maximally entangled orthonormal bases are generalisations of Bell basis [2,3]. Our goal is to synthesize some natural algebraic relations between the operator groups and maximally entangled bases. These notions play essential roles in the development of superdense coding and teleportation. We will use the notation [[i, j]] to denote the set of integers {i, i + 1, . . . , j − 1, j}.

Case of two qubits
Let H 1 = C 2 be the 2-dimensional complex Hilbert space and let H 2 = H 1 ⊗ H 1 be its tensor power of exponent 2. We consider the vectors displayed at Table 1 where e ij = e i ⊗ e j is the ij-th canonical vector of H 2 .

Case of three qubits
where L (b) is the one-dimensional space, or ray, spanned by vector b, and , be the "controlled-not" map, and define recursively, For each integer n ≥ 1, CN n is a balanced map, and consequently card CN −1 n (0) = 2 n = card CN −1 n (1).
One can also see that, for n = 4, in H 4 :

Several levels quantum states
Let k ≥ 2 be a positive integer, which counts the number of truth values, or quantum levels. Let ρ k = e i 2π k be the primitive k-th root of unity. We denote by H (k) 1 = C k the k-dimensional complex Hilbert space, and, again, by e 0 , . . ., e k−1 the vectors in its canonical basis.

Case of two quantum states
We will refer to the elements of the unit sphere in H where, as usual, δ ij is Kroenecker's delta. Then,  Table 4. For n = 4, each "signed" Bell vector appears just once, but each direction appears 4 times.
be the matrix representing the linear map given as the "rotation" of the canonical basis e ν → e (ν+1) mod k . Then, from relation (4) we may see that Besides, Consequently, U 10 C p k = ρ p k C p k U 10 and U q 10 C p k = ρ q k C p k U q 10 , which implies ∀m, n, p, q : U mn U pq = ρ nq k U (m+p) mod k,(n+q) mod k .
The relation (5) determines a subgroup structure, in U(k), generated by the set U k = (U mn ) mn∈[[0,k−1]] 2 , with unit U 00 = 1 k (see (4)). This last class of maps is indeed a subset of SU(k) when k is odd. Thus, and ∀m, n, p, q :

Case of k quantum states
Let k ≥ 2 be a positive integer, and let H is an orthonormal basis of H (k) k . As in eq. (6), we have which gives an equivalent form of eq. (7) to recover a map 1 k ⊗ k−1 =1 U p q transforming a given register in the Bell basis into another one in the same basis.
However, at present case there are k 2 k−1 = k 2k−2 maps of the form 1 k ⊗ k−1 =1 U p q and there are k k registers in the Bell basis. One can see that for any two Bell registers b Indeed, for any index sequence n 0 n 1 · · · n k−1 ∈ [[0, k − 1]] (k) let for any two maps n 0 n 1 ···n k−1 .
Then, according to (8), Hence, the index of the equivalence relation "∼ n 0 n 1 ···n k−1 " is k k , the cardinality of the Bell basis. Thus, by fixing canonical representatives in each equivalence class, for instance, the first map 1 k ⊗ k−1 =1 U p q according to the lexical ordering of the index sequence p 1 q 1 · · · p k−1 q k−1 , superdense coding can be performed univocally.

Discussion
Superdense coding was originally presented in the literature in the 2-dimensional case. Modeled with k quantum levels, superdense coding allows the communication of 2 log 2 k classical bits among two parts with the transmission of just one k-level quantum state. Departing from the protocol presented in [4], we generalise it to many participants, in a purely algebraic way: a central part, Alice, is receiving independent pieces of information, from several correspondent participants, and we remark direct dependencies in order to recover the unitary operations performed by each participant when codifying his information pieces. Alice is able to receive k log 2 k classical bits by receiving (k − 1) k-level quantum states, each from each correspondent. Although the generalisation of the proposed multipart superdense protocol enables univocity in the decoding process for the case of k = 2, the univocity is lost for k > 2. However an equivalence relation is well determined among the generalised operators, thus, by fixing a notion of canonical representatives, superdense can be fixed as well.
In [5] a nonoblivious encoding is proposed in order to produce superdense coding using twodimensional qubits. In [6] a superdense coding is proposed by mixing GHZ-quantum states and Bell states. In [7,8] a mixture of the so called W -states and Bell states is used to produce a form of superdense coding. Instead, in our approach we consider just direct generalisations of Bell states.

Conclusions
We have analised the subgroup structure by the operators analogous of Pauli operators for kquantum levels states. The main interest in this study consists in the algebraic analysis of the involved operator compositions.