Eutactic quantum codes

The increased experimental feasibility to manipulate single or few particle quantum states, and the theoretical concentration on the algebraic properties of the mathematical models underlying quantum mechanics, have stimulated a wealth of applications in information and computation theory [1,2]. In this line of reasoning, we shall consider quantized systems which are in a coherent superposition of constituent states in such a way that only the coherent superposition of these pure states is in a predefined state; whereas one or all of the constituent states are not. Heuristically speaking, only the coherently combined states yield the “encoded message,” the constituents or “shares” do not. This feature could be compared to “quantum secret sharing” schemes [3–7], as well as to “entangled entanglement” scenarios [8,9]. There, mostly entangled multipartite system are investigated. Thus, while the above cases concentrate mainly on quantum entanglement, in what follows quantum coherence will be utilized: in the secret-sharing scheme proposed here, one party receives part of a quantum state and the other party receives the other part. The parts are components of a vector lying in subspaces of a higher-dimensional Hilbert space. While the possible quantum states to be sent are orthogonal, the parts are not, so that the parties must put their parts together to decipher the message. We shall deal with the general case first and consider examples later. Consider an orthonormal basis E = he1 , . . . ,enj of the n-dimensional real Hilbert space Rn [whose origin is at s0, . . . ,0d]. Every point x in Rn has a coordinate representation xi= kx ueil, i=1, . . . ,n with respect to the basis E. Hence, any vector from the origin v=x has a representation in terms of the basis vectors given by v =oi=1 n kv ueilei=voi=1 n feieig, where the matrix notation has been used, in which ei and v are row vectors and “T” indicates transposition. (k·u · l and the matrix feieig stand for the scalar product and the dyadic product of the vector ei with itself, respectively). Hence, oi=1 n feieig= In, where In is the n-dimensional identity matrix. Next, consider more general, redundant, bases consisting of systems of “well-arranged” linear dependent vectors F = hf1 , . . . , fmj with m.n, which are the orthogonal projections of orthonormal bases of m(i.e., higher-than-n-) dimensional Hilbert spaces. Such systems are often referred to as eutactic stars [10–14]. When properly normed, the sum of the dyadic products of their vectors yields unity, i.e., oi=1 m ffifig= In, giving raise to redundant eutactic coordinates xi8= kx u fil, i=1, . . . ,m.n. Indeed, many properties of operators and tensors defined with respect to standard orthonormal bases directly translate into eutactic coordinates [14]. In terms of m-ary (radix m) measures of quantum information based on state partitions [15], k elementary m-state systems can carry k nits [16–18]. A nit can be encoded by the one-dimensional subspaces of Rm spanned by some orthonormal basis vectors E8= he1 , . . . ,emj. In the quantum logic approach pioneered by Birkhoff and von Neumann (e.g., Refs. [19–22]), every such basis vector corresponds to the physical proposition that “the system is in a particular one of m different states.” All the propositions corresponding to orthogonal base vectors are comeasurable. On the contrary, the propositions corresponding to the eutactic star