We consider the problem of minimizing the ancillary systems required to realize an arbitrary strategy of a quantum protocol, with the assistance of classical memory. For this purpose we introduce the notion of memory cost of a strategy to measure the resources required in terms of ancillary dimension. We provide a condition for the cost to be equal to a given value, and we use this result to evaluate the cost in some special cases. As an example, we show that any covariant protocol for the cloning of a unitary transformation requires at most one ancillary qubit. We also prove that the memory cost has to be determined globally and cannot be calculated by optimizing the resources independently at each step of the strategy.
Since quantum measurement is universal for quantum computation (Nielsen [3]), the minimization of the resources required for measurement‐based quantum computation is a crucial point ([3, 1, 4, 5, 6]). We have shown in [6] that a set of observables composed of one two‐qubit measurement and three one‐qubit measurements form a universal set for quantum computation if one ancillary qubit is available as an additional resource. This set is minimal in terms of two‐qubit measurements, because at least one multi‐qubit operation is needed to create entanglement, and is also minimal in terms of ancillary qubits, because at least one ancillary qubit is required to simulate reversible transformations with projective measurements only. We discuss in this paper the required number of one‐qubit measurements for a set of observables to be universal. A set composed of one two‐qubit measurement and only two one‐qubit measurements is proved universal, at the cost of a second ancillary qubit. Thus an additional ancillary qubit seems to be required to obtain a smaller universal set of measurements. One may conclude to a trade off between ancillary qubits and observables in measurement‐based quantum computation.
We addressed the question of optimality of private quantum channels. We have shown that the Shannon entropy of the classical key necessary to securely transfer the quantum information is lower bounded by the entropy exchange of the private quantum channel
E and von Neumann entropy of the ciphertext state
ϱ(0). Based on these bounds we have shown that decomposition of private quantum channels into orthogonal unitaries (if exists) is optimizing the entropy. For non-ancillary single qubit PQC we have derived the optimal entropy for arbitrary set of plaintexts. In particular, we have shown that except when the (closure of the) set of plaintexts contains all states, one bit key is sufficient. We characterized and analyzed all the possible single qubit private quantum channels for arbitrary set of plaintexts. For the set of plaintexts consisting of all qubit states we have characterized all possible approximate private quantum channels and we have derived the relation between the security parameter and the corresponding minimal entropy.