On the convex characterisation of the set of unital quantum channels

In this paper, we consider the convex set of $d$ dimensional unital quantum channels. In particular, we parametrise a family of maps and through this parametrisation we provide a partial characterisation of the set of unital quantum maps with respect to this family of channels. For the case of qutrit channels, we consider the extreme points of the set and their classification with respect to the Kraus rank. In this setting, we see that the parametrised family of maps corresponds to maps with Kraus rank three. Furthermore, we introduce a novel family of qutrit unital quantum channels with Kraus rank four to consider the extreme points of the set over all possible Kraus ranks. We construct explicit examples of these two families of channels and we consider the question of whether these channels correspond to extreme points of the set of quantum unital channels. Finally, we demonstrate how well-known channels relate to the examples presented.Comment: 18 page

On a generalized quantum SWAP gate

The SWAP gate plays a central role in network designs for qubit quantum computation. However, there has been a view to generalize qubit quantum computing to higher dimensional quantum systems. In this paper we construct a generalized SWAP gate using only instances of the generalized controlled-NOT gate to cyclically permute the states of d qudits for d prime

Towards an optimal swap gate

We present a novel approach that generalizes the well known quantum SWAP gate to higher dimensions and construct a regular quantum gate composed entirely in terms of the generalized CNOT gate that cyclically permutes the states of d qudits for d prime. We also investigate the case for d other than prime. A key feature of the construction design relates to the periodicity evaluation for a family of linear recurrences which we achieve by exploiting generating functions and their factorization over the complex reals

On swapping the states of two qudits

The SWAP gate has become an integral feature of quantum circuit architectures and is designed to permute the states of two qubits through the use of the well-known controlled-NOT gate. We consider the question of whether a two-qudit quantum circuit composed entirely from instances of the generalised controlled-NOT gate can be constructed to permute the states of two qudits. Arguing via the signature of a permutation, we demonstrate the impossibility of such circuits for dimensions $d \equiv 3$ (mod 4)

Weak randomness completely trounces the security of QKD

In usual security proofs of quantum protocols the adversary (Eve) is expected to have full control over any quantum communication between any communicating parties (Alice and Bob). Eve is also expected to have full access to an authenticated classical channel between Alice and Bob. Unconditional security against any attack by Eve can be proved even in the realistic setting of device and channel imperfection. In this Letter we show that the security of QKD protocols is ruined if one allows Eve to possess a very limited access to the random sources used by Alice. Such knowledge should always be expected in realistic experimental conditions via different side channels