An Epistemic Measurement System for Quantum Security

We develop a formal system to reason about knowledge properties of quantum security protocols. The formalism is obtained via a marriage of measurement calculus [3], an algebraic framework for measurementbased quantum computing [7], with the algebra of epistemic actions and their appearance maps [1, 8]. Measurement calculus has also proven to be a proper language to describe and to analyse distributed quantum protocols [4]. Protocols (here referred to as measurement pattern) are described by a combination of commands: 1-qubit preparations Ni (prepares qubit i in state |+〉i), 2-qubit entanglement operators Eij := ∧Zij (controlled-Z operator), 1-qubit measurementsMα i , and 1-qubit Pauli corrections Xi and Zi, where i, j represent the qubits on which each of these operations apply, and α ∈ [0, 2π). Measurement Mα i is defined by orthogonal projections P |+α〉 i (with outcome si = 0) and P |−α〉 i (with outcome si = 1). 1 Dependent corrections, used to control non-determinism, will be written Xj i and Z sj i , with X 0 i = Z 0 i = I , X1 i = Xi, and Z 1 i = Zi. Any pattern can be put in a standard form, where all the preparation and entanglement can be done first, followed by local measurements and corrections and classical communications. The initial entanglement state is the distributed global memory shared among the agents at the beginning of the protocol and the classical outcome of measurements represents the classical communication of agents. The starting point is our main result that proves the well-defined measurement patterns with flow [2] form a quantale Q. Recall that a quantale is a sup-monoid (Q,≤, ∨ , •, ) where in our case the monoid multiplication is the sequential composition (or juxtaposition) of measurement pattern commands. In the fragment of measurement patterns with flow, probabilities of each branch of measurement are equal, as a result a measurement can be written as the non-deterministic choice of its projections, that is Mα i = P |+α〉 i ∨ P |−α〉 i . The induced order is the non-deterministic order of information between each projection of the measurement, that is P |+α〉 i ≤ Mα i and P |−α〉 i ≤ Mα i . We add agents A ∈ A to our quantale by endowing it with a family of lax quantale endomorphisms f A : Q → Q, one for each agent A ∈ A. We interpret f A (q) as appearance of agent A about action q, that is all the actions that agent A considers as happening when action q is happening in reality. Since f A preserves all joins, it has a Galois right adjoints that preserves all meets. The adjunction is denoted by f A a 2 Q A and we read 2 Q A q as ‘agent A knows that action q is happening’. These maps and their relation to the traditional notions of knowledge and belief in epistemic logic have been studied in [1, 8]. Each action has an owner that generates the action, we encode the owner as the map gen : Q → A and whenever gen(q) = Q, we use the shorthand qA. For example, the generator of an entanglement action is the source that creates the entangled state. In this case, we distinguish the agents that share the state from the source via the shorthand E i,j where inv(E C,A,B i,j ) = (A,B), gen(E C,A,B i,j ) = C and inv : Q → A×A is defined partially on the entanglement actions. We use these maps to assign appearances to action for each agent. For instance, all actions appear as identity to their generators, that is fA(M α,A i ) = M α,A i . We assume that the right module M of our quantale Q is the lattice of results of measurements. The action of the quantale on the module−·− : M ×Q→M stands for the change of the state of a system as a