Quantum History Cannot Be Copied

We show that unitarity does not allow cloning of any two points in a ray. This has implication for cloning of the geometric phase information in a quantum state. In particular, the quantum history which is encoded in the geometric phase during cyclic evolution of a quantum system cannot be copied. We also prove that the generalized geometric phase information cannot be copied by a unitary operation. We argue that our result also holds in the consistent history formulation of quantum mechanics. In quantum theory state of a single quantum is represented by not just a vector |ψ〉 in a separable Hilbert space H, but by a ray in the ray space R. A ray is a set of equivalence classes of states that differ from each other by complex numbers of unit modulus. Thus the ray space R is defined as R = {|ψ〉 : |ψ〉 ∼ |ψ〉 = c|ψ〉}, where c ∈ C is a group of non-zero complex numbers and |c| = 1. Given a quantum state |ψ〉 we can generate a ray by the application of the ‘ray operator’ R(c) = exp[iArg (c)|ψ〉〈ψ|] = I + (c− 1)|ψ〉〈ψ| such that R(c)|ψ〉 = |ψ〉. Geometrically, we can represent all these equivalent classes of states as points in a ray and all of them represent the same physical state. The set of rays of the Hilbert space H is called the projective Hilbert space P = H/U(1). If we have a continuous unitary time evolution of a quantum system |ψ〉 → U(t)|ψ〉 then the evolution can be represented as an open curve Γ : t → |ψ(t)〉 in R whose projection in P is also an open curve Γ̂. The quantum state at different times can belong to different rays. If the quantum state at two different times belongs to the same ray, then it may trace an open curve C in R, but its projection in P is a closed curve Ĉ. Such an evolution is called a cyclic evolution. In quantum information theory we view a quantum state |ψ〉 as the carrier of both classical and quantum information. The fundamental unit of classical information is a bit and that of quantum information is a qubit. Classical bit can be copied but quantum bit or qubit cannot be copied. It is the linearity of quantum theory that does not allow us to produce a copy of an arbitrary quantum state [1,2]. Using unitarity one can prove that two non-orthogonal states cannot be copied either [3]. However, orthogonal quantum states like ∗Email: [email protected]