A discrete-time quantum walk taking continuous steps

This note introduces a quantum random walk in R and proves the weak convergence of its rescaled n-step densities. Quantum walks of the type we consider in this note were introduced in [1], which defined and analyzed the Hadamard quantum walk on Z, and a “new type of convergence theorem” for such quantum walks on Z was discovered by Konno [4, 5]. A much simpler proof of Konno’s theorem has recently appeared in [3], allowing the theorem to be generalized to quantum walks in Zd. Inspired by the technique of [3], I have proven that Konno’s theorem also holds for an analog of the quantum walk that takes steps in R d instead of Zd. This is shown below, but first I review the Hadamard quantum walk of [1] in order to motivate the definition of the “Plancherel” quantum walk. The Hadamard random walker steps along the lattice Z, carrying with her a “quantum coin.” Formally, the Walker&Coin state is specified by a unit vector in l2(Z)⊗ C2; the standard basis vectors of the auxilliary “coin space” C2 will be denoted |H〉 for “Heads” and |T 〉 for “Tails.” A complete measurement of the walker’s position would find her at j ∈ Z with probability P (j;ψ) = ∣∣ 〈 (j ⊗H) ∣ψ 〉∣∣ 2 + ∣∣ 〈 (j ⊗ T ) ∣ψ 〉∣∣ 2 (1) if the state of the Walker&Coin is ψ ∈ l2(Z) ⊗ C2. But the walker walks unobserved, and her position will become entangled with the state of her coin. To take a step, the quantum walker flips her coin by a Hadamard transform |H〉 7−→ 1 √ 2 (|H〉+ |T 〉) |T 〉 7−→ 1 √ 2 (|H〉 − |T 〉) (2)