System-time entanglement in a discrete-time model

We present a model of discrete quantum evolution based on quantum correlations between the evolving system and a reference quantum clock system. A quantum circuit for the model is provided, which in the case of a constant Hamiltonian is able to represent the evolution over $2^n$ time steps in terms of just $n$ time qubits and $n$ control gates. We then introduce the concept of system-time entanglement as a measure of distinguishable quantum evolution, based on the entanglement between the system and the reference clock. This quantity vanishes for stationary states and is maximum for systems jumping onto a new orthogonal state at each time step. In the case of a constant Hamiltonian leading to a cyclic evolution it is a measure of the spread over distinct energy eigenstates and satisfies an entropic energy-time uncertainty relation. The evolution of mixed states is also examined. Analytical expressions for the basic case of a qubit clock, as well as for the continuous limit in the evolution between two states, are provided.

Figure 1

Schematic circuit representing the generation of the system-time pure state, (1). The control gate performs the operation $U_t$ on $S$ if $T$ is in state $|t\rangle$, while the Hadamard-type gate $H$ creates the superposition $\propto {\sum }_{t=0}^{N-1}|t\rangle$.

Figure 2

Circuit representing the generation of system-time state (1) for $U_t={\left(U\right)}^t$ and $N=2^n$. The $n$ control gates perform the operation $U^t=U^{{\sum }_{j=1}^n{t}_j{2}^{j-1}}$ on the system after writing $t$ in the binary form $t={\sum }_{j=1}^{n-1}{t}_j{2}^{j-1}$, while the $n$ Hadamard gates lead to a coherent sum over all values of the $t_j$'s, i.e., over all $t$'s from 0 to $2^n-1$.