Unitary evolution and the distinguishability of quantum states

The study of quantum systems evolving from initial states to distinguishable, orthogonal final states is important for information processing applications such as quantum computing and quantum metrology. However, for most unitary evolutions and initial states the system does not evolve to an orthogonal quantum state. Here, we ask what proportion of quantum states evolve to nearly orthogonal systems as a function of the dimensionality of the Hilbert space of the system, and numerically study the evolution of quantum states in low-dimensional Hilbert spaces. We find that, as well as the speed of dynamical evolution, the level of maximum distinguishability depends critically on the Hamiltonian of the system.

Figure 1

The probability that a randomly chosen state $|\psi \rangle$ evolves to a state $\left|\psi \right(t)\rangle$ with maximum distinguishability $D\ge 1-\epsilon$. The solid curve is the theoretical value of Eq. (9), while the triangles are obtained by a Monte Carlo approach.

Figure 2

Relative populations of different maximum distinguishability $D$ for representative dimensions (logarithmic scale) of systems with a harmonic Hamiltonian. The figure shows the ratio ${log}_{10}P\left(D\right)/{log}_{10}{P}_{\mathrm{total}}$, where $P\left(D\right)$ is the population in bin $D$ and $P_{\mathrm{total}}$ is the total population. The bin size $\Delta D$ is $0.01$. Note the inflection point occurring at $N=4$.

Figure 3

The probability of picking a state with maximum distinguishability $D\ge 1-\epsilon$ as a function of $\epsilon$ for different dimensions $N$ with a harmonic Hamiltonian.

Figure 4

Lowest common multiple of the set $\{1^2,2^2,...,N^2\}$ (logarithmic scale, base 10).

Figure 5

Relative populations of different maximum distinguishability $D$ for representative dimensions (logarithmic scale) of systems with an atomic Hamiltonian. The figure shows the ratio ${log}_{10}P\left(D\right)/{log}_{10}{P}_{\mathrm{total}}$, where $P\left(D\right)$ is the population in bin $D$ and $P_{\mathrm{total}}$ is the total population. The bin size $\Delta D$ is $0.01$. The inflection point occurs again at $N=4$, but states achieve near-orthogonality significantly faster compared to harmonic systems.

Figure 6

The probability of picking a state with maximum distinguishability $D\ge 1-\epsilon$ as a function of $\epsilon$ for different dimensions $N$.

Figure 7

The average maximum distinguishability $\langle D\rangle$ as a function of the dimensionality of the harmonic (left plot) and atomic (right plot) systems, respectively. Note how $\langle D\rangle$ approaches 1 faster for atomic systems, but the fluctuations in $D$ are also larger for atomic systems compared to harmonic systems. The dashed horizontal line is there to assist the eye in the comparison of the two graphs.