Phase-random states: Ensembles of states with fixed amplitudes and uniformly distributed phases in a fixed basis

Motivated by studies of typical properties of quantum states in statistical mechanics, we introduce phase-random states, an ensemble of pure states with fixed amplitudes and uniformly distributed phases in a fixed basis. We first give a sufficient condition for canonical states to typically appear in subsystems of phase-random states, which reveals a trade-off relation between the initial state in the bounded energy subspace and the energy eigenstates that define that subspace. We then investigate the simulatability of phase-random states, which is directly related to that of time evolution in closed systems, by studying their entanglement properties. We find that, starting from a separable state, time evolutions under Hamiltonians composed of only separable eigenstates generate extremely high entanglement and are difficult to simulate with matrix-product states. We also show that random quantum circuits consisting of only two-qubit diagonal unitaries can generate an ensemble with the same average entanglement as phase-random states.

Figure 1

The distributions for $N=8$ of the amount of entanglement for two ensembles: (a) random states and (b) phase-random states with equal amplitudes and a separable basis, using the Meyer-Wallach measure of entanglement, $E_{\mathrm{MW}}\left(\right|\phi \rangle ):=(2/N){\sum }_{k=1}^N{E}_L^{\left(\right\{k\left\}\right)}\left(\right|\phi \rangle )$, where $k$ labels single-qubit subsystems [11]. The number of samples is ${10}^4$, binned in intervals of $0.002$.

Figure 2

Phase-random circuit composed of two-qubit unitaries $W_t(i_t,j_t)$ acting on randomly selected pairs of qubits $(i_t,j_t)$.

Figure 3

An example of some irreducible sets of $\mathcal{M}$ when $N=3$. Directed lines imply that the transition occurs with a fixed probability. The probability of blue (dotted), red (solid), green (dashed), and purple (dash-dotted) lines is $1/6$, $1/12$, $1/3$, and 1, respectively. Elements such as 000, $z00$, and so on are invariant under the Markov process.

Figure 4

Graph of the Markov chain ${\tilde{\mathcal{M}}}_{\Gamma }$. Colored lines with arrows imply that the transition occurs with a fixed probability. Transition probabilities are given in Proposition 3.