Gluing Randomness via Entanglement: Tight Bound from Second Rényi Entropy

Abstract

The efficient generation of random quantum states is a long-standing challenge, motivated by their diverse applications in quantum information processing tasks. In this work, we identify entanglement as the key resource that enables local random unitaries to generate global random states by effectively gluing randomness across the system. Specifically, we demonstrate that approximate random states can be produced from an entangled state $|ψ\rangle$ through the application of local random unitaries. We show that the resulting ensemble forms an approximate state design with an error saturating as $Θ(e^{-\mathcal{N}_2(ψ)})$, where $\mathcal{N}_2(ψ)$ is the second Rényi entanglement entropy of $|ψ\rangle$. Furthermore, we prove that this tight bound also applies to the second Rényi entropy of coherence when the ensemble is constructed using coherence-free operations. These results imply that, when restricted to resource-free gates, the quality of the generated random states is determined entirely by the resource content of the initial state. Notably, we find that among all $α$-Rényi entropeis, the second Rényi entropy yields the tightest bounds. Consequently, these second Rényi entropies can be interpreted as the maximal capacities for generating randomness using resource-free operations. Finally, moving beyond approximate state designs, we utilize this entanglement-assisted gluing mechanism to present a novel method for generating pseudorandom states in multipartite systems from a locally entangled state via pseudorandom unitaries in each of parties.

Figures (3)

• Figure 1: Approximate designs from entangled states and local operations. (a) Alice (left) and Bob (right) initially share a state $\ket{\phi}$ having a non-vanishing second Rényi entanglement entropy. They apply local random unitary operators to scramble their systems. (b) Global random states are generated by random circuits. States generated in (a) global approximate random states in (b). (c,d) Alice (left) and Bob (mid) as well as Bob and Charlie (right) initially share states $\ket{\phi_{AB}}$ and $\ket{\phi_{BC}}$ or the GHZ state. These initial states have non-trivial second Rényi entanglement entropies. They then apply local random unitary operators and generate approximate random states.
• Figure 2: Gluing local randomness. (a) Small approximate unitary designs $U_{AB}$ and $U_{BC}$ glue other approximate unitary designs $U_A$, $U_B$, and $U_C$ to give a global approximate unitary design. (B) Bell pairs glue local approximate unitary designs $U_A$, $U_B$, and $U_C$ and enable the generation of a global approximate state design by a single layer of local random unitaries.
• Figure 3: Quantifying maximal achievable randomness. (a,b) $\mathcal{N}$, $\mathcal{M}$, and $\mathcal{C}$ represent amounts of entanglement, magic, and coherence in arbitrary fixed measures. $M_\psi$ is the set of states that can be constructed by applying entanglement-free operators $U_A\otimes U_B$ to $\ket{\psi}$. $\rho^{(t)}_\mathrm{Haar}$ is the $t$-th moments of the ensemble of Haar random states. (a) The ensemble $\mathcal{E}_\mathrm{ent.}(\psi)$ of states in $M_\psi$ with the uniform distribution according to the Haar measure has the minimum deviation $\epsilon_\mathrm{min}$ from $\rho^{(t)}_\mathrm{Haar}$. States in this ensemble typically have near maximal magic and coherence. (b) The ensemble of states in $M_\psi$ with a non-uniform distribution has a larger deviation $\epsilon_>$ from $\rho^{(t)}_\mathrm{Haar}$, i.e., $\|\epsilon_>\|_1\geq\|\epsilon_\mathrm{min}\|_1$.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Theorem 5
• proof
• Theorem 6
• Theorem 7: Informal
• Theorem 8
• Theorem 9