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Efficient Generation of Generic Entanglement

R. Oliveira, O. C. O. Dahlsten, M. B. Plenio

Abstract

We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process generates the average entanglement of the uniform (Haar) measure in at most $O(N^3)$ steps for $N$ qubits. This is despite an exponentially growing number of such gates being necessary for generating that measure fully on the state space. Numerics furthermore show a variation cut-off allowing one to associate a specific time with the achievement of the uniform measure entanglement distribution. Various extensions of this work are discussed. The results are relevant to entanglement theory and to protocols that assume generic entanglement can be achieved efficiently.

Efficient Generation of Generic Entanglement

Abstract

We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process generates the average entanglement of the uniform (Haar) measure in at most steps for qubits. This is despite an exponentially growing number of such gates being necessary for generating that measure fully on the state space. Numerics furthermore show a variation cut-off allowing one to associate a specific time with the achievement of the uniform measure entanglement distribution. Various extensions of this work are discussed. The results are relevant to entanglement theory and to protocols that assume generic entanglement can be achieved efficiently.

Paper Structure

This paper contains 2 theorems, 7 equations, 2 figures.

Key Result

Theorem 1

Suppose that $N_B-N_A=t\geq 0$ and that some arbitrary $\varepsilon\in (0,1)$ is given. Then for a number $n$ of gates in $C_n$ satisfying

Figures (2)

  • Figure 1: Typical numerical simulation using the random circuit. The entanglement average of the uniform measure is reached to an accuracy $\varepsilon$ in $n$ steps. We prove that it suffices with $n=O(N^3)$ to achieve a fixed $arbitrary$$\varepsilon$ accuracy when increasing $N$.
  • Figure 2: Observe a variation Cut-Off of the entanglement probability distribution compared with that of the uniform measure as determined numerically. The state space has been discretised by rounding off entanglement values to the nearest integer. We observe that $TV\simeq 1$ for a while and then falls. Finally there is a stage where $TV\simeq 0$. The effect becomes more pronounced with increasing $N$. The results for $N>8$ are done using the stabilizer random circuit.

Theorems & Definitions (2)

  • Theorem 1
  • Lemma 1