dq.measurements_sse_to_sme

measurements_sse_to_sme(
    measurements: ArrayLike, tsave: ArrayLike, etas: ArrayLike, key: PRNGKeyArray
) -> Array

Post-process measurements from a diffusive SSE simulation to obtain measurements for a diffusive SME simulation.

The SME measurements are obtained from the SSE measurements by adding additional Gaussian noise.

Note

More precisely, the SME measurement record \(\dd Y\) for a specific jump operator is defined as follows from the SSE measurement record \(\dd \tilde Y\): $$ \dd Y = \sqrt{\eta}\,\dd \tilde Y + \sqrt{1-\eta}\,\dd W $$ where \(\dd W\) is another independent Wiener process sampled for the post-processing.

The SME time-averaged measurement \(I(t_n, t_{n+1})\) is then defined as follows from the SSE time-averaged measurements \(\tilde I(t_n, t_{n+1})\): $$ I(t_n, t_{n+1}) = \frac{1}{\Delta t_n}\int_{t_n}^{t_{n+1}} \dd Y(t) = \sqrt{\eta}\, \tilde I(t_n, t_{n+1}) + \sqrt{1-\eta}\,\frac{\Delta W}{\sqrt{\Delta t_n}}, $$ where \(\Delta t_n=t_{n+1}-t_n\) and \(\Delta W\sim \mathcal{N}(0, 1)\) is a standard Gaussian random variable.

Parameters

• measurements –

  See result of dq.dssesolve().

• tsave –

  See dq.dssesolve().

• etas –

  See dq.dsmesolve().

• key –

  PRNG key used to sample the added noise for the post-processing.

Returns

(array of shape (...)) SME measurements. The shape is the same as measurements, except that the dimension corresponding to the number of jump operators measured may be smaller, if the corresponding efficiency is null.