Timeline for Fisher information of parametric channel
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 16 at 15:07 | vote | accept | Quantastic | ||
| Jan 15 at 11:04 | comment | added | Quantastic | @QuantumMechanic I don't mind the dependency on $\rho$. I meant that I'm looking for some explicit expression, if it exists, involving operations on $u_\theta$ and $\rho$ as matrices. | |
| Jan 15 at 2:33 | answer | added | Quantum Mechanic | timeline score: 2 | |
| Jan 15 at 1:36 | comment | added | Quantum Mechanic | @Quantastic it will always depend on the initial state $\rho$: imagine starting with the vacuum, then $(\rho\otimes \tau)$ is an eigenstate of $U_\theta$ (because $U$ conserves excitation number for the two modes due to the $u_\theta$ that you've given), then $\partial_\theta \rho_\theta=0$, while other states have nonzero derivatives for the same $U$. As for the inequalities, those are just other properties of QFI from other formulations, of which there are a whole bunch | |
| Jan 14 at 23:15 | comment | added | Quantastic | @QuantumMechanic We can assume $\rho_S$ to be pure and just take $\tau_E$ as the empty ket. I'm by any means not an expert in quantum metrology; can you tell me where that inequality is coming from? Also, I don't suppose it can be restated as something that only involves the matrix $u_\theta$? | |
| Jan 14 at 23:10 | comment | added | Quantastic | @flippiefanus Yes, $\rho_\theta = \Phi_\theta(\rho)$. | |
| Jan 14 at 18:13 | comment | added | Quantum Mechanic | Some possible simplifications: we know $U_\theta=\exp(i \mathbf{r}_\theta\cdot \mathbf{G})$ for the vector of generators $\mathbf{G}$ of SU($2n$). If $\mathbf{r}_\theta=\theta\mathbf{n}$ for fixed $\mathbf{n}$, as is often the case, the QFI will be $\leq$ Var${}_{\rho_\theta}(\mathbf{n}\cdot\mathbf{G})$. Alternatively, if $\rho_S$ and $\tau_E$ are pure, then we know QFI equals the max QFI over purifications of $\rho_\theta$, one of which is the pure state $U_\theta(\rho\otimes \tau)U_\theta^\dagger$, so the QFI $\leq$ (and maybe saturates) Var${}_{\rho\otimes \tau} (-i dU/d\theta U^\dagger)$ | |
| Jan 14 at 18:02 | comment | added | Quantum Mechanic | To rephrase: this is a bosonic system subject to a (linear optics) unitary transformation and loss; is there a simple expression for QFI in that case? | |
| Jan 14 at 18:00 | comment | added | Quantum Mechanic | @flippiefanus $\rho_\theta=\Phi_\theta(\rho)$ I presume | |
| Jan 14 at 3:15 | comment | added | flippiefanus | Can you perhaps provide a definition for $\rho_{\theta}$? | |
| Jan 13 at 18:46 | history | asked | Quantastic | CC BY-SA 4.0 |