Calculating trace distances of bosonic states in Krylov subspace

Abstract

Continuous-variable quantum systems are central to quantum technologies, with Gaussian states playing a key role due to their broad applicability and simple description via first and second moments. Distinguishing Gaussian states requires computing their trace distance, but no analytical formula exists for general states, and numerical evaluation is difficult due to the exponential cost of representing infinite-dimensional operators. We introduce an efficient numerical method to compute the trace distance between a pure and a mixed Gaussian state, based on a generalized Lanczos algorithm that avoids explicit matrix representations and uses only moment information. The technique extends to non-Gaussian states expressible as linear combinations of Gaussian states. We also show how it can yield lower bounds on the trace distance between mixed Gaussian states, offering a practical tool for state certification and learning in continuous-variable quantum systems.

Figures (4)

• Figure 1: (Top) Trace distance between a single-mode squashed statemartinez2023classical, $\bm{r} = \bm{0}$ and $\bm{V} = \frac{\hbar}{2}(1001 + 4\bar{n})$, and the vacuum state, $\bm{r} = \bm{0}$ and $\bm{V} = \frac{\hbar}{2}\mathbb{I}_2$, as a function of the mean number of photons, $\bar{n}$. (Bottom) Trace distance between the $(M=10)$-mode pure Gaussian state parametrized by $\bm{r}_1=\bm{0}$ and $\bm{V}_1=\frac{\hbar}{2}(e^{-2s}\mathbb{I}_M)\oplus(e^{2s}\mathbb{I}_M)$, and the mixed state represented by $\bm{r}_2=\bm{0}$ and $\bm{V}_2=\frac{\hbar}{2}([(1-\eta)e^{-2s}+\eta]\mathbb{I}_M)\oplus([(1-\eta)e^{2s}+\eta]\mathbb{I}_M)$. Results are presented as a function of the loss parameter $\eta\in[0.0,1.0]$, for $s=0.5$. In both figures, the estimations running Lanczos algorithm for $\ell = 10$ steps are shown in yellow circles. In the single-mode case, results using the diagonalization of $|\psi\rangle\langle\psi|-\hat{\varrho}$, in Fock basis with cutoff $c=100$, are represented by a solid black line. Several bounds on the trace distance weedbrook2012gaussian, including the variational bound derived in the Supplemental Material, are shown in dotted lines.
• Figure 2: Trace distance between the single-mode cat states$|\mathcal{C}_{\pm}^{(p)}\rangle = (\sqrt{N_{\pm}})^{-1/2}\sum_{j=0}^{p-1}(\pm 1)^{j}|\alpha e^{i2\pi j/p}\rangle$, and the mixed states $\hat{\varrho}_\eta^{(\pm,\,p)}=\sum_{j,k=0}^{p-1}b_{j,k}^{\pm}|\sqrt{1-\eta}\alpha e^{i2\pi j/p}\rangle\langle \sqrt{1-\eta}\alpha e^{i2\pi k/p}|$, with $b_{j,k}^{\pm} = (\pm1)^{j+k}e^{-\eta|\alpha|^2}\exp(\eta|\alpha|^2e^{i2\pi (j-k)/p})/N_{\pm}$, which are obtained after sending the cat states through a loss channel, $\mathcal{L}_\eta[\cdot]$, with loss parameter $\eta$, i.e., $\hat{\varrho}_\eta^{(\pm,\,p)}=\mathcal{L}_{\eta}[|\mathcal{C}_{\pm}^{(p)}\rangle\langle\mathcal{C}_{\pm}^{(p)}|]$. Here, $|\alpha\rangle$ is a coherent state, $\bm{r}=\sqrt{2\hbar}(\mathrm{Re}(\alpha), \mathrm{Im}(\alpha))$ and $\bm{V}=\frac{\hbar}{2}\mathbb{I}_2$, and $N_{\pm}$ are normalization constants. The estimated results are shown as a function of $\eta\in[0.0, 1.0]$ for $\alpha = 2.0$, and $p\in\{2, 4, 6, 8\}$. Solid lines correspond to the diagonalization of $\hat{\varrho}_0^{(\pm,\,p)}-\hat{\varrho}_\eta^{(\pm,\,p)}$ in Fock basis with cutoff $c=100$. Circles correspond to an estimation running Lanczos algorithm for $\ell = 10$ steps.
• Figure 3: Illustration of the lower bounds on the trace distance obtained by extending Lanczos algorithm to the case in which both Gaussian states are mixed. (Top) Trace distance between the states $\hat{\varrho}_{\pm}=\mathcal{L}_\eta[\hat{D}(\pm\alpha)\hat{S}(r)|0\rangle\langle 0|\hat{S}^\dagger(r)\hat{D}^\dagger(\pm\alpha)]$, where $\mathcal{L}_\eta[\cdot]$ is a loss channel with transmission $1-\eta$, $\hat{D}(\alpha)$ is a displacement operator, and $\hat{S}(r)$ is a squeezing operator. The parametrization of these Gaussian states is $\bm{r}_{\pm}=\pm\sqrt{2\hbar(1-\eta)}(\mathrm{Re}(\alpha), \mathrm{Im}(\alpha))$ and $\bm{V}_{\pm} = \frac{\hbar}{2}(1-\eta)(e^{-2r}00e^{2r})+\frac{\hbar}{2}\eta\mathbb{I}_2$. The estimated results are shown as a function of $\eta\in[0.5, 1.0]$ for $\alpha = 0.8$, and $r\in\{0.05, 0.3, 1.5\}$ (or, equivalently, squeezing values of $\{0.43\,\text{dB}, 2.61\,\text{dB}, 13.03\,\text{dB}\}$). Solid lines correspond to the diagonalization of $\hat{\varrho}_+-\hat{\varrho}_-$ in Fock basis with cutoff $c=100$. Circles and crosses correspond to an estimation running Lanczos algorithm for $\ell = 5$ steps using a Gaussian trial vector with $\bm{r}=(1.5, 1.5)$ and $\bm{V}=\frac{\hbar}{2}\mathbb{I}_2$. (Bottom) Trace distance between the $(M=5)$-mode mixed Gaussian state parametrized by $\bm{r}_1=\bm{0}$ and $\bm{V}_1=\frac{\hbar}{2}(\mathbb{I}_M)\oplus([(1-\eta)(1 + 4 \sinh^2(s)) + \eta]\mathbb{I}_M)$, and the mixed state represented by $\bm{r}_2=\bm{0}$ and $\bm{V}_2=\frac{\hbar}{2}([(1-\eta)e^{-2s}+\eta]\mathbb{I}_M)\oplus([(1-\eta)e^{2s}+\eta]\mathbb{I}_M)$. Results are presented as a function of the loss parameter $\eta\in[0.0,1.0]$, for $s=0.5$. Circles and crosses correspond to an estimation running Lanczos algorithm for $\ell = 4$ steps using a Gaussian trial vector with $\bm{r}=\bm{1}_{2M}$ and $\bm{V}=\frac{\hbar}{2}\mathbb{I}_{2M}$, where $\bm{1}_{2M}$ is a $2M$-length vector whose components are all equal to 1. Several bounds on the trace distance weedbrook2012gaussian are shown in dotted lines.
• Figure 4: Illustration of the lower bounds on the trace distance obtained by extending Lanczos algorithm to the case in which both mixed states can be written as linear combinations of outer products of pure Gaussian states. We estimate the trace distance between the mixed states $\hat{\varrho}_\eta^{(\pm,\,p)}=\sum_{j,k=0}^{p-1}b_{j,k}^{\pm}|\sqrt{1-\eta}\alpha e^{i2\pi j/p}\rangle\langle \sqrt{1-\eta}\alpha e^{i2\pi k/p}|$, where $b_{j,k}^{\pm} = (\pm1)^{j+k}e^{-\eta|\alpha|^2}\exp(\eta|\alpha|^2e^{i2\pi (j-k)/p})/N_{\pm}$. Here, $|\alpha\rangle$ is a coherent state, $\bm{r}=\sqrt{2\hbar}(\mathrm{Re}(\alpha), \mathrm{Im}(\alpha))$ and $\bm{V}=\frac{\hbar}{2}\mathbb{I}_2$, $\eta$ is a loss parameter, and $N_{\pm}$ are normalization constants. The results are shown as a function of $\eta\in[0.0, 1.0]$ for $\alpha = 2.0$, and $p\in\{2, 4, 6, 8\}$. Solid lines correspond to the diagonalization of $\hat{\varrho}_\eta^{(+,\,p)}-\hat{\varrho}_\eta^{(-,\,p)}$ in Fock basis with cutoff $c=100$. Circles and crosses correspond to an estimation running Lanczos algorithm for $\ell = 10$ steps using $|\alpha\rangle$ as trial vector.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2: Courant-Fischer
• proof
• Proposition 1
• proof
• Proposition 2
• proof