Quantifying the performance of approximate teleportation and quantum error correction via symmetric 2-PPT-extendible channels

The ideal realization of quantum teleportation relies on having access to a maximally entangled state; however, in practice, such an ideal state is typically not available and one can instead only realize an approximate teleportation. With this in mind, we present a method to quantify the performance of approximate teleportation when using an arbitrary resource state. More specifically, after framing the task of approximate teleportation as an optimization of a simulation error over one-way local operations and classical communication (LOCC) channels, we establish a semidefinite relaxation of this optimization task by instead optimizing over the larger set of 2-PPT-extendible channels (where PPT denotes positive partial transpose). The main analytical calculations in our paper consist of exploiting the unitary covariance symmetry of the identity channel to establish a significant reduction of the computational cost of this latter optimization. Next, by exploiting known connections between approximate teleportation and quantum error correction, we also apply these concepts to establish bounds on the performance of approximate quantum error correction over a given quantum channel. Finally, we evaluate our bounds for various examples of resource states and channels.

Figure 1

Comparison between 2-PPT-extendiblity and PPT constraints for bounding the simulation error in approximate teleportation, when using the resource state $p{\mathrm{\Phi }}_{\widehat{A}\widehat{B}}+(1-p){\pi }_{\widehat{A}}\otimes {\sigma }_{\widehat{B}}$, where $p\in [0,1]$. The plot shows that 2-PPT-extendibility gives slightly better bounds for $p<0.5$. For higher values of $p$, the two curves become indistinguishable.

Figure 2

Comparison between 2-PPT-extendiblity and PPT constraints for bounding the simulation error in approximate quantum error correction when using the resource channel with Choi state $p{\mathrm{\Phi }}_{\widehat{A}\widehat{B}}+(1-p){\pi }_{\widehat{A}}\otimes {\sigma }_{\widehat{B}}$, where $p\in [0,1]$ and ${\sigma }_{\widehat{B}}$ is defined in (222). PPTNS and 2PENS are the curves obtained using the SDPs in (223) and Proposition t9, respectively, giving lower bounds on the error in approximate quantum error correction. There is no significant difference in the numerical values obtained from these two conditions. PPT and 2PE are the curves obtained using the same SDPs but without the nonsignaling constraints, hence, giving lower bounds on the error in one-way LOCC-assisted approximate error correction.

Figure 3

Comparison between bounds on the simulation error for approximate teleportation when using a two-dimensional special mixed state and a three-dimensional special mixed state as a resource to teleport a two dimensional state. The resource state is of the form $p{\mathrm{\Phi }}_{\widehat{A}\widehat{B}}+(1-p){\pi }_{\widehat{A}}\otimes {\sigma }_{\widehat{B}}$, where $p\in [0,1]$ and ${\sigma }_{\widehat{B}}$ is chosen to be (225) when $d_{\widehat{B}}=2$ and (226) when $d_{\widehat{B}}=3$. The bounds on the simulation error are calculated using both the 2PE constraints given in Proposition t3 and the PPT constraints given in (220). There is no significant difference in the numerical values obtained from both constraints for $d_{\widehat{B}}=2$.

Figure 4

Lower bounds on the simulation error of approximate quantum error correction for depolarizing channels when simulating a two-dimensional identity channel. The bounds are calculated using the SDP in Proposition t9 with 2-PPT-extendibility constraints, and the SDP in (223) with PPT constraints only, for different dimensions of the depolarizing channel ($d_{\widehat{B}}=2$ and $d_{\widehat{B}}=3$). There is no significant difference in the numerical values obtained from PPT and 2-PPT-extendibility constraints for $d_{\widehat{B}}=2$. The bounds are obtained without the nonsignaling constraints, hence, corresponding to one-way LOCC simulation.

Figure 5

Lower bounds on the simulation error of unideal teleportation when using the two-mode squeezed vacuum state as the resource state. The parameter $\lambda =tanh\left(r\right)$, where $r$ is the squeezing parameter. Larger values of $\lambda$ correspond to larger values of entanglement, which leads to a smaller error in simulating the identity channel.

Figure 6

Action of an amplitude damping channel on a three-level quantum system. The parameters ${\gamma }_{10}, {\gamma }_{20}$, and ${\gamma }_{21}$ represent decay rates between the respective levels.

Figure 7

Lower bounds on the simulation error of approximate quantum error correction when using a three-level amplitude damping channel. The parameters ${\gamma }_{10}, {\gamma }_{21}$, and ${\gamma }_{20}$ are decay parameters for the labeled states. The dimension $d$ of the target identity channel is set equal to the input and output dimensions (equal to three) of the amplitude-damping channel.