Beating the classical limits of information transmission using a quantum decoder

Encoding schemes and error-correcting codes are widely used in information technology to improve the reliability of data transmission over real-world communication channels. Quantum information protocols can further enhance the performance in data transmission by encoding a message in quantum states; however, most proposals to date have focused on the regime of a large number of uses of the noisy channel, which is unfeasible with current quantum technology. We experimentally demonstrate quantum enhanced communication over an amplitude damping noisy channel with only two uses of the channel per bit and a single entangling gate at the decoder. By simulating the channel using a photonic interferometric setup, we experimentally increase the reliability of transmitting a data bit by greater than $20\%$ for a certain damping range over classically sending the message twice. We show how our methodology can be extended to larger systems by simulating the transmission of a single bit with up to eight uses of the channel and a two-bit message with three uses of the channel, predicting a quantum enhancement in all cases.

Figure 1

Numerical simulations for transmitting one bit over an amplitude damping channel encoded in the state of one, two, four, and eight qubits. (a) The classical scheme of duplicating a bit $M$ times before transmission over an ADC. (b) The coherent scheme that uses separable coherent quantum states to encode the data with a local rotation ${\theta }_{\gamma }$, which is numerically optimized for each $\gamma$ value. (c) The quantum scheme with local encoding and an entangling measurement at the decoder. For this one-bit (two code word) case, the trace distance between code words after the ADC yields the maximum success probability of an optimal quantum decoder, measurement, and mapping. (d)–(g) The success probability for the classical, coherent, and quantum schemes for $\gamma =[0,1]$ and $M=1,2,4$, and 8. (h)–(k) The corresponding gain over the classical scheme, calculated as $\frac{P-P_c}{P_c}$, where $P$ is the quantum or coherence success probability and $P_c$ is the classical probability.

Figure 2

Experimentally implemented quantum scheme for transmitting one bit over an amplitude damping channel encoded in the state of two qubits. (a) The circuit schematic for the entangling decoder we implement. This circuit is approximately optimal for large $\gamma$. (b) Our experimental implementation based on polarization photonic qubits. The components used are as follows: HWP, half wave plate; PBS, polarizing beam splitter; BS, 50/50 beam splitter; BB, beam block; PPBS, partially polarizing beam splitter (100% horizontal transmission; 66% vertical transmission). The $\phi$ rotation is a $\sqrt{H}$ gate. (c) Experimental results for the classical, coherent, and quantum schemes. The points are experimentally measured results and lines are calculated from circuit simulations. We also plot the theoretical maximum as a black dashed line and the shaded area is where we measure a quantum enhancement in message recovery. (d) The gain over the classical scheme calculated as $\frac{P-P_c}{P_c}$. The points are calculated using experimental data from the classical, coherent, and quantum schemes. The lines are calculated from the ideal curves. The red area highlights the advantage of the quantum scheme. The gray area is where the classical scheme achieves a higher success probability and therefore the gain is negative.

Figure 3

Numerical simulations for transmitting two bits over an amplitude damping channel encoded in the state of three qubits. (a) The optimal classical scheme for transmitting a two bit message with three channel uses. (b) The optimal coherent scheme, where local encoding and decoding is used. (c) The maximum success probability with a quantum decoder. As there are more than two code words, the trace distance no longer directly calculates the optimal success probability. Therefore, a semidefinite program is solved to find the success probability for the optimal quantum decoder across all code words. (d) A designed quantum decoder that approximates the optimal decoder for large $\gamma$. (e) The success probability of each scheme. The entangling decoder surpasses the coherent scheme for $\gamma >0.55$ and achieves the theoretical maximum for large $\gamma$. (f) The gain over the classical scheme for the coherent, analytic maximum and our quantum decoder schemes.

Figure 4

Spontaneous parametric down conversion source of indistinguishable horizontally polarized photon pairs.

Figure 5

(a) Quantum scheme including local encoder and entangling decoder for transmitting a one bit message over two qubits. (b) A scheme including an entangling encoder and decoder. (c) The success probability for each of these schemes, demonstrating that the entangling encoder adds no benefit to the success probability.

Figure 6

(a) Quantum encoding scheme with a local decoder. (b) A local encoding scheme with a quantum decoder. (c) The success probability for each of these schemes where we optimize for all $\gamma$ values simultaneously, to maximize the average success probability. (d) The success probability where we have optimized for each $\gamma$ value independently.