Optical quantum teleportation with known amplitude distorting factors of teleported qubits

Abstract

We develop a quantum teleportation protocol of an unknown optical single rail qubit using a hybrid quantum channel composed of continuous variable (CV) states of certain parity. The quantum channel is characterized by two parameters: a squeezing parameter of single-mode squeezed vacuum (SMSV) state and the beam splitter (BS) parameter used to implement it. The CV part of the hybrid state belongs to Alice, while discrete variable (DV) half is controlled by Bob. The third parameter of the protocol is a parameter of the beam splitter, used to mix the CV components of the hybrid quantum state with unknown optical single-rail qubit. Even though the number of measurement results Alice sends may increase, Bob can obtain the original qubit half the time with an appropriate choice of parameter values. In almost half the remaining cases, Bob obtains the original qubit with distorted amplitudes, and both participants know the value of the distortion factors. This means that as the amount of classical information transmitted by Alice increases, they both gain greater access to partial information about the unitary transformations that the teleported qubits undergo, allowing Bob to continue using them or attempt to recover them to improve the protocol's efficiency. The proposed method is a generalization of quantum teleportation with a nonlocal photon used as a quantum channel and unknown single-rail optical qubit.

Figures (4)

• Figure 1: Schematic representation of the quantum teleportation protocol of photonic single-rail state \ref{['eq:2']} using optical hybrid quantum channel (1). The part of the quantum channel containing the CV states is mixed with the teleported qubit on balanced beam splitter, followed by detection of the measurement outcomes by two PNR detectors. Four measurement results 01,10,20,02 form the basis for Alice to compose two-bit messages (highlighted in green) for Bob. For certain values of the parameters of the quantum channel $S_{SMSV}\,\,(dB)$ and $B_0$, Bob receives the initial qubit with probability $P_{pt}=0.5$ provided that Alice measured 01 and 10. With the other measurement results from Alice, Bob receives a qubit with an amplitude distortion, i.e., the one on which the unitary transformation in equations (\ref{['eq:10']},\ref{['eq:11']}) has been performed. To help Bob determine which amplitude-distorted qubit he received, Alice should increase the number of bits kl transmitted (highlighted in dark blue). Given partial information about his qubit, Bob can attempt to reconstruct the original qubit using additional photon states (this part of the diagram is highlighted by the turquoise dotted line).
• Figure 2: (a-d). (a) and (c) Contour plots of the probabilities $P_{01}^{(odd)}$ and $P_{10}^{(odd)}$ as functions of the quantum channel parameters $S_{SMSV}\,\,(dB)$ and $B_0$. In the selected range of $S_{SMSV}$ and $B_0$ variation, the probabilities can be monitored using a graduated color scale up to 0.25. The contour graph in (b) and (d) show dependences of the BS parameter $B_{01}$ and $B_{10}$, used in the quantum teleportation in \ref{['Figure.1']}, on the parameters $S_{SMSV}\,\,(dB)$ and $B_0$ in the same rectangle of their change as for (a) and (c). The straight lines $B=1$ on both graphs (b) and (d) are the same.
• Figure 3: (a-d). (a) and (c) Contour plots of the probabilities $P_{20}^{(odd)}$ and $P_{02}^{(odd)}$ as functions of the parameters $S_{SMSV}\,\,(dB)$ and $B_0$ of the quantum channel. These dependencies are displayed in the same rectangle of $S_{SMSV}\,\,(dB)$ and $B_0$ as in the graphs in \ref{['Figure.2']}, the parameter $B_{20}$ takes values less than 1 ($B_{20}<1$), while $B_{02}$ becomes greater than one ($B_{02}>1$) to ensure elimination of the corresponding amplitude distorting factor.
• Figure 4: Dependence of amplitude distortion multipliers $b_{20}^{(even)}(B=1)=\lvert b_{02}^{(even)} (B=1)\rvert$ on the squeezing parameter $S_{SMSV}\,\,(dB)$ of the quantum channel (1). When constructing the curve, those values of $S_{SMSV}\,\,(dB)$ and $B_0$ are used that ensure a straight line $B=1$ in the \ref{['Figure.2']}. The amplitude distortion factors increase with increasing $S_{SMSV}\,\,(dB)$ but remain less than one.