Asymmetric quantum dialogue in noisy environment

Abstract

A notion of asymmetric quantum dialogue (AQD) is introduced. Conventional protocols of quantum dialogue are essentially symmetric as the users (Alice and Bob) can encode the same amount of classical information. In contrast, the proposed scheme for AQD provides different amount of communication powers to Alice and Bob. The proposed scheme offers an architecture, where the entangled state to be used and the encoding scheme to be shared between Alice and Bob depend on the amount of classical information they want to exchange with each other. The general structure for the AQD scheme has been obtained using a group theoretic structure of the operators introduced in Shukla et al. (Phys Lett A 377:518, 2013). The effect of different types of noises (e.g., amplitude damping and phase damping noise) on the proposed scheme is investigated, and it is shown that the proposed scheme for AQD is robust and it uses an optimized amount of quantum resources.

Appendix: Modified Pauli groups and the notation used

Modified Pauli groups and the notations used to denote them in the present work were introduced earlier in [20, 39]. Here, for consistency, we briefly summarize the definition and the notation used.

It is easy to verify that the set of Pauli operators $\{I_{2}, \sigma _{x}, i\sigma _{y}, \sigma _{z}\}$ forms a group under multiplication $G_{1}^{\prime }=\{ \pm I_{2},\pm iI_{2},\pm \sigma _{x},\pm i\sigma _{x},\pm \sigma _{y},\pm i\sigma _{y},\pm \sigma _{z},\pm i\sigma _{z}\}$ (cf. Section 10.5.1 of [40]). The closure property of $G_{1}^{\prime }$ is satisfied under normal matrix multiplication because of the inclusion of ±1 and ±i. However, if any of the operators $\sigma _{i},-\sigma _{i}, i\sigma _{i}$ or $-i\sigma _{i}$ operates on a quantum state, the effect would be the same. Keeping this in mind, if global phase is ignored from the product of matrices (which is consistent with quantum mechanics), we obtain a modified Pauli group $G_{1}=\{I_{2}, \sigma _{x}, i\sigma _{y}, \sigma _{z}\}=\{I_{2},X, iY, Z\}$. Clearly, under the above-defined multiplication rule, $G_{1}$ is an Abelian group of order 4 and its generators are $\langle X,Z\rangle ,\langle X,iY\rangle$ and $\langle iY,Z\rangle$. Similarly, we may define the modified generalized Pauli group $G_{n}=G_{1}^{\otimes n}$ as a group of order $2^{2^{n}}=4^{n}$ and whose elements are all n-fold tensor products of Pauli matrices [20]. For example,

$$\begin{aligned} G_{2}= & {} G_{1}\otimes G_{1}=\{I_{2}, X, iY,\,Z\}\otimes \{I_{2},X,\,iY,Z\}\nonumber \\= & {} \left\{ I_{2}\otimes I_{2},I_{2}\otimes X,I_{2}\otimes iY,I_{2}\otimes Z,X\otimes I_{2},X\otimes X,\right. \nonumber \\&X\otimes iY,X\otimes Z,iY\otimes I_{2},iY\otimes X,iY\otimes iY,\nonumber \\&\left. iY\otimes Z,Z\otimes I_{2},Z\otimes X,Z\otimes iY,Z\otimes Z\right\} . \end{aligned}$$

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In Ref. [20], it was discussed in detail how to construct subgroups of $G_{2}$. Here, we list 11 subgroups of $G_{2}$, which are used in this paper (each is of order 8):

$$\begin{aligned} G_{2}^{1}(8)= & {} \left\{ I_{2}\otimes I_{2}, X\otimes I_{2}, iY\otimes I_{2}, Z\otimes I_{2}, I_{2}\otimes X, X\otimes X, iY\otimes X, Z\otimes X\right\} ,\nonumber \\ G_{2}^{2}(8)= & {} \left\{ I_{2}\otimes I_{2}, X\otimes I_{2}, iY\otimes I_{2}, Z\otimes I_{2}, I_{2}\otimes iY, X\otimes iY, iY\otimes iY, Z\otimes iY\right\} ,\nonumber \\ G_{2}^{3}(8)= & {} \left\{ I_{2}\otimes I_{2}, X\otimes I_{2}, iY\otimes I_{2}, Z\otimes I_{2}, I_{2}\otimes Z, X\otimes Z, iY\otimes Z, Z\otimes Z\right\} ,\nonumber \\ G_{2}^{4}(8)= & {} \left\{ I_{2}\otimes I_{2}, I_{2}\otimes X, I_{2}\otimes iY, I_{2}\otimes Z, X\otimes I_{2}, X\otimes X, X\otimes iY, X\otimes Z\right\} \nonumber \\ G_{2}^{5}(8)= & {} \left\{ I_{2}\otimes I_{2}, I_{2}\otimes X, I_{2}\otimes iY, I_{2}\otimes Z, iY\otimes I_{2}, iY\otimes X, iY\otimes iY, iY\otimes Z\right\} \nonumber \\ G_{2}^{6}(8)= & {} \left\{ I_{2}\otimes I_{2}, I_{2}\otimes X, I_{2}\otimes iY, I_{2}\otimes Z, Z\otimes I_{2}, Z\otimes X, Z\otimes iY, Z\otimes Z\right\} \nonumber \\ G_{2}^{7}(8)= & {} \left\{ I_{2}\otimes I_{2},I_{2}\otimes Z,Z\otimes I_{2},Z\otimes Z,X\otimes X,iY\otimes X,X\otimes iY,iY\otimes iY\right\} ,\nonumber \\ G_{2}^{8}(8)= & {} \left\{ I_{2}\otimes I_{2},Z\otimes Z,X\otimes iY,iY\otimes X,I_{2}\otimes X, Z\otimes iY, iY\otimes I_{2}, X\otimes Z\right\} ,\nonumber \\ G_{2}^{9}(8)= & {} \left\{ I_{2}\otimes I_{2},Z\otimes Z,X\otimes iY,iY\otimes X,X\otimes I_{2},iY\otimes Z,Z\otimes X,I_{2}\otimes iY\right\} ,\nonumber \\ G_{2}^{10}(8)= & {} \left\{ I_{2}\otimes I_{2},X\otimes I_{2},I_{2}\otimes X,X\otimes X,Z\otimes Z,iY\otimes Z,Z\otimes iY,iY\otimes iY\right\} ,\nonumber \\ G_{2}^{11}(8)= & {} \left\{ I_{2}\otimes I_{2},iY\otimes I_{2},I_{2}\otimes iY,iY\otimes iY,Z\otimes Z,Z\otimes X,X\otimes Z,X\otimes X\right\} ,\nonumber \\ \end{aligned}$$

(15)

where $G_{n}^{j}(m)$ denotes jth subgroup of order $m<4^{n}$ of the group $G_{n}$ whose order is $4^{n}.$