High-dimensional quantum teleportation under noisy environments

We study the protocol of qudit teleportation using quantum systems subjected to several kinds of noise for arbitrary dimensionality $d$. We consider four classes of noise: dit-flip, $d$-phase-flip, dit-phase-flip, and depolarizing noise, each of them corresponding to a family of Weyl operators, introduced via Kraus formalism. We derive a general expression for the average fidelity of teleportation in arbitrary dimension $d$ for any combination of noise on the involved qudits. Under a different approach we derive the average fidelity of teleportation for a more general scenario involving the $d$-dimensional generalization of amplitude damping noise as well. We show that all possible scenarios may be classified in four different behaviors and discuss the cases in which it is possible to improve the fidelity by increasing the associated noise fractions. All our results are in agreement with previous analysis for the case of qubits [R. Fortes and G. Rigolin, Phys. Rev. A 92, 012338 (2015)].

Figure 1

Teleportation scheme: Alice and Bob share a channel composed by two entangled qudits prepared in a state ${\widehat{\rho }}_{\text{ch}}$ (red dashed). Alice performs a joint measurement on qudits $I$ and $A$ (blue dotted). Under appropriate conditions (maximal entanglement) and after LOCC (green dot-dashed), corresponding to transmission of a pair of dits $(m,n)$ using a classical channel and subsequent application of a local unitary operation ${\widehat{U}}_{mn}$ on qudit $B$, it finally holds in the initial state of the input qudit $I$.

Figure 2

Normalized quantum contribution to the fidelity $f_Q^{\prime }[=(d+1)f_Q/(d-1)]$ as a function of the amount of entanglement in the channel (normalized to $logd$), for a maximally entangled measurement basis and a set of $N$ random pure states of two entangled qudits, for several values of $d$. (a) $d=2$, $N={10}^4$. (b) $d=3$, $N={10}^5$. (c) $d=4$, $N={10}^5$. (d) $d=5$, $N={10}^6$. Each line corresponds to a particular family of “boundary” states ${\phi }_{\mu }$, given by Eq. (5): Red dotted, $\mu =1$; green dashed, $\mu =2$; blue solid, $\mu =3$; and cyan dash-dotted, $\mu =4$. Note also that any intersection point between two family lines corresponds to a maximally symmetric $\nu$-rank state ${\varphi }_{\nu }$ [Eq. (6)].

Figure 3

Weyl operators ${\widehat{U}}_{mn}$ and their relation with Kraus operators for several kinds of noise on $d$-dimensional systems. The blue row represents dit-flip-like operators, the yellow column represents $d$-phase-flip-like operators, and the pink squares are related to matrices corresponding to dit-phase-flip-like noise. Note that the three classes mentioned before are employed to define depolarizing noise (blue dashed line). In addition, the set of Pauli matrices corresponds to the nontrivial operators for $d=2$ (red dashed line).

Figure 4

Solid: Optimal fidelity of teleportation for the case in which only one of the qudits may suffer $d$-phase-flip noise. Dashed: Classical fidelity. (a) Calculations of $\langle F_p\rangle$ for $2\le d\le 5$. (b) Optimal fidelity of teleportation for arbitrary dimension $d$.

Figure 5

Fidelity of teleportation under a scenario in which the qudits are affected by dit-phase-flip, dit-flip, and $d$-phase-flip noises, respectively $(FP,F,P)$, and $d=3$, for configurations leading to values above the classical limit, $\langle F\rangle >f_C=1/2$. It is possible to notice that the larger the noise fraction the lower the fidelity of teleportation, as expected.

Figure 6

Fidelity of teleportation for the scenario in which all the parts are affected by dit-flip noise, and $d=3$. The axes represent the noise fraction associated to each qudit involved in the teleportation process. Note that fidelity of teleportation values above the classical limit are obtained either if all the noise fractions are lower or if two of them are higher than the noise threshold $p^{*}=1-1/d$.

Figure 7

Fidelity of teleportation for the scenario of one qutrit affected by depolarizing and the other two affected by trit-flip noise, for configurations leading to values above the classical limit. Note that there is a region in which the fidelity attains values above $f_C$, even for large noise fractions. See Table 1 for all possible noise configurations presenting this behavior.

Figure 8

Fidelity of teleportation for a scenario in which all parts are affected by amplitude damping noise $(AD,AD,AD)$ and $d=3$, as a function of the noise fractions associated to each qudit in the channel. Note that in the limit $p_I\to 0$ only a small region of noise parameters leads to fidelities below its classical value, $f_C=1/2$. Furthermore, as the amount of noise in the channel increases, the fidelity approaches its classical limit.