Quantum cost of dense coding and teleportation

Quantum cost is a key ingredient in evaluating the quality of quantum protocols from a practical viewpoint. We show that the quantum cost of a $d$-dimensional dense coding protocol is equal to $d+3$ when transmitting the classical message (0,0) and $d+4$ when transmitting another classical message. It appears as linear growth with the dimension and thus makes sense for implementation. In contrast, the quantum cost of high-dimensional teleportation protocols is equal to 13, which is the maximum value of the cost for the two-dimensional case. As an application, we establish the relation between the quantum cost and fidelity of dense coding protocols in terms of four typical noise scenarios.

Figure 1

Protocol for the $d$-dimensional dense coding. The two left gates are used to prepare Bell states $|{\varphi }_1\rangle$ with the qudit $|0,0\rangle$. The gates to the right of the dashed line are used by Bob to recover the message.

Figure 2

Teleportation protocol via the $d$-dimensional Bell state $|{\varphi }_1\rangle =\frac{1}{\sqrt{d}}{\sum }_{k=1}^{d-1}|k,k\rangle$. The unitary gates to recover the message are shown to the right of the dashed line. Three kinds of basic gates are used in the circuit: $H_d$, cnot, and $P_{0,d}$.

Figure 3

Teleportation protocol via the $d$-dimensional Bell state $|{\varphi }_d\rangle$. The unitary gates to recover the message are shown to the right of the dashed line. Four kinds of basic gates are used in the circuit: $H_d$, cnot, $P_{0,d}$, and $P_{1,d}$.

Figure 4

Fidelities of dense coding under a scenario in which the channel is affected by dit-flip ${\mathcal{F}}_F, d$-phase-flip ${\mathcal{F}}_P$, dit-phase-flip ${\mathcal{F}}_{FP}$, and depolarizing noise ${\mathcal{F}}_D$, respectively. The axes represent the fidelity, error probability $p$, and the quantum cost $D$, for $0\le p\le 1$ and $6\le D\le 20$.

Figure 5

Value of $-\frac{b}{a_i}$ ($i=1,2$) with respect to the quantum cost $D$. The blue dashed line and red line with squares represent the values of $-\frac{b}{a_1}$ and $-\frac{b}{a_2}$, respectively.