Optimal quantum operations at zero energy cost

Quantum technologies are developing powerful tools to generate and manipulate coherent superpositions of different energy levels. Envisaging a new generation of energy-efficient quantum devices, here we explore how coherence can be manipulated without exchanging energy with the surrounding environment. We start from the task of converting a coherent superposition of energy eigenstates into another. We identify the optimal energy-preserving operations, both in the deterministic and in the probabilistic scenario. We then design a recursive protocol, wherein a branching sequence of energy-preserving filters increases the probability of success while reaching maximum fidelity at each iteration. Building on the recursive protocol, we construct efficient approximations of the optimal fidelity-probability trade-off, by taking coherent superpositions of the different branches generated by probabilistic filtering. The benefits of this construction are illustrated in applications to quantum metrology, quantum cloning, coherent state amplification, and ancilla-driven computation. Finally, we extend our results to transitions where the input state is generally mixed and we apply our findings to the task of purifying quantum coherence.

Figure 1

Manipulating quantum information with limited energy resources. (a) Quantum device powered by a battery. The device contains an information register, where data are stored, and a battery, representing the energy resource(s) used to operate on the data. The information register and the battery are allowed to interact with the surrounding environment (possibly including ancillas), as long as the interaction involves no exchange of energy. The situation where the device uses energy from the environment to aliment its battery can be included in the model by formally regarding the energy sources in the environment as part of the battery. (b) Quantum device in low-power mode. In this mode, some quantum operations are performed without the aid of the battery, i.e., relying only on the interaction between the information register (here denoted simply as “the system”) and the environment (possibly including ancillas). The evolution of the system is described by an energy-preserving operation.

Figure 2

Probabilistic phase estimation via the recursive protocol and its coherent coarse-graining. The figure shows the trade-off between success probability and average gain for phase estimation with the qubit state $|{\phi }_{\theta }{\rangle }^{\otimes N}$ for $N=30$. The green solid line (with numerics represented by red dots) shows the trade-off between estimation gain and success probability for a recursive protocol with $K=27$ rounds, with the $T\mathrm{th}$ point corresponding to the first $T$ steps. The blue solid line (with numerics represented by the black dots) shows the trade-off for filters generated by coherent coarse-graining, with the $T\mathrm{th}$ point corresponding to the coherent coarse-graining of the first $T$ steps. Note that the gain for the coherent coarse-graining remains higher than the optimal deterministic estimation's gain (the black dashed line) even when the protocol becomes “almost deterministic” (i.e., the probability of success tends to one), while the gain for the recursive protocol drops down quickly with the growth of the success probability.

Figure 3

Scaling of the gain and success probability for coherently coarse-grained protocols. Panel (a) shows the estimation gain as a function of the number of copies $N$, for three coherently coarse-grained protocols corresponding to different values of $T$, including $T=1$ (black line with black dots), $T=3$ (green line with red dots), and $T=5$ (blue line with purple dots). The dashed line with black dots represents the optimal deterministic gain $\langle G_{\det }\rangle$. Panel (b) shows the decrease of the total success probability as a function of $N$ for different values of $T$, including $T=1$ (black line with black dots), $T=3$ (green line with red dots), and $T=5$ (blue line with purple dots).

Figure 4

Probabilistic phase estimation via the recursive protocol and its coherent coarse-graining. The figure shows the trade-off between success probability and average gain for phase estimation with maximally coherent states with $N=61$. The green solid line (with numerics represented by red dots) shows the probability-gain trade-off for $K=17$ rounds of the recursive protocol. At the first round the protocol reaches the maximum possible gain, equal to $G_{max}=99.9\%$, in agreement with the analytical expression of Eq. (75). The blue solid line (with numerics represented by the black dots) shows the trade-off for filters generated by coherent coarse-graining, with the $T\mathrm{th}$ point corresponding to the coherent coarse-graining of the first $T$ steps of the recursive protocol. For the first $K=17$ rounds the estimation gain of coherent coarse-graining remains approximately equal to $G_{max}=99.9\%$, although eventually it is bound to decrease to the optimal deterministic value $\langle G_{\det }\rangle =99.2\%$ (black dashed line).

Figure 5

Scaling of the estimation gain and success probability for the recursive protocol. Panels (a) and (b) show the average gain $G$ and the total success probability (b) as a function of the energy scale $N$ for different values of $T$, including $T=1$ (black line with black dots), $T=2$ (green line with red dots), and $T=3$ (blue line with purple dots).

Figure 6

Energy-preserving cloning of quantum clocks via the recursive protocol and its coherent coarse-graining. The figure shows the trade-off between success probability and fidelity for the $N$-to-$M$ cloning of the clock state $|{\psi }_t\rangle =\left(e^{i\omega t/2}\right|0\rangle +e^{-i\omega t/2}|1\rangle )/\sqrt{2}$ with $M=400$ and $N=80$. The green solid line (with numerics represented by red dots) shows the probability-fidelity trade-off for a recursive protocol with $K=32$ rounds. The blue solid line (with numerics represented by the black dots) shows the trade-off for filters generated by coherent coarse-graining, with the $T\mathrm{th}$ point corresponding to the coherent coarse-graining of the first $T$ steps of the recursive protocol. Finally, the black dashed line represents the fidelity for the optimal deterministic cloning protocol. Notice that the recursive protocol maintains fidelity larger than the optimal deterministic fidelity for all steps up to the last.

Figure 7

Energy-preserving amplification of coherent light via the recursive protocol and its coherent coarse-graining. The figure shows the trade-off between success probability and the average fidelity for the amplification $|r_1\rangle \to |r_2\rangle$ with $r_1=1, r_2=1.5$, and $N=80$. The green solid line (with numerics represented by red dots) shows the probability-fidelity trade-off for a recursive protocol with $K=81$ rounds. The blue solid line (with numerics represented by the black dots) shows the trade-off for filters generated by coherent coarse-graining. Note that the difference between the two curves becomes large as the success probability tends to 1. For unit probability the recursive protocol and its coherent coarse-graining give fidelities $F_{\det }=49.9\%$ and $F_{\det }^{\prime }={\left|\langle r_1|r_2\rangle \right|}^2=77.9\%$, respectively.

Figure 8

Energy-preserving correction of a quantum operation via the recursive protocol and its coherent coarse-graining. The figure shows the trade-off between success probability and average fidelity for unlearning the quantum operation with Kraus operator $M_x={\sum }_{n=1}^d{\mu }^{n/2}|n\rangle \langle n|$ with $d=100$ and $\mu =0.9$. The green solid line (with numerics represented by red dots) shows the probability-fidelity trade-off for a recursive protocol with $K=70$ rounds. The blue solid line (with numerics represented by the black dots) shows the trade-off for filters generated by coherent coarse-graining.

Figure 9

Optimal purification of coherence via energy-preserving operations. Panel (a) shows the optimal fidelities as function of the number of input copies $N$. Here different colors represent different values of $\beta$—specifically, $\beta =0.4$ (blue line with purple dots), $\beta =0.8$ (black line with black dots), and $\beta =1.2$ (green line with red dots). The dashed lines represent the deterministic fidelities, while the solid lines represent the probabilistic fidelities. While the probabilistic fidelities increase with $N$, the deterministic fidelities decrease, due to the restriction imposed by energy preservation. Panel (b) shows the maximum success probability for the quantum operations with maximum fidelity. The color code is the same as in panel (a).

Figure 10

Hierarchy of quantum channels. Energy-preserving channels are contained in the class of time-covariant channels, corresponding to the special case of channels that arise from an interaction that leaves the environment inside a fixed eigenspace at every time. For Hamiltonians with nondegenerate spectrum, covariant channels are a special case of incoherent channels, i.e., channels that do not generate coherence across the eigenbasis of the energy.