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Quantum speed limits—primer, perspectives, and potential future directions

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Abstract

Fundamental physical limits on the speed of state evolution in quantum systems exist in the form of the Mandelstam–Tamm and the Margolus–Levitin inequalities. We give an expository review of the development of these quantum speed limit (QSL) inequalities, including extensions to different energy statistics and generalizations to mixed system states and open and multipartite systems. The QSLs expressed by these various inequalities have implications for quantum computation, quantum metrology, and control of quantum systems. These connections are surveyed, and some important open questions are noted.

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Correspondence to Michael R. Frey.

Appendix

Appendix

We derive here the LC QSL inequality (35) for pure initial states. Our derivation is an amalgam of arguments found in [21, 29, 30]. Once proved for pure initial states, the LC QSL inequality then readily extends to general mixed states by the argument of Giovannetti et al. [15] reviewed in Sect. 2. We also find here conditions under which this LC QSL is tight by presenting states that saturate (35).

Consider an autonomous quantum system with Hamiltonian \({\mathbf {H}}\) initially in the pure state \(|\psi _0\rangle \). The overlap between \(|\psi _0\rangle \) and the system state \(|\psi _t\rangle \) at the later time \(t>0\) is \(\langle \psi _0 |\psi _t\rangle =\langle \exp (-it{\mathbf {H}}/\hbar )\rangle \) so

$$\begin{aligned} |\langle \psi _0 |\psi _t\rangle | \ge \textit{Re} \langle \psi _0 |\psi _t\rangle = \textit{Re} \langle \exp (-it{\mathbf {H}}/\hbar )\rangle = \left\langle \cos \frac{t {\mathbf {H}}}{\hbar } \right\rangle . \end{aligned}$$
(83)

Define

$$\begin{aligned} A_\nu =\max _{x\ge 0} \frac{1-\cos x}{x^\nu }; \end{aligned}$$
(84)

\(A_\nu \) is plotted in Fig. 8. Then, as illustrated in Fig. 9, for \(\nu \in [0,2]\), \(\cos x \ge 1-A_\nu |x|^\nu \) for all x, so

$$\begin{aligned} \sqrt{F(t)} \ge 1-A_\nu \left( \frac{t}{\hbar }\right) ^\nu \langle |{\mathbf {H}}|^\nu \rangle \end{aligned}$$
(85)

where \(F(t) = |\langle \psi _0 |\psi _t\rangle |^2\) is the fidelity of state \(|\psi _t\rangle \). The reference energy for \({\mathbf {H}}\) has no physical meaning and can be shifted by any chosen amount without invalidating (85). Choosing the reference energy \(E_\nu ^*\) that minimizes \(M_\nu ^*= \langle |\mathbf {H}-E_\nu ^*|^\nu \rangle \), we have

$$\begin{aligned} \sqrt{F(t)} \ge 1-A_\nu \left( \frac{t}{\hbar }\right) ^\nu M_\nu ^*. \end{aligned}$$
(86)

Let \(t_\epsilon \) be the time necessary to reach a state with fidelity \(\epsilon =F(t_\epsilon )\). We have then from (86) that

$$\begin{aligned} t_\epsilon \ge \hbar \left( \frac{1-\sqrt{\epsilon }}{A_\nu M_\nu ^*} \right) ^{1/\nu } \end{aligned}$$
(87)

for any \(\nu \in [0,2]\) for which \(M_\nu ^*= \langle |\mathbf {H}-E_\nu ^*|^\nu \rangle \) exists. This establishes the LC QSL inequality for pure initial states.

Fig. 8
figure 8

\(A_\nu \) is a factor in the LC QSL time (87) defined by the centered moment of order \(\nu \). \(a_\nu \) defines the state that saturate the LC QSL

Fig. 9
figure 9

The dotted curve is \(1-\cos x\), and the solid curves are \(A_\nu |x|^\nu \) for various moment orders \(\nu \), illustrating the elementary inequality \(\cos x \ge 1-A_\nu |x|^\nu \) for \(\nu \in [0,2]\) used to derive the LC QSL

To see conditions under which inequality (87) is tight, consider a quantum system with energy eigenstates \(|- E\rangle , |0\rangle , |E\rangle \) and corresponding eigenenergies \(-E,0,E\). Let \(\mathcal{T}=\{ (\nu ,\epsilon )\in [0,2]\times [0,1]: \sqrt{\epsilon }\ge \cos a_\nu \}\) be the set shown in Fig. 4, and for \((\nu ,\epsilon )\in \mathcal{T}\) define the class of initial states

$$\begin{aligned} |\psi _0\rangle = \sqrt{\frac{\alpha }{2}} |- E\rangle + \sqrt{1-\alpha } |0\rangle + \sqrt{\frac{\alpha }{2}} |E\rangle \end{aligned}$$
(88)

where

$$\begin{aligned} \alpha = \frac{1-\sqrt{\epsilon }}{1-\cos a_\nu }, \end{aligned}$$
(89)

with

$$\begin{aligned} a_\nu =\mathop {\arg \,\max }\limits _{x\ge 0} \frac{1-\cos x}{x^\nu } \end{aligned}$$
(90)

as plotted in Fig. 8. The restriction \((\nu ,\epsilon )\in \mathcal{T}\) ensures that \(\alpha \le 1\) in (88). The overlap of \(|\psi _0\rangle \) in (88) with a later state \(|\psi _t\rangle \) is

$$\begin{aligned} \langle \psi _0 |\psi _t\rangle = 1-\alpha + \alpha \cos \frac{Et}{\hbar }, \end{aligned}$$
(91)

so for the smallest time \(t_\epsilon \) such that the fidelity is \(F(t_\epsilon )=\epsilon \),

$$\begin{aligned} 1-\sqrt{\epsilon } = \alpha \left( 1- \cos \frac{Et_\epsilon }{\hbar }\right) . \end{aligned}$$
(92)

Substitute (89) for \(\alpha \) in (92). Then (92) becomes simply \(t_\epsilon = a_\nu \hbar /E\). But, for \(|\psi _0\rangle \) in (88), \(E_\nu ^*=0\) and \(M_\nu ^*= \alpha E^\nu \) for \(\nu \in [0,2]\), so

$$\begin{aligned} t_\epsilon = \hbar \left( \frac{\alpha a_\nu ^\nu }{M_\nu ^*} \right) ^{1/\nu } = \hbar \left( \frac{a_\nu ^\nu }{1-\cos a_\nu } \frac{1-\sqrt{\epsilon }}{M_\nu ^*} \right) ^{1/\nu }. \end{aligned}$$
(93)

Since \(A_\nu \), \(a_\nu \) in (84) and (90) satisfy \(1-\cos a_\nu = A_\nu a_\nu ^\nu \), this proves that \(t_\epsilon = \tau _\epsilon \) where \(\tau _\epsilon \) is the QSL time in (87). Specifically, this proves that QSL (87) is tight for \((\nu ,\epsilon )\) pairs in the hatched region \(\mathcal{T}\) in Fig. 4. In particular, (87) is not tight for \(\nu =2\). Indeed, comparing the MT QSL time \(\tau _\epsilon ^{\mathrm {MT}}\) with the QSL time \(\tau _\epsilon ^{\mathrm {LC}}\) in (87) for \(\nu =2\), we find

$$\begin{aligned} \frac{\tau _\epsilon ^{\mathrm {MT}}}{\tau _\epsilon ^{\mathrm {LC}}} =\frac{\frac{h}{4}\frac{\beta (\epsilon )}{\Delta E}}{\hbar \left( \frac{1-\sqrt{\epsilon }}{\Delta E^2}\right) ^{1/2}} =\frac{\mathrm {Cos}^{-1} \sqrt{\epsilon }}{\sqrt{\frac{1-\sqrt{\epsilon }}{2}}} \ge 2. \end{aligned}$$
(94)

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Frey, M.R. Quantum speed limits—primer, perspectives, and potential future directions. Quantum Inf Process 15, 3919–3950 (2016). https://doi.org/10.1007/s11128-016-1405-x

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