Quantum Speed Limits For Adiabatic Evolution, Loschmidt Echo and Beyond

Abstract

One often needs to estimate how fast an evolving state of a quantum system can depart from some target state or target subspace of a Hilbert space. Such estimates are known as quantum speed limits. We derive a quantum speed limit for a general time-dependent target subspace. When the target subspace is an instantaneous invariant subspace of a time-dependent Hamiltonian, the obtained quantum speed limit bounds the adiabatic fidelity, which is a figure of merit of quantum adiabaticity. We also compare two states evolving under two different Hamiltonians and derive a bound on the Loschmidt echo.

Appendices

Appendix A: Norm of a One–Dimensional Projector

Here we prove the equality

$$ ||\dot{P}_{{\phi}_{t}}||=\sqrt{\langle\dot{\phi}_{t}|\dot{\phi}_{t}\rangle-|\langle\dot{\phi}_{t}|{\phi}_{t}\rangle|^{2}}. $$

(41)

valid for any normalised vector ϕ_t. To this end we introduce a normalised vector ${\phi }_{t}^{\bot }=(\mathbb {I}-P_{{\phi }_{t}})\dot {\phi }_{t}/\|(\mathbb {I}-P_{{\phi }_{t}})\dot {\phi }_{t}\|$ which is orthogonal to ϕ_t, and expand $\dot {\phi }_{t}$:

$$ \dot{\phi}_{t}=P_{{\phi}_{t}}\dot{\phi}_{t}+(\mathbb{I}-P_{{\phi}_{t}}) \dot{\phi}_{t}=\langle {\phi}_{t}| \dot {\phi}_{t} \rangle {\phi}_{t}+ \|(\mathbb{I}-P_{{\phi}_{t}})\dot{\phi}_{t}\| {\phi}_{t}^\bot. $$

(42)

This implies

$$ \dot{P}_{{\phi}_{t}}=\langle \dot{\phi}_{t}|{\phi}_{t}\rangle P_{{\phi}_{t}}+ |{\phi}_{t}^\bot \rangle\langle{\phi}_{t}|\|(\mathbb{I}-P_{{\phi}_{t}})\dot{\phi}_{t}\|+h.c.. $$

(43)

Due to the normalization condition 〈ϕ_t|ϕ_t〉 = 1 the first term and its complex conjugate cancel: $\langle \dot {\phi }_{t}|{\phi }_{t}\rangle P_{{\phi }_{t}}+\langle {\phi }_{t}|\dot {\phi }_{t}\rangle P_{{\phi }_{t}}=\left (\frac {d}{dt}\langle {\phi }_{t}|{\phi }_{t}\rangle \right ) P_{{\phi }_{t}}=0$. Thus

$$ ||\dot{P}_{{\phi}_{t}}||=\|(\mathbb{I}-P_{{\phi}_{t}})\dot{\phi}_{t}\|=\sqrt{\langle\dot{\phi}_{t}|\dot{\phi}_{t}\rangle-|\langle\dot{\phi}_{t}|{\phi}_{t}\rangle|^{2}}. $$

(44)

Appendix B: Proof of the Bound (29)

Consider π_t generated by some unitary W_t, ${\Pi }_{t}=W_{t} {\Pi }_{0} W_{t}^{\dag }$. This evolution can be described by a Schrödinger equation with a fictitious Hamiltonian ${\mathscr{H}}_{t}=i\dot {W}_{t}W_{t}^{\dag }$,

$$ i\dot{\Pi}_{t}=[\mathcal{H}_{t},{\Pi}_{t}]. $$

(45)

To derive this equation one should use the fact that $W_{t}W_{t}^{\dag }=W_{t}^{\dag } W_{t}=1$ and, hence, $\dot {W_{t}}W_{t}^{\dag }+W_{t} \dot {W_{t}}^{\dag }= \dot {W_{t}}^{\dag } W_{t}+W_{t}^{\dag } \dot {W_{t}}=0$. Observe that

$$ \|\mathcal{H}_{t}\|=\|\dot{W}_{t}\|. $$

(46)

From (12) and (45) we obtain

$$ \dot{\Pi}_{t}=i {\Pi}_{t}\mathcal{H}_{t}(\mathbb{I}-{\Pi}_{t})-i(\mathbb{I}-{\Pi}_{t})\mathcal{H}_{t} {\Pi}_{t}. $$

(47)

Since $\dot {\Pi }_{t}$ is self-adjoint operator we can estimate its norm as

$$ \begin{array}{@{}rcl@{}} \|\dot{\Pi}_{t}\|&=&\sup\limits_{\|\varphi\|=1}\langle\varphi|\dot{\Pi}_{t}|\varphi\rangle= 2\sup\limits_{\|\varphi\|=1}\text{Im}\langle\varphi|(\mathbb{I}-{\Pi}_{t})\mathcal{H}_{t} {\Pi}_{t}|\varphi\rangle\\ &\leqslant&2\sup\limits_{\|\varphi\|=1}\|(\mathbb{I}-{\Pi}_{t})\varphi\|\|\mathcal{H}_{t}\|\|{\Pi}_{t}\varphi\|. \end{array} $$

(48)

Further, since π_t is a projector and ∥φ∥ = 1, we have $\|(\mathbb {I}-{\Pi }_{t})\varphi \|=\sqrt {1-\|{\Pi }_{t}\varphi \|^{2}}$. Therefore

$$ \|\dot{\Pi}_{t}\|\leqslant2\|\mathcal{H}_{t}\|\sup\limits_{\|\varphi\|=1} \sqrt{\|{\Pi}_{t}\varphi\|^{2}(1-\|{\Pi}_{t}\varphi\|^{2})}. $$

(49)

As $\sup _{x\in [0,1]}\sqrt {x(1-x)}=1/2$ then

$$ \|\dot{\Pi}_{t}\|\leqslant\|\mathcal{H}_{t}\|. $$

(50)

In view of (46) this proves the bound (29).

We note that the equality in (50) can be reached for ${\mathscr{H}}_{t}={\mathscr H}_{t}\equiv i[\dot {\Pi }_{t},{\Pi }_{t}]$ introduced in Section 3. Let us prove this fact. First, one verifies that ${\mathscr H}_{t}$ indeed generates π_t via (45), see (12), and hence

$$ \|\dot{\Pi}_{t}\|\leqslant\|{\mathscr H}_{t}\|. $$

(51)

On the other hand

$$ {\mathscr H}_{t}=i(\mathbb{I}-{\Pi}_{t})\dot{\Pi}_{t} {\Pi}_{t}-i {\Pi}_{t}\dot{\Pi}_{t}(\mathbb{I}-{\Pi}_{t}) $$

(52)

and we get

$$ \|{\mathscr H}_{t}\|\leqslant\|\dot{\Pi}_{t}\| $$

(53)

analogously to (50). Inequalities (51) and (53) imply

$$ \|\dot{\Pi}_{t}\|=\|{\mathscr H}_{t}\|. $$

(54)