**1. Introduction**

An essential step in the development of viable quantum technologies is to achieve precise control over quantum dynamics [1,2]. In many situations, optimal performance relies on the ability to create particular target states. However, in dynamically reaching such states, the quantum adiabatic theorem [3] poses a formidable challenge since finitetime driving inevitably causes parasitic excitations [4–7]. Acknowledging and addressing this issue, the field of "shortcuts to adiabaticity" (STA) [8–11] has developed a variety of techniques that permit to facilitate effectively adiabatic dynamics in finite time.

Recent years have seen an explosion of work on, for instance, counterdiabatic driving [12–19], the fast-forward method [20–23], time-rescaling [24,25], methods based on identifying the adiabatic invariant [26–29], and even generalizations to classical dynamics [30–32]. For comprehensive reviews of the various techniques, we refer to the recent literature [9–11].

Among these different paradigms, counterdiabatic driving (CD) stands out, as it is the only method that forces evolution through the adiabatic manifold at all times. However, experimentally realizing the CD method requires applying a complicated control field, which often involves non-local terms that are hard to implement in many-body systems [15,17]. This may be particularly challenging if the system is not readily accessible, due to, for instance, geometric restrictions of the experimental set-up.

In the present paper, we propose an alternative method to achieve transitionless quantum driving by leveraging the system's (realistically) inevitable interaction with the environment. Our novel paradigm is inspired by "envariance," which is short for entanglement-assisted invariance. Envariance is a symmetry of composite quantum systems, first described by Wojciech H. Zurek [33]. Consider a quantum state |*ψ*SE - that lives on a composite quantum universe comprising the system, S, and its environment, E. Then, |*ψ*SE - is called envariant under a unitary map *u*S ⊗ IE if and only if there exists another unitary IS ⊗ *u*E acting on E such that the composite state remains unaltered after applying both maps, i.e., (*<sup>u</sup>*S ⊗ IE )|*ψ*SE - = |*φ*SE - and (<sup>I</sup>S ⊗ *u*E )|*φ*SE - = |*ψ*SE -. In other words, the state is envariant if the action of a unitary on S can be inverted by applying a unitary on E.

**Citation:** Touil, A.; Deffner, S. Environment-Assisted Shortcuts to Adiabaticity. *Entropy* **2021**, *23*, 1479. https://doi.org/10.3390/e23111479

Academic Editor: Ronnie Kosloff

Received: 16 October 2021 Accepted: 5 November 2021 Published: 9 November 2021

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Envariance was essential to derive Born's rule [33,34], and in formulating a novel approach to the foundations of statistical mechanics [35]. Moreover, experiments [36,37] showed that this inherent symmetry of composite quantum states is indeed a physical reality, or rather a "quantum fact of life" with no classical analog [34]. Drawing inspiration from envariance, we develop a novel method for transitionless quantum driving. In the following, we will see that instead of inverting the action of a unitary on S, we can suppress undesirable transitions in the energy eigenbasis of S by applying a control field on the environment E. In particular, we consider the unitary evolution of an ensemble of composite states {| *ψ*SE -} on a Hilbert space HS ⊗ HE of arbitrary dimension, and we determine the general analytic form of the time-dependent driving on HE , which suppresses undesirable transitions in the system of interest S. This general driving on the environment E guarantees that the system S evolves through the adiabatic manifold at all times. We dub this technique environment-assisted shortcuts to adiabaticity, or "EASTA" for short. In addition, we prove that the cost associated with the EASTA technique is exactly equal to that of counterdiabatic driving. We illustrate our results in a simple two-qubit model, where the system and the environment are each described by a single qubit. Finally, we conclude with discussing a few implications of our results in the general context of decoherence theory and quantum Darwinism.

#### **2. Counterdiabatic Driving**

We start by briefly reviewing counterdiabatic driving to establish notions and notations. Consider a quantum system S, in a Hilbert space HS of dimension *d*S, driven by the Hamiltonian *<sup>H</sup>*0(*t*) with instantaneous eigenvalues {*En*(*t*)}*n*<sup>∈</sup>-0, *d*S−<sup>1</sup> and eigenstates {|*n*(*t*)-}*n*<sup>∈</sup>-0, *d*S−<sup>1</sup>. For slowly varying *<sup>H</sup>*0(*t*), according to the quantum adiabatic theorem [3], the driving of S is transitionless. In other words, if the system starts in the eigenstate |*n*(0)-, at *t* = 0, it evolves into the eigenstate |*n*(*t*)- at time *t* (with a phase factor) as follows:

$$|\psi\_n(t)\rangle \equiv \mathcal{U}(t)|n(0)\rangle = e^{-\frac{i}{\hbar}\int\_0^t \mathbb{E}\_n(s)ds - \int\_0^t \langle n|\partial\_s n\rangle ds}|n(t)\rangle \equiv e^{-\frac{i}{\hbar}f\_n(t)}|n(t)\rangle. \tag{1}$$

For arbitrary driving *<sup>H</sup>*0(*t*), namely for driving rates larger than the typical energy gaps, the system undergoes transitions. However, it was shown [12–14] that the addition of a counterdiabatic field *<sup>H</sup>*CD(*t*) forces the system to evolve through the adiabatic manifold. Using the following total Hamiltonian,

$$H = H\_0(t) + H\_{\infty}(t) = H\_0(t) + i\hbar \sum\_{n} (|\partial\_l n\rangle\langle n| - \langle n \mid \partial\_l n\rangle\langle n| \langle n|) ),\tag{2}$$

the system evolves with the corresponding unitary *<sup>U</sup>*CD(*t*) = ∑*n* |*ψn*(*t*)*n*(0)| such that the following holds:

$$
\langle \mathcal{U}\_{\rm cp}(t) | n(0) \rangle = e^{-\frac{i}{\hbar} f\_n(t)} | n(t) \rangle. \tag{3}
$$

This evolution is exact no matter how fast the system is driven by the total Hamiltonian. However, the counterdiabatic driving (CD) method requires adding a complicated counterdiabatic field *<sup>H</sup>*CD(*t*) involving highly non-local terms that are hard to implement in a many-body set-up [15,17]. Constructing this counterdiabatic field requires determining the instantaneous eigenstates {|*n*(*t*)-}*n*<sup>∈</sup>-0, *d*S−<sup>1</sup> of the time-dependent Hamiltonian *<sup>H</sup>*0(*t*). Moreover, changing the dynamics of the system of interest (i.e., adding the counterdiabatic field) requires direct access and control on S.

In the following, we will see how (at least) the second issue can be circumvented by relying on the environment E that inevitably couples to the system of interest. In particular, we make use of the entanglement between system and environment to avoid any transitions in the system. To this end, we construct the unique driving of the environment E that counteracts the transitions in S.

#### **3. Open System Dynamics and STA for Mixed States**

We start by stating three crucial assumptions: (i) the joint state of the system S and the environment E is described by an initial wave function |*ψ*SE (0)- evolving unitarily, according to the Schrödinger equation; (ii) the environment's degrees of freedom do not interact with each other; (iii) the S-E joint state belongs to the ensemble of singly branching states [38]. These branching states have the following general form:

$$|\psi\_{\mathcal{SE}}\rangle = \sum\_{n=0}^{N-1} \sqrt{p\_n} |n\rangle \bigotimes\_{l=1}^{N\_{\mathcal{E}}} |\mathcal{E}\_n^l\rangle\_{\prime} \tag{4}$$

where *pn* ∈ [0, 1] is the probability associated with the *n*th branch of the wave function, with orthonormal states |*n*-∈HS and *<sup>N</sup>*E *l*=1 |E*ln*-∈HE .

Without loss of generality, we can further assume √*pn* = 1/√*N* for all *n* ∈ -0, *N* − 1 since if √*pn* = 1/√*N* we can always find an extended Hilbert space [33,34] such that the state |*ψ*SE - becomes even. Thus, we can consider branching states |*ψ*SE - of the simpler form as follows:

$$|\psi\_{\mathcal{SE}}\rangle = \frac{1}{\sqrt{N}} \sum\_{n=0}^{N-1} |n\rangle \bigotimes\_{l=1}^{N\_{\mathcal{E}}} |\mathcal{E}\_n^l\rangle. \tag{5}$$

In the following, we will see that EASTA can actually only be facilitated for even states (5). In Appendix B, we show that EASTA cannot be implemented for arbitrary probabilities (i.e., (∃ *n*); √*pn* = 1/√*N*).

#### *3.1. Two-Level Environment* E

We start with the instructive case of a two-level environment, cf. Figure 1. To this end, consider the following branching state:

$$|\psi\_{\mathcal{SE}}(0)\rangle = \frac{1}{\sqrt{2}}|\mathbf{g}(0)\rangle \otimes |\mathcal{E}\_{\mathbb{X}}(0)\rangle + \frac{1}{\sqrt{2}}|\varepsilon(0)\rangle \otimes |\mathcal{E}\_{\varepsilon}(0)\rangle,\tag{6}$$

where the states |E*g*(0)- and |E*e*(0)- form a basis on the environment E, and the states |*g*(0)- and |*e*(0)- represent the ground and excited states of S at *t* = 0, respectively.

It is then easy to see that there exists a unique unitary *U* such that the system evolves through the adiabatic manifold in each branch of the wave function as follows:

$$(\exists!\,\mathcal{U}');\ (\mathcal{U}\otimes\mathcal{U}')|\psi\_{\mathcal{S}\mathcal{E}}(0)\rangle = (\mathcal{U}\_{\text{cp}}\otimes\mathbb{I}\_{\mathcal{E}})|\psi\_{\mathcal{S}\mathcal{E}}(0)\rangle.\tag{7}$$

Starting from the above equality, we obtain the following:

$$\begin{split} \langle \mathcal{U}|\mathcal{G}(0)\rangle \otimes \mathcal{U}'|\mathcal{E}\_{\mathcal{S}}(0)\rangle + \mathcal{U}|\varepsilon(0)\rangle \otimes \mathcal{U}'|\mathcal{E}\_{\varepsilon}(0)\rangle &= e^{-\frac{i}{\hbar}f\_{\mathcal{S}}(t)}|\mathcal{G}(t)\rangle \otimes |\mathcal{E}\_{\mathcal{S}}(0)\rangle \\ &+ e^{-\frac{i}{\hbar}f\_{\varepsilon}(t)}|\varepsilon(t)\rangle \otimes |\mathcal{E}\_{\varepsilon}(0)\rangle. \end{split} \tag{8}$$

Projecting the environment E into the state "|E*g*(0)-", we have

$$
\langle \mathcal{U} | \mathcal{G}(0) \rangle \langle \mathcal{E}\_{\mathcal{S}}(0) | \mathcal{U}' | \mathcal{E}\_{\mathcal{S}}(0) \rangle + \mathcal{U} | \varepsilon(0) \rangle \langle \mathcal{E}\_{\mathcal{S}}(0) | \mathcal{U}' | \mathcal{E}\_{\mathcal{E}}(0) \rangle = e^{-\frac{1}{\hbar} f\_{\mathcal{S}}(t)} | \mathcal{g}(t) \rangle,\tag{9}
$$

equivalently written as

$$
\langle (\mathcal{U}'\_{\mathcal{S}\circ \mathcal{S}}) \mathcal{U} | \mathcal{g}(0) \rangle + (\mathcal{U}'\_{\mathcal{S}^c}) \mathcal{U} | \varepsilon(0) \rangle = e^{-\frac{i}{\hbar} f\_{\mathcal{S}}(t)} | \mathcal{g}(t) \rangle,\tag{10}
$$

which implies the following:

$$(\mathcal{U}'\_{\mathcal{S},\emptyset})|\mathcal{g}(0)\rangle + (\mathcal{U}'\_{\mathcal{S},\mathcal{e}})|\mathcal{e}(0)\rangle = e^{-\frac{i}{\hbar}f\_{\mathcal{S}}(t)}\mathcal{U}^{\dagger}|\mathcal{g}(t)\rangle. \tag{11}$$

Therefore,

$$\mathcal{U}'\_{\mathcal{G},\mathcal{E}} = e^{-\frac{i}{\hbar}f\_{\mathcal{E}}(t)} \langle \mathcal{g}(0) | \mathcal{U}^{\dagger} | \mathcal{g}(t) \rangle,\text{ and }\mathcal{U}'\_{\mathcal{G},\mathcal{E}} = e^{-\frac{i}{\hbar}f\_{\mathcal{E}}(t)} \langle e(0) | \mathcal{U}^{\dagger} | \mathcal{g}(t) \rangle. \tag{12}$$

Additionally, by projecting E into the state "|E*e*(0)-" we obtain the following:

$$\mathcal{U}\_{\varepsilon,\xi}^{\prime} = e^{-\frac{i}{\hbar}f\_{\varepsilon}(t)} \langle g(0) | \mathcal{U}^{\dagger} | e(t) \rangle, \text{ and } \mathcal{U}\_{\varepsilon,\varepsilon}^{\prime} = e^{-\frac{i}{\hbar}f\_{\varepsilon}(t)} \langle e(0) | \mathcal{U}^{\dagger} | e(t) \rangle. \tag{13}$$

It is straightforward to check that the operator *U*, which reads as follows:

$$\mathcal{U}' = \begin{pmatrix} \mathcal{U}'\_{\mathcal{K}:\mathcal{S}} & \mathcal{U}'\_{\mathcal{K}:\mathcal{E}} \\ \mathcal{U}'\_{\mathcal{C}:\mathcal{S}} & \mathcal{U}'\_{\mathcal{C}:\mathcal{E}} \end{pmatrix} \tag{14}$$

is indeed a unitary on E.

In conclusion, we have constructed a unique unitary map that acts only on E, but counteracts transitions in S. Note that coupling the system and environment implies that the state of the system is no longer described by a wave function. Hence the usual counterdiabatic scheme evolves the density matrix *ρ*S (0) to another density *ρ*S (*t*) such that both matrices have the same populations and coherence in the instantaneous eigenbasis of *<sup>H</sup>*0(*t*) (which is what EASTA accomplishes, as well).

(**b**) Environment-assisted shortcut scheme.

**Figure 1.** Sketch of the two different schemes of applying STA in a branching state of the form presented in Equation (6). In both panels, the state preparation involves a Hadamard gate (H) applied on S, and coupling with the environment through a c-not operation. In panel (**a**), we describe the "usual" counterdiabatic scheme. As shown in Section 3, local driving on E suppresses any transitions of S in the instantaneous eigenbasis of *<sup>H</sup>*0(*t*). The latter scheme is illustrated in panel (**b**).

#### *3.2. N-Level Environment* E

We can easily generalize the two-level analysis to an *N*-level environment. Similar to the above description, coupling the system to the environment leads to a branching state of the following form:

$$|\psi\_{S\mathcal{E}}(0)\rangle = \frac{1}{\sqrt{N}}\sum\_{n=0}^{N-1}|n(0)\rangle \otimes |\mathcal{E}\_n(0)\rangle,\tag{15}$$

where the states {|E*n*(0)-}*n* form a basis on the environment E. We can then construct a unique unitary *U* such that the system evolves through the adiabatic manifold in each branch of the wave function as follows:

$$<\langle \exists! \, \mathcal{U} \rangle ; \ (\mathcal{U} \otimes \mathcal{U}') | \psi\_{\mathcal{S} \mathcal{E}}(0) \rangle = (\mathcal{U}\_{\text{cp}} \otimes \mathbb{I}\_{\mathcal{E}}) | \psi\_{\mathcal{S} \mathcal{E}}(0) \rangle. \tag{16}$$

The proof follows the exact same strategy as the two-level case, and we find the following:

$$(\forall \ (m, n) \in [0, N - 1]^2); \ l\!\!/^{\prime}\_{m, n} = e^{-\frac{i}{\hbar}f\_m(t)} \langle n(0) | \!\!/ \!L^{\dagger} | m(t) \rangle. \tag{17}$$

The above expression of the elements of the unitary *U* is our main result, which holds for any driving *<sup>H</sup>*0(*t*) and any *N*-dimensional system.

#### *3.3. Process Cost*

Having established the general analytic form of the unitary applied on the environment, the next logical step is to compute and compare the cost of both schemes: (a) the usual counterdiabatic scheme and (b) the environment-assisted shortcut scheme presented above (cf. Figure 1). More specifically, we now compare the time integral of the instantaneous cost [39] for both driving schemes [39–43], (a) *<sup>C</sup>*CD(*t*)=(1/*τ*) *t* 0 \$*<sup>H</sup>*CD(*s*)\$*ds* and (b) *<sup>C</sup>*env(*t*)=(1/*τ*) *t* 0 \$*<sup>H</sup>*env(*s*)\$*ds* (\$.\$ is the operator norm), where the driving Hamiltonian on the environment can be determined from the expression of *U* (*t*), *<sup>H</sup>*env(*t*) = *ih*¯ *dU* (*t*) *dt<sup>U</sup>*†(*t*).

In fact, from Equation (17) it is not too hard to see that the field applied on the environment *<sup>H</sup>*env(*t*) has the same eigenvalues as the counterdiabatic field *<sup>H</sup>*CD(*t*), since there exists a similarity transformation between *<sup>H</sup>*env(*t*) and *<sup>H</sup>*<sup>∗</sup>CD(*t*). Therefore, the cost of both processes is exactly the same, *C*CD = *C*env, for any arbitrary driving *<sup>H</sup>*0(*t*). Details of the derivation can be found in Appendix A. Note that for *t* = *τ*, the above definition of the cost becomes the total cost for the duration "*τ*" of the process.
