Lindbladian purification

Noncommutativity is one of the key features of quantum mechanics. The order in which operations and/or measurements are performed influences the outcomes of an experiment. In particular, in the Lie-theoretical approach to quantum control theory [1], the noncommutativity plays an important role. The goal of quantum control is to steer a quantum system to realise a desired transformation on it by shaping classical time-dependent fields [2]. Here, the noncommutativity of the generators associated with the control fields influences the complexity of the resulting dynamics. For instance, for two commuting Hamiltonians, H₁ and H₂, that can be switched on and off by external control fields, the resulting unitary evolution is just equivalent to the one generated by a linear combination of H₁ and H₂. On the contrary, by properly concatenating transformations induced by two noncommuting Hamiltonians, one can produce effective evolutions associated with generators which are linearly independent of the original ones, enabling the system to explore more 'directions.' For a finite-dimensional closed quantum system Q, the set of effective evolutions that can be implemented in this way is formed by the unitaries of the Lie group ${e}^{{\mathfrak{L}}({\bf{H}})}$ generated by the dynamical Lie algebra ${\mathfrak{L}}({\bf{H}})$ associated to the set of control Hamiltonians ${\bf{H}}:= \{{H}_{1},{H}_{2},\ldots \}$, i.e., the real vector space spanned by all possible linear combinations of the elements of ${\bf{H}}$ and their iterated commutators [1, 3, 4]. Accordingly, a closed system Q characterised by a control set ${\bf{H}}$ is said to be fully controllable if ${e}^{{\mathfrak{L}}({\bf{H}})}$ includes all possible unitary transformations on Q, or equivalently, if the dynamical Lie algebra ${\mathfrak{L}}({\bf{H}})$ spans the whole operator algebra of Q, this last property of ${\bf{H}}$ being also referred to as accessibility.

When it comes to open quantum systems, the characterisation of the reachable (realisable) operations, as well as the associated notion of controllability, becomes more complicated since the allowed operations do not possess a group structure and the notion of dynamical generators is typically lost [5, 6]. A partial exception is provided by the subset of Markov processes, which are equipped with a semigroup structure and admit the notion of dynamical generators, i.e., the super-operators of Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form [7, 8] (Lindbladians in the following). Still, also in this case, determining which dynamics can be activated by controlling a given collection ${\bf{L}}:= \{{{ \mathcal L }}_{1},{{ \mathcal L }}_{2},\ldots \}$ of Lindbladians is a difficult unsolved problem. One would be tempted to tackle it by studying the Lie algebra ${\mathfrak{L}}({\bf{L}})$ generated by ${\bf{L}}$ and the corresponding Lie group ${e}^{{\mathfrak{L}}({\bf{L}})}$. However, at variance with the closed system scenario, linear combinations and commutators of elements of ${\bf{L}}$ will in general produce super-operators that are no longer allowed dynamical generators (e.g., they cannot be cast in the GKLS form), or said it differently, ${e}^{{\mathfrak{L}}({\bf{L}})}$ will include transformations which are unphysical. Furthermore, even for the elements of ${e}^{{\mathfrak{L}}({\bf{L}})}$ which are physically allowed, it is in general not clear if it would be possible to implement them by simply playing with the control fields. In view of these facts for open quantum systems, one distinguishes the physical notion of controllability, i.e., the ability of using ${\bf{L}}$ and the classical fields which activate them to perform all physically allowed quantum transformations, from the weaker notion of accessibility, which in this case corresponds to have ${\mathfrak{L}}({\bf{L}})$ equal to whole Lie algebra generated by arbitrary Lindbladians. In the following, we will refer to this as the 'GKLS algebra' noting, however, that it contains many elements which are not of GKLS form. Different from the closed quantum system case, it is indeed possible that a control set ${\bf{L}}$ is accessible but not controllable. Still studying the accessibility of a collection of Lindbladians is a well-posed mathematical problem, which can also shed light on the controllability issue, with accessibility being a necessary condition for controllability. Furthermore, accessibility implies that the reachable set has non-zero volume and therefore has physical relevance: the short time dynamics explores a high dimensional space and is therefore of high complexity.

It turns out that almost all control sets ${\bf{L}}$ are accessible [7]. Analogously to the case of closed systems, the key ingredient of this result can be identified with the noncommutativity of the elements of ${\bf{L}}$, but what about models where ${\bf{L}}$ includes only mutually commutating Lindbladians? Is there a way to expand their algebra ${\mathfrak{L}}({\bf{L}})$ to cover the full GKLS algebra? For closed quantum systems it has been observed that one can substantially change the dimension of the dynamical Lie algebra ${\mathfrak{L}}({\bf{H}})$ through frequently observing a part of the system [9], or by tampering it with a strong dissipative process that exhibits decoherence-free subspaces [10] (the gain being exponential in some cases). As a matter of fact, on the basis of the quantum Zeno effect [11], starting with a set of commuting control Hamiltonians ${\bf{H}}$, noncommutativity can be enforced through frequently projecting out part of the system onto a subspace where accessibility and hence full controllability is achieved. Also, it has been observed that the projection trick can be reversed: specifically, starting from a set of noncommutative Hamiltonians ${\bf{H}}$, one can construct a new set $\tilde{{\bf{H}}}$ formed by commutative elements on an extended Hilbert space which under projection reduces to the original one. This mechanism was studied in great detail in [12], where, borrowing from the notion of purification of mixed quantum states [13], the term Hamiltonian purification was introduced.

One may then ask whether a similar procedures can be applied to the algebra of a set ${\bf{L}}$ of Lindbladians, namely, if it is possible to enlarge ${\mathfrak{L}}({\bf{L}})$ by means of some projection mechanisms and, on the contrary, if Lindbladian purification is always achievable. In this paper, we address these issues showing that indeed any set ${\bf{L}}$ of Lindbladians can be 'purified,' i.e., can be made commutative with each other, by embedding them in a larger space (note that the term 'pure' was already used in [14] for Markovian generators in a slightly different way). To this end, we need to employ a different scheme from those for the Hamiltonian purification introduced in [12], since the naive application of the latter trivially violates some structural properties of GKLS generators (more details in the following). Our construction allows one to make Lindbladians and Hamiltonians commutative on an extended space by means of an auxiliary system, which, through frequent non-selective measurements, yields the original noncommutative dynamics. Moreover, we present a universal pair of Lindbladians that generate the full GKLS dynamical Lie algebra for generic system sizes, the analysis providing us with a short and elementary proof of the generic accessibility [7]. Applying hence the Lindbladian purification procedure to such a universal set, we then show that almost all open systems become accessible even though their generators are commutative with each other, by performing frequent non-selective measurements on a part of the system.

This paper is organised as follows. Along the lines of [12], we begin in section 2 by reviewing the definition of Hamiltonian purification and presenting an explicit construction for purifying an arbitrary number of Lindbladians and Hamiltonians. In section 3, we consider the accessibility of controlled master equations. Concluding remarks are given in section 4 and some details on the derivation of the projected dynamics and the proof of accessibility are provided in the appendices.

To begin with, we first review the definition of Hamiltonian purification [12]. Suppose that we have n control Hamiltonians, which are switched on and off to steer a d-dimensional quantum system Q. Let ${\bf{H}}=\{{H}_{1},\ldots ,{H}_{n}\}$ be the set of the control Hamiltonians acting on the Hilbert space ${{ \mathcal H }}_{d}$ of Q, and $\tilde{{\bf{H}}}=\{{\tilde{H}}_{1},\ldots ,{\tilde{H}}_{n}\}$ be a corresponding set of Hamiltonians acting on an extended Hilbert space ${{ \mathcal H }}_{{d}_{E}}$ of dimension ${d}_{E}(\gt d)$, which includes ${{ \mathcal H }}_{d}$ as a proper subspace. We call $\tilde{{\bf{H}}}$ a purifying set of ${\bf{H}}$ if all the elements of $\tilde{{\bf{H}}}$ commute with each other,

$$\begin{eqnarray}&&{\tilde{H}}_{i}{\tilde{H}}_{j}={\tilde{H}}_{j}{\tilde{H}}_{i},\quad \forall \,i,j\in \{1,\ldots ,n\},\end{eqnarray} \tag{ 1 }$$

and they are related to those from ${\bf{H}}$ through

$$\begin{eqnarray}&&{H}_{j}=P{\tilde{H}}_{j}P,\quad \forall \,j\in \{1,\ldots ,n\},\end{eqnarray} \tag{ 2 }$$

with P being the projection onto ${{ \mathcal H }}_{d}$. For a generic set ${\bf{H}}$ consisting of n linearly independent Hamiltonians, it can be shown [12] that there always exists an $\tilde{{\bf{H}}}$ where the minimal dimension ${d}_{E}^{(\min )}$ of the extended Hilbert space is bounded above by ${d}_{E}^{(\min )}\leqslant {nd}$. For instance, for the case with n  =  2 Hamiltonians H₁ and H₂, proposition 1 of [12] states that a purifying set can be constructed on ${{ \mathcal H }}_{{d}_{E}}={{ \mathcal H }}_{d}\otimes {{ \mathcal H }}_{{d}_{A}}$ with an auxiliary single qubit Hilbert space ${{ \mathcal H }}_{{d}_{A}}$, the purifications and the projector being

$$\begin{eqnarray}{\tilde{H}}_{1} & = & {H}_{1}\otimes {{\mathbb{1}}}_{2}+{H}_{2}\otimes {\sigma }_{x},\\ {\tilde{H}}_{2} & = & {H}_{2}\otimes {{\mathbb{1}}}_{2}+{H}_{1}\otimes {\sigma }_{x},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&P={{\mathbb{1}}}_{d}\otimes \displaystyle \frac{{{\mathbb{1}}}_{2}+{\sigma }_{z}}{2},\end{eqnarray} \tag{ 4 }$$

with ${\sigma }_{x}$, ${\sigma }_{z}$, and ${{\mathbb{1}}}_{2}$ the Pauli and the identity operators of the auxiliary qubit, respectively. The mapping ${\tilde{H}}_{j}\to {H}_{j}$ can finally be realised through the quantum Zeno effect [11, 15] by frequently monitoring the extended system via a von Neumann measurement which projects the system onto ${{ \mathcal H }}_{d}$, i.e.,

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{N\to \infty }{({{Pe}}^{-i{\tilde{H}}_{j}t/N}P)}^{N}={e}^{-{{iH}}_{j}t}P.\end{eqnarray} \tag{ 5 }$$

The question arises if an analogous construction can be extended to the case of Lindbladians. Specifically, consider a set ${\bf{L}}=\{{{ \mathcal L }}_{1},\ldots ,{{ \mathcal L }}_{n}\}$ of n GKLS generators operating on a target system Q,

$$\begin{eqnarray}&&{{ \mathcal L }}_{j}={{ \mathcal K }}_{j}+{{ \mathcal D }}_{j},\quad j\in \{1,\ldots ,n\},\end{eqnarray} \tag{ 6 }$$

with ${{ \mathcal K }}_{j}$ and ${{ \mathcal D }}_{j}$ being the Hamiltonian and dissipator contributions, i.e., the super-operators

$$\begin{eqnarray}&&{{ \mathcal K }}_{j}(\cdots )=-i[{H}_{j},\cdots ],\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}{{ \mathcal D }}_{j}(\cdots ) & = & \displaystyle \sum _{\alpha }[2{L}_{j,\alpha }(\cdots ){L}_{j,\alpha }^{\dagger }-{L}_{j,\alpha }^{\dagger }{L}_{j,\alpha }(\cdots )-(\cdots ){L}_{j,\alpha }^{\dagger }{L}_{j,\alpha }],\end{eqnarray} \tag{ 8 }$$

${L}_{j,\alpha }$ being the Lindblad operators acting on the Hilbert space ${{ \mathcal H }}_{d}$ of Q. We ask whether if it is possible to associate with ${\bf{L}}$ a purifying set $\tilde{{\bf{L}}}=\{{\tilde{{ \mathcal L }}}_{1},\ldots ,{\tilde{{ \mathcal L }}}_{n}\}$ formed by GKLS generators possibly acting on an extended system, which are mutually commuting, i.e.,

$$\begin{eqnarray}&&{\tilde{{ \mathcal L }}}_{i}\,\circ \,{\tilde{{ \mathcal L }}}_{j}={\tilde{{ \mathcal L }}}_{j}\,\circ \,{\tilde{{ \mathcal L }}}_{i},\quad \forall \,i,j\in \{1,\ldots ,n\},\end{eqnarray} \tag{ 9 }$$

from which one can recover the original elements via a projective mapping that should mimic equation (5). (In the above expressions, we used the symbol '$\circ$' to indicate the composition of super-operators).

A natural guess for identifying $\tilde{{\bf{L}}}$ and the projective mapping would be to simply transporting the purification schemes of [12] at super-operator level, or equivalently, to represent the ${{ \mathcal L }}_{j}{\rm{s}}$ as operators in Liouville space [16] and then simply applying to them the Hamiltonian purification scheme. This simple trick, however, does not work because, for instance, mapping as equation (3) will take positive operators into non-positive one, hence spoiling one fundamental property of GKLS generators. Another problem comes from the fact that for Markovian open systems described by Lindbladians, the quantum Zeno effect, which as we have seen is responsible for the implementation of the mapping ${\tilde{H}}_{j}\to {H}_{j}$, does not take place: a Markovian system can leak from one subspace specified by the projection operator belonging to a measurement outcome even in the limit of infinitely frequent projective measurements. In spite of these issues, however, a Lindbladian purification scheme can be obtained with the following simple construction:

A purifying set $\tilde{{\bf{L}}}$ can always be constructed by introducing an auxiliary Hilbert space ${{ \mathcal H }}_{n}$ of dimension n and identifying the Hamiltonians $\{{\tilde{H}}_{j}\}$ and the Lindblad operators $\{{\tilde{L}}_{j,\alpha }\}$ of the purifying element ${\tilde{{ \mathcal L }}}_{j}={\tilde{{ \mathcal K }}}_{j}+{\tilde{{ \mathcal D }}}_{j}$ as

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\tilde{H}}_{j}={{nH}}_{j}\otimes | j\rangle \langle j| ,\\ {\tilde{L}}_{j,\alpha }=\sqrt{n}\,{L}_{j,\alpha }\otimes | j\rangle \langle j| ,\end{array}\right.\quad j\in \{1,\ldots ,n\},\end{eqnarray} \tag{ 10 }$$

with $\{| j\rangle \}\,{}_{j=1}^{n}$ being an orthonormal basis for ${{ \mathcal H }}_{n}$.

Obviously, through such a construction the operators $\{{\tilde{H}}_{j}\}$ and $\{{\tilde{L}}_{j,\alpha }\}$ commute with each other for different j, trivially ensuring the requirement (equation (9)). Regarding the analogue of equation (5), we focus on non-selective projective measurement [17, 18] operating on the auxiliary system, i.e., the completely positive and trace preserving (CPTP) mapping of the form

$$\begin{eqnarray}&&{ \mathcal P }(\cdots )=\displaystyle \sum _{k}{P}_{k}(\cdots ){P}_{k},\end{eqnarray} \tag{ 11 }$$

given in terms of a complete set of orthonormal projection operators $\{{P}_{k}\}$ corresponding to measurement outcomes and satisfying ${P}_{k}{P}_{k^{\prime} }={\delta }_{{kk}^{\prime} }{P}_{k}$ and ${\sum }_{k}{P}_{k}={\mathbb{1}}$. Notice that if we perform $(N+1)$ of such non-selective measurements at regular time intervals t/N during the evolution driven by a Lindbladian ${ \mathcal L }$, the system will evolve according to the CPTP transformation

$$\begin{eqnarray}{{\rm{\Phi }}}_{t,N}^{({ \mathcal L })} & := & {({ \mathcal P }\circ {e}^{{ \mathcal L }t/N}\circ { \mathcal P })}^{N}\\ & = & {\left[\mathrm{id}+({ \mathcal P }\circ { \mathcal L }\circ { \mathcal P })\tfrac{t}{N}+O\left(\tfrac{{t}^{2}}{{N}^{2}}\right)\right]}^{N}\,\circ \,{ \mathcal P },\end{eqnarray} \tag{ 12 }$$

which in the limit of $N\to \infty$ converges to

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{t,\infty }^{({ \mathcal L })}=\mathop{\mathrm{lim}}\limits_{N\to \infty }{{\rm{\Phi }}}_{t,N}^{({ \mathcal L })}={e}^{({ \mathcal P }\circ { \mathcal L }\circ { \mathcal P })t}\,\circ \,{ \mathcal P },\end{eqnarray} \tag{ 13 }$$

where $\mathrm{id}$ is the identity map and where we used the idempotent property ${ \mathcal P }={ \mathcal P }\,\circ \,{ \mathcal P }={{ \mathcal P }}^{2}$ of (11). Equation (13) can also be derived following a pertubative approach with a strong amplitude-damping channel inducing the projection ${ \mathcal P }$ [19–21]. In our construction, equation (13) is the formal counterpart of the Zeno limit (equation (5)): it shows that alternating the dynamics induced by a GKLS generator ${ \mathcal L }$ with ${ \mathcal P }$ induces on the system an evolution which can be effectively described in terms of an effective dynamical generator described by the projected super-operator ${ \mathcal P }\,\circ \,{ \mathcal L }\,\circ \,{ \mathcal P }$. It should be stressed that the latter is not in GKLS form, i.e., it is not a Lindbladian. Indeed it acts as a proper Lindbladian only within the subspace specified by the superprojector ${ \mathcal P }$, but the map ${e}^{({ \mathcal P }\circ { \mathcal L }\circ { \mathcal P })t}$ itself is not CPTP (an explicit example of this fact is provided in appendix A). Still, we are going to identify equation (13) with the mechanism that yields the original Lindbladians ${{ \mathcal L }}_{j}\in {\bf{L}}$ expressed in the form of equations (6)–(8) from their purified counterparts ${\tilde{{ \mathcal L }}}_{j}$ of with equation (10). For this purpose, we assume the projectors P_k in equation (11) to be of the form

$$\begin{eqnarray}&&{P}_{k}={\mathbb{1}}\otimes | {\phi }_{k}\rangle \langle {\phi }_{k}| ,\end{eqnarray} \tag{ 14 }$$

where $\{| {\phi }_{k}\rangle \}\,{}_{k=1}^{{d}_{A}}$ is an orthonormal basis for the auxiliary Hilbert space which is chosen to be mutually unbiased [22] against the orthonormal basis $\{| j\rangle \}\,{}_{j=1}^{{d}_{A}}$ used for the purification (10). Then, as shown in appendix B, one can verify that under the transformation (equation (13)) a generic density operator ${\rho }_{Q}(0)$ for the original system, obtained by taking the trace over the auxiliary Hilbert space ${{ \mathcal H }}_{{d}_{A}}$, evolves according to

$$\begin{eqnarray}&&{\rho }_{Q}(t)={e}^{{{ \mathcal L }}_{j}t}{\rho }_{Q}(0),\end{eqnarray} \tag{ 15 }$$

recovering hence the original dynamics generated by the unpurified Lindbladian ${{ \mathcal L }}_{j}$.

As a simple example of Lindbladian purification, we consider amplitude damping (AD) and pure dephasing (PD) in x direction of a single qubit. Within the Born–Markov approximation, the corresponding Lindbladians ${{ \mathcal L }}_{{\rm{AD}}}$ and ${{ \mathcal L }}_{{\rm{PD}}}$ are typically used to describe the main noise sources in two level systems [13, 23]. The Lindblad operators read ${L}_{{\rm{AD}}}=\sigma ,\,{L}_{{\rm{PD}}}=\sigma +{\sigma }^{\dagger }$, where $\sigma =| 0\rangle \langle 1|$ is the atomic lowering operator and we note that ${{ \mathcal L }}_{{\rm{AD}}}$ and ${{ \mathcal L }}_{{\rm{PD}}}$ do not commute. According to equation (10), a purifying set $\{{\tilde{{ \mathcal L }}}_{{\rm{AD}}},{\tilde{{ \mathcal L }}}_{{\rm{PD}}}\}$ is obtained through the purified Lindblad operators

$$\begin{eqnarray}&&{\tilde{L}}_{{\rm{AD}}}=\sigma \otimes \sigma {\sigma }^{\dagger },\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\tilde{L}}_{{\rm{PD}}}=(\sigma +{\sigma }^{\dagger })\otimes {\sigma }^{\dagger }\sigma ,\end{eqnarray} \tag{ 17 }$$

and a frequent non-selective measurement (equation (11)) with projectors ${P}_{\pm }={\mathbb{1}}\otimes | \pm \rangle \langle \pm |$ where $| \pm \rangle =(1/\sqrt{2})(| 0\rangle \pm | 1\rangle )$ recovers the unpurified dynamics. We note here that Lindblad operators similar to the purified versions, equations (16) and (17), were introduced in [24] for bosonic systems.

We now turn our attention to the question on how frequent non-selective measurements can enrich the algebra ${\mathfrak{L}}({\bf{L}})$ of a Markovian open quantum system described by a collection ${\bf{L}}$ of controlled generators. Specifically, we shall focus on systems driven by master equations of the form

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\rho (t)={ \mathcal L }(t)\rho (t),\end{eqnarray} \tag{ 18 }$$

where the super-operator ${ \mathcal L }(t)={ \mathcal K }(t)+{ \mathcal D }$ is provided by a constant dissipative part represented by Lindblad operators L_α, and by a time-dependent Hamiltonian term ${ \mathcal K }(t)(\cdots )=-i[H(t),\cdots ]$ with

$$\begin{eqnarray}&&H(t)={H}_{0}+\displaystyle \sum _{k=1}^{m}{u}_{k}(t){H}_{k},\end{eqnarray} \tag{ 19 }$$

${\{{u}_{k}(t)\}}_{k=1}^{m}$ being classical control fields that can be operated to switch on and off m control Hamiltonians ${\{{H}_{k}\}}_{k=1}^{m}$. This corresponds to having a control set ${\bf{L}}:= \{{{ \mathcal L }}_{0},{{ \mathcal K }}_{1},\ldots ,{{ \mathcal K }}_{m}\}$ consisting of a drift (unmodulated) term,

$$\begin{eqnarray}&&{{ \mathcal L }}_{0}={{ \mathcal K }}_{0}+{ \mathcal D },\end{eqnarray} \tag{ 20 }$$

that includes both the dissipative part ${ \mathcal D }$ and the Hamiltonian contribution ${{ \mathcal K }}_{0}(\cdots )=-i[{H}_{0},\cdots ]$, and of the set of Hamiltonian control generators

$$\begin{eqnarray}&&{{ \mathcal K }}_{k}(\cdots )=-i[{H}_{k},\cdots ],\quad k\in \{1,\ldots ,m\}.\end{eqnarray} \tag{ 21 }$$

As already mentioned in the introduction, for a closed quantum system, i.e., without the dissipative part ${ \mathcal D }$, the algebra ${\mathfrak{L}}({\bf{L}})$ associated with ${\bf{L}}$ (i.e., the set of all real linear combinations and iterated commutators of these elements, drift term included) will fully characterise the set of unitary operations that can be implemented through shaping the control functions ${\{{u}_{k}(t)\}}_{k=1}^{m}$. For an open quantum system described by the master equation (18), instead, ${\mathfrak{L}}({\bf{L}})$ only characterises the accessibility of the system. For a detailed analysis of the general structure of ${\mathfrak{L}}({\bf{L}})$ and simple examples, we refer to [7]. Here, we focus instead on studying how the purification mechanism can influence the dimension of ${\mathfrak{L}}({\bf{L}})$. In particular we shall see how a set of commutative Lindbladians can be turned into a new set of noncommutative Lindbladians which grant accessibility to the full GKLS algebra via the projection through frequent measurements.

To show this, we start by showing that it is possible to identify a set ${\bf{L}}$ formed by just a pair of Lindbladians whose algebra ${\mathfrak{L}}({\bf{L}})$ spans the full GKLS algebra. We therefore first prove that the pair

$$\begin{eqnarray}&&{{ \mathcal L }}_{0}=-i\,{\mathrm{ad}}_{{H}_{0}}+{{ \mathcal D }}_{| 1\rangle \langle 2| },\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{ \mathcal K }=-i\,{\mathrm{ad}}_{| 1\rangle \langle 1| }\end{eqnarray} \tag{ 23 }$$

with

$$\begin{eqnarray}&&{H}_{0}=\displaystyle \sum _{j=1}^{d-1}| j\rangle \langle j+1| +{\rm{h}}.{\rm{c}}\,.,\end{eqnarray} \tag{ 24 }$$

where

$$\begin{eqnarray}&&-i\,{\mathrm{ad}}_{H}(\cdots )=-i[H,\cdots ],\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{{ \mathcal D }}_{L}(\cdots )=2L(\cdots ){L}^{\dagger }-{L}^{\dagger }L(\cdots )-(\cdots ){L}^{\dagger }L,\end{eqnarray} \tag{ 26 }$$

does the job, namely, every possible Lindbladian can be generated by linear combinations and iterated commutators of equations (22) and (23). We only sketch the main steps here, whereas the details can be found in appendix C. In the following we also use the notations

$$\begin{eqnarray}&&{{ \mathcal D }}_{A,B}(\cdots )=2B(\cdots ){A}^{\dagger }-{A}^{\dagger }B(\cdots )-(\cdots ){A}^{\dagger }B,\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&{\mathrm{Ad}}_{U}(\cdots )=U(\cdots ){U}^{\dagger }.\end{eqnarray} \tag{ 28 }$$

We first note that terms of the form $-i\,{\mathrm{ad}}_{| j\rangle \langle j| }$ commute with the dissipative part ${{ \mathcal D }}_{| 1\rangle \langle 2| }$ and according to [25] we can generate every element in $-i\,{\mathrm{ad}}_{{\mathfrak{u}}(d)}$ with ${\mathfrak{u}}(d)$ the Lie algebra of d × d hermitian matrices. Using ${\mathrm{Ad}}_{U(d)}=\exp (-i\,{\mathrm{ad}}_{{\mathfrak{u}}(d)})$ with $U(d)$ being the unitary group, we can get

$$\begin{eqnarray}&&{\mathrm{Ad}}_{U}\,\circ \,{{ \mathcal D }}_{| 1\rangle \langle 2| }\,\circ \,{\mathrm{Ad}}_{{U}^{\dagger }}={{ \mathcal D }}_{U| 1\rangle \langle 2| {U}^{\dagger }}\end{eqnarray} \tag{ 29 }$$

for any $U\in U(d)$. Now we consider unitaries U that act as $U| j\rangle ={\sum }_{k\in { \mathcal I }}{c}_{k}^{(j)}| k\rangle$ for $j=1,2$ and ${ \mathcal I }=\{1,2,3,4\}$. We numerically verified that, from ${{ \mathcal D }}_{U| 1\rangle \langle 2| {U}^{\dagger }}$, thus created together with $-i{\mathrm{ad}}_{H}$ for all Hamiltonians $H={\sum }_{i,j\in { \mathcal I }}{h}_{{ij}}| i\rangle \langle j|$ having support on ${ \mathcal I }$, all the operators of the form

$$\begin{eqnarray}&&\left\{\begin{array}{l}{{ \mathcal D }}_{| i\rangle \langle j| ,| k\rangle \langle l| }+{{ \mathcal D }}_{| k\rangle \langle l| ,| i\rangle \langle j| },\\ i({{ \mathcal D }}_{| i\rangle \langle j| ,| k\rangle \langle l| }-{{ \mathcal D }}_{| k\rangle \langle l| ,| i\rangle \langle j| }),\end{array}\right.\quad i,j,k,l\in { \mathcal I }\end{eqnarray} \tag{ 30 }$$

can be generated. Doing the same for different quartets ${ \mathcal I }=\{i,j,k,l\}$, we are able to provide linearly independent operators (equation (30)) for all $i,j,k,l\in \{1,\ldots ,d\}$. Since any Lindbladian can be written in the Kossakowski form as a linear combination of those operators, it means that every Lindbladian can be generated through iterated commutators and linear combinations of the pair of generators $\{{{ \mathcal L }}_{0},{ \mathcal K }\}$ in equations (22) and (23). Given that this specific pair of Lindbladians is accessible, it then follows from the standard argument (see, e.g., [9]) that almost all pairs are. This was shown previously in a more abstract way by Kurniawan [7].

Now that we have found a pair ${\bf{L}}=\{{{ \mathcal L }}_{0},{ \mathcal K }\}$ that describes an accessible quantum system in arbitrary dimensions, we can make them commutative using a two-dimensional (${d}_{A}=2$) auxiliary Hilbert space, i.e., we can purify them to

$$\begin{eqnarray}&&{\tilde{{ \mathcal L }}}_{0}=-2i{\mathrm{ad}}_{{H}_{0}\otimes | 2\rangle \langle 2| }+\sqrt{2}\,{{ \mathcal D }}_{| 1\rangle \langle 2| \otimes | 2\rangle \langle 2| },\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&\tilde{{ \mathcal K }}=-2i{\mathrm{ad}}_{| 1\rangle \langle 1| \otimes | 1\rangle \langle 1| }.\end{eqnarray} \tag{ 32 }$$

Obviously on the extended Hilbert space, the Lie algebra associated with the set $\tilde{{\bf{L}}}=\{{\tilde{{ \mathcal L }}}_{0},\tilde{{ \mathcal K }}\}$ is just two-dimensional, $\dim {\mathfrak{L}}(\tilde{{\bf{L}}})=2$, and the system is not accessible. If we perform frequent non-selective projective measurements on the auxiliary system described by the superprojector (equation (11)) with ${P}_{\pm }={\mathbb{1}}\otimes | \pm \rangle \langle \pm |$, where $| \pm \rangle$ are defined at the end of section 2, the original dynamics is recovered as equation (15) and the system becomes accessible. The existence of such a specific setup allows us to conclude [9] that almost all open quantum systems become accessible by Zeno measurements.

We generalised the work [12] on Hamiltonian purification by establishing a new and simple purification scheme for Lindbladians, which is also applicable to Hamiltonians. Given n Lindbladians, they can be made commutative by adding an n-dimensional auxiliary system to extend the Lindblad operators with hermitian projectors that form an orthonormal basis for the auxiliary space. Through the projection by Zeno measurements for semigroup dynamics, the original possibly noncommutative dynamics can be recovered by frequently measuring the auxiliary system in a non-selective way. The purification of more general dynamical maps with non-Markovian dynamics is left for future studies.

Moreover, we have proven that the pair of Lindbladians (equations (22) and (23)) describe an accessible open quantum system for generic system sizes, which tells us that generally a nonaccessible open quantum system is turned into an accessible one by frequent non-selective measurements. The model has also potential applications in simulating an arbitrary Markovian open-system dynamics [26, 27] by steering it through control fields.

Clearly, the presented purification scheme also works for observables and density operators, although, except for the partial trace, an operational way that allows us to recover the original observables and states is not known to us. Since the noncommutativity is a unique feature of quantum mechanics, and in fact it was argued in [28, 29] that the noncommutativity distinguishes between quantum and classical mechanics, it is tempting to say that every quantum system can be made classical by purifying it to a larger space. However, we remark here that this is only the case in a dynamical sense, i.e. the dynamics can be made commutative but the observable algebra associated with the system still remains noncommutative.

We would like to thank John Gough for fruitful discussions. This work was supported by the Top Global University Project from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. DB acknowledges support from EPSRC grant EP/M01634X/1. KY was supported by the Grant-in-Aid for Scientific Research (C) (No. 26400406) from the Japan Society for the Promotion of Science (JSPS) and by the Waseda University Grant for Special Research Project- (No. 2016K-215). HN was supported by the Waseda University Grant for Special Research Project (No. 2016B-173).

As an example of the fact that the projected counterpart ${ \mathcal P }\,\circ \,{ \mathcal L }\,\circ \,{ \mathcal P }$ of a Lindbladian ${ \mathcal L }$ does not generate proper quantum dynamics, consider for instance the case where ${ \mathcal L }$ describes a qubit amplitude damping with fixed point $| 0\rangle \langle 0|$ (this is characterised by a null Hamiltonian term H  =  0 and a unique Lindblad operator ${L}_{\alpha }=| 0\rangle \langle 1|$) and where the transformation ${ \mathcal P }$ is the dephasing map [13] associated with the canonical qubit base, i.e.,

$$\begin{eqnarray}&&{ \mathcal P }(\cdots )=| 0\rangle \langle 0| (\cdots )| 0\rangle \langle 0| +| 1\rangle \langle 1| (\cdots )| 1\rangle \langle 1| .\end{eqnarray} \tag{ A1 }$$

Accordingly for an arbitrary density matrix ρ we have

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{t\to \infty }({e}^{({ \mathcal P }\circ { \mathcal L }\circ { \mathcal P })t}\circ { \mathcal P })\rho =| 0\rangle \langle 0| ,\end{eqnarray} \tag{ A2 }$$

while on the contrary

$$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{t\to \infty }{e}^{({ \mathcal P }\circ { \mathcal L }\circ { \mathcal P })t}\rho & = & | 0\rangle \langle 0| +(\mathrm{id}-{ \mathcal P })\rho \\ & = & | 0\rangle \langle 0| +| 0\rangle \langle 0| \rho | 1\rangle \langle 1| +| 1\rangle \langle 1| \rho | 0\rangle \langle 0| ,\end{eqnarray} \tag{ A3 }$$

which in general is not a valid state.

We start by noticing that given a generic non-selective measurement ${ \mathcal P }$ as in equation (11) and the unitary generator ${ \mathcal K }$ with Hamiltonian H, the following identity holds

$$\begin{eqnarray}&&({ \mathcal P }\,\circ \,{ \mathcal K }\,\circ \,{ \mathcal P })(\cdots )=-i\displaystyle \sum _{k}[{H}^{(k)},{P}_{k}(\cdots ){P}_{k}],\end{eqnarray} \tag{ B1 }$$

where ${H}^{(k)}={P}_{k}{{HP}}_{k}$. Similarly, given a dissipator ${ \mathcal D }$ characterised by Lindblad operators ${L}_{\alpha }$ we have

$$\begin{eqnarray}({ \mathcal P }\,\circ \,{ \mathcal D }\,\circ \,{ \mathcal P })(\cdots ) & = & \displaystyle \sum _{\alpha ,k,k^{\prime} }[2{L}_{\alpha }^{({kk}^{\prime} )}(\cdots ){L}_{\alpha }^{({kk}^{\prime} )\dagger }-{L}_{\alpha }^{({kk}^{\prime} )\dagger }{L}_{\alpha }^{({kk}^{\prime} )}(\cdots ){P}_{k^{\prime} }-{P}_{k^{\prime} }(\cdots ){L}_{\alpha }^{({kk}^{\prime} )\dagger }{L}_{\alpha }^{({kk}^{\prime} )}],\end{eqnarray} \tag{ B2 }$$

with ${L}_{\alpha }^{({kk}^{\prime} )}={P}_{k}{L}_{\alpha }{P}_{k^{\prime} }$. Assume next H and L_α as those associated with the Lindbladian ${\tilde{{ \mathcal L }}}_{j}$ with equation (10), and P_k as in equation (14). Since $\{| {\phi }_{k}\rangle \}$ is mutually unbiased with respect to $\{| j\rangle \}$, the following identity holds,

$$\begin{eqnarray}&&\langle j| {\phi }_{k}\rangle ={e}^{-i{\varphi }_{{jk}}}/\sqrt{{d}_{A}},\end{eqnarray} \tag{ B3 }$$

with ${\varphi }_{{jk}}$ generic phases, hence

$$\begin{eqnarray}&&{\tilde{H}}_{j}^{(k)}={P}_{k}{\tilde{H}}_{j}{P}_{k}={H}_{j}\otimes | {\phi }_{k}\rangle \langle {\phi }_{k}| ={H}_{j}{P}_{k},\end{eqnarray} \tag{ B4 }$$

$$\begin{eqnarray}&&{\tilde{L}}_{j,\alpha }^{({kk}^{\prime} )}={P}_{k}{\tilde{L}}_{j,\alpha }{P}_{k^{\prime} }={e}^{i({\varphi }_{{jk}}-{\varphi }_{{jk}^{\prime} })}{L}_{j,\alpha }\otimes | {\phi }_{k}\rangle \langle {\phi }_{k^{\prime} }| /\sqrt{{d}_{A}},\end{eqnarray} \tag{ B5 }$$

$$\begin{eqnarray}&&{\tilde{L}}_{j,\alpha }^{({kk}^{\prime} )\dagger }{\tilde{L}}_{j,\alpha }^{({kk}^{\prime} )}={L}_{j,\alpha }^{\dagger }{L}_{j,\alpha }\otimes | {\phi }_{k^{\prime} }\rangle \langle {\phi }_{k^{\prime} }| ={L}_{j,\alpha }^{\dagger }{L}_{j,\alpha }{P}_{k^{\prime} }/{d}_{A}.\end{eqnarray} \tag{ B6 }$$

Inserting these into (B1) and (B2) we then obtain

$$\begin{eqnarray}&&({ \mathcal P }\,\circ \,{\tilde{{ \mathcal K }}}_{j}\,\circ \,{ \mathcal P })(\cdots )=({{ \mathcal K }}_{j}\,\circ \,{ \mathcal P })(\cdots ),\end{eqnarray} \tag{ B7 }$$

$$\begin{eqnarray}({ \mathcal P }\,\circ \,{\tilde{{ \mathcal D }}}_{j}\,\circ \,{ \mathcal P })(\cdots ) & = & ({{ \mathcal D }}_{j}\,\circ \,{ \mathcal P })(\cdots )+2\displaystyle \sum _{\alpha }{L}_{j,\alpha }{ \mathcal T }(\cdots ){L}_{j,\alpha }^{\dagger },\end{eqnarray} \tag{ B8 }$$

that is

$$\begin{eqnarray}&&({ \mathcal P }\,\circ \,{\tilde{{ \mathcal L }}}_{j}\,\circ \,{ \mathcal P })(\cdots )=({{ \mathcal L }}_{j}\,\circ \,{ \mathcal P })(\cdots )+2\displaystyle \sum _{\alpha }{L}_{j,\alpha }{ \mathcal T }(\cdots ){L}_{j,\alpha }^{\dagger },\end{eqnarray} \tag{ B9 }$$

where ${ \mathcal T }$ is the super-operator

$$\begin{eqnarray}&&{ \mathcal T }(\cdots )=\displaystyle \frac{1}{{d}_{A}}{{\mathbb{1}}}_{A}{\mathrm{Tr}}_{A}[(\cdots )]-{ \mathcal P }(\cdots ),\end{eqnarray} \tag{ B10 }$$

with ${\mathrm{Tr}}_{A}[(\cdots )]$ indicating the partial trace over the auxiliary system A and ${{\mathbb{1}}}_{A}$ being the identity operator on the associated Hilbert space.

To prove equation (15), let us now focus on the evolution induced by CPTP map ${{\rm{\Phi }}}_{t,\infty }^{({{ \mathcal L }}_{j})}$ in equation (13) associated with the jth element of ${\bf{L}}$ on a generic density matrix $\rho (0)$ of the joint system Q + A, i.e., $\rho (t)={{\rm{\Phi }}}_{t,\infty }^{({{ \mathcal L }}_{j})}\rho (0)$. We are interested in the dynamics of the reduced density matrix of Q, i.e.,

$$\begin{eqnarray}&&{\rho }_{Q}(t)={\mathrm{Tr}}_{A}[\rho (t)]={\mathrm{Tr}}_{A}[{{\rm{\Phi }}}_{t,\infty }^{({{ \mathcal L }}_{j})}\rho (0)].\end{eqnarray} \tag{ B11 }$$

By taking the first derivative with respect to t and using equation (B9) we obtain

$$\begin{eqnarray}\displaystyle \frac{\partial }{\partial t}{\rho }_{Q}(t) & = & {\mathrm{Tr}}_{A}[({ \mathcal P }\,\circ \,{\tilde{{ \mathcal L }}}_{j}\,\circ \,{ \mathcal P })\rho (t)]\\ & = & {{ \mathcal L }}_{j}({\mathrm{Tr}}_{A}[{ \mathcal P }\rho (t)])+2\displaystyle \sum _{\alpha }{L}_{j,\alpha }{\mathrm{Tr}}_{A}[{ \mathcal T }\rho (t)]{L}_{j,\alpha }^{\dagger },\end{eqnarray} \tag{ B12 }$$

which finally yields the thesis

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}{\rho }_{Q}(t)={{ \mathcal L }}_{j}{\rho }_{Q}(t)\ \Longrightarrow \ {\rho }_{Q}(t)={e}^{{{ \mathcal L }}_{j}t}{\rho }_{Q}(0),\end{eqnarray} \tag{ B13 }$$

by noticing that

$$\begin{eqnarray}&&{\mathrm{Tr}}_{A}[{ \mathcal P }\rho (t)]={\rho }_{Q}(t),\quad {\mathrm{Tr}}_{A}[{ \mathcal T }\rho (t)]=0.\end{eqnarray} \tag{ B14 }$$

Here we show that the pair of Lindbladians $\{{{ \mathcal L }}_{0},{ \mathcal K }\}$ in (22) and (23) generate an accessible system. We use the notations (25)–(28). First, we show that ${ \mathcal K }=-i{\mathrm{ad}}_{| j\rangle \langle j| }$ commutes with the dissipative part ${{ \mathcal D }}_{| 1\rangle \langle 2| }$ of ${{ \mathcal L }}_{0}$ in equation (22). Using an identity [30]

$$\begin{eqnarray}&&[-i{\mathrm{ad}}_{H},{{ \mathcal D }}_{A}]=\displaystyle \frac{1}{2}{{ \mathcal D }}_{A-i[H,A]}-\displaystyle \frac{1}{2}{{ \mathcal D }}_{A+i[H,A]},\end{eqnarray} \tag{ C1 }$$

we have

$$\begin{eqnarray}[-i{\mathrm{ad}}_{| j\rangle \langle j| },{{ \mathcal D }}_{| 1\rangle \langle 2| }] & = & \displaystyle \frac{1}{2}{{ \mathcal D }}_{| 1\rangle \langle 2| -i| 1\rangle \langle 2| {\delta }_{j1}+i| 1\rangle \langle 2| {\delta }_{j2}}-\displaystyle \frac{1}{2}{{ \mathcal D }}_{| 1\rangle \langle 2| +i| 1\rangle \langle 2| {\delta }_{j1}-i| 1\rangle \langle 2| {\delta }_{j2}}.\end{eqnarray} \tag{ C2 }$$

For $j\ne 1,2$ it trivially vanishes, while for $j=1,2$ we get $\tfrac{1}{2}(| 1\mp i{| }^{2}-| 1\pm i{| }^{2}){{ \mathcal D }}_{| 1\rangle \langle 2| }=0$, where we have used ${{ \mathcal D }}_{\alpha A}=| \alpha {| }^{2}{{ \mathcal D }}_{A}$. This commutativity implies that we can generate every $-i{\mathrm{ad}}_{{\mathfrak{u}}(d)}$ (see [25]), thus every ${\mathrm{Ad}}_{U}\,\circ \,{{ \mathcal D }}_{| 1\rangle \langle 2| }\,\circ \,{\mathrm{Ad}}_{{U}^{\dagger }}={{ \mathcal D }}_{U| 1\rangle \langle 2| {U}^{\dagger }}$ for any $U\in U(d)$. Taking unitaries $U| j\rangle ={\sum }_{k\in { \mathcal I }}{c}_{k}^{(j)}| k\rangle$ for $j=1,2$ and ${ \mathcal I }=\{1,2,3,4\}$, we have

$$\begin{eqnarray}&&{{ \mathcal D }}_{U| 1\rangle \langle 2| {U}^{\dagger }}=\displaystyle \sum _{i,j,k,l\in { \mathcal I }}{({c}_{i}^{(1)})}^{* }{c}_{j}^{(2)}{c}_{k}^{(1)}{({c}_{l}^{(2)})}^{* }{{ \mathcal D }}_{| i\rangle \langle j| ,| k\rangle \langle l| }.\end{eqnarray} \tag{ C3 }$$

We numerically verified that all the operators of the form equation (30) for ${ \mathcal I }=\{1,2,3,4\}$ can be obtained by linear combinations of ${{ \mathcal D }}_{U| 1\rangle \langle 2| {U}^{\dagger }}$ and $-i{\mathrm{ad}}_{H}$ with different U and $H={\sum }_{i,j\in { \mathcal I }}{h}_{{ij}}| i\rangle \langle j|$ on ${ \mathcal I }=\{1,2,3,4\}$. The same argument applies to any quartets ${ \mathcal I }=\{i,j,k,l\}$, and all the operators of the form equation (30) for all ${ \mathcal I }$ are available. Then, every Lindbladian can be given as a linear combination of those operators, i.e.,

$$\begin{eqnarray}{ \mathcal L } & = & \displaystyle \sum _{i,j,k,l}[{c}_{{ijkl}}^{+}\,({{ \mathcal D }}_{| i\rangle \langle j| ,| k\rangle \langle l| }+{{ \mathcal D }}_{| k\rangle \langle l| ,| i\rangle \langle j| })+{c}_{{ijkl}}^{-}i({{ \mathcal D }}_{| i\rangle \langle j| ,| k\rangle \langle l| }-{{ \mathcal D }}_{| k\rangle \langle l| ,| i\rangle \langle j| })]\end{eqnarray} \tag{ C4 }$$

with some coefficients ${c}_{{ijkl}}^{\pm }$.