# A geometrical description of non-Hermitian dynamics:
speed limits in finite rank density operators

Niklas Hörnedal  0000-0002-2005-8694 Department of Physics and Materials
Science, University of Luxembourg, L-1511 Luxembourg, G. D. Luxembourg  Oskar
A. Prośniak  0000-0003-4550-7561 Department of Physics and Materials Science,
University of Luxembourg, L-1511 Luxembourg, G. D. Luxembourg  Adolfo del
Campo  0000-0003-2219-2851 Department of Physics and Materials Science,
University of Luxembourg, L-1511 Luxembourg, G. D. Luxembourg  Donostia
International Physics Center, E-20018 San Sebastián, Spain  Aurélia Chenu
0000-0002-4461-8289 Department of Physics and Materials Science, University of
Luxembourg, L-1511 Luxembourg, G. D. Luxembourg

###### Abstract

Non-Hermitian dynamics in quantum systems preserves the rank of the state
density operator. We use this insight to develop its geometrical description.
In particular, we identify mutually orthogonal coherent and incoherent
directions and give their physical interpretation. This understanding allows
us to optimize the success rate for implementing non-Hermitian driving along
prescribed trajectories. We show its significance for shortcuts to
adiabaticity. We introduce the geometrical interpretation of a speed limit for
non-Hermitian Hamiltonians and analyze its tightness. We derive the explicit
expression that saturates such a speed limit and illustrate our results on a
minimal example of a dissipative qubit.

##  1 Introduction

Non-Hermitian dynamics naturally arises in a variety of applications in
physics, engineering, and computer science [1, 2]. In quantum systems, it
describes the evolution of the state of the system conditioned to remain in a
given subspace and can be rigorously justified using the Feshbach projection
approach. Non-Hermitian evolution also arises in the context of continuous
quantum measurements, in the description of a subensemble of quantum
trajectories via post-selection, e.g., in the absence of quantum jumps (null-
measurement conditioning) [3, 4]. Non-Hermitian descriptions have also been
used since the early days of quantum mechanics to describe decay rates
phenomenologically. They find further applications in numerical methods for
quantum dynamics and the theory of chemical reaction rates, among other
examples [1].

In addition, an arbitrary differentiable trajectory of fixed-rank density
matrices can be associated with a generator of evolution that is non-
Hermitian. The associated equation of motion involves a nonlinear term to
render the evolution trace-preserving [5].

This work focuses on a geometric approach to non-Hermitian quantum evolution.
The description of dynamics in geometric terms has a long history that has
shed new light on the foundations of physics. It is central in information
geometry, whether classical [6] or quantum [7]. In quantum physics, it plays a
key role in the understanding of quantum adiabatic dynamics and geometric
phases [8, 9, 10], proofs of the adiabatic theorem, and shortcuts to
adiabaticity (STA) [11, 12, 13]. A geometric understanding of quantum dynamics
is also instrumental in studying time-energy uncertainty relations and their
refinement in the form of quantum speed limits, i.e., bounds on the minimum
time for a process to unfold [14, 15].

Here, we describe how the set of reachable states under non-Hermitian dynamics
forms a differentiable manifold. This observation allows for a geometrical
interpretation of non-Hermitian dynamics. Specifically, the local effects of
the dynamics can be described through a tangent space, which can be decomposed
into a unitary and commutative subspaces. We show that these subspaces are
maximally distinguishable with respect to any monotone metric. This
geometrical picture sheds a new understanding on STAs generalizing
counterdiabatic driving [16, 17, 18, 19] to open quantum systems [20, 5, 21].
We then focus on the Bures metric and explicitly describe the geodesics on the
fixed rank manifolds. We show applications to derive quantum speed limits for
non-Hermitian systems and illustrate the control dynamics of a dissipative
qubit. The notions of differential geometry that we use are described in
Appendix A.

##  2 Non-Hermitian dynamics and fixed-rank manifold

In this section, we describe how non-Hermitian state dynamics arises as a
special case of a continuous implementation of effective measurements. We
review how this dynamics gives rise to a group action on the state space by
the general linear group and describe the differential structure of its
orbits.

###  2.1 Measurement and non-Hermitian dynamics

From an axiomatic viewpoint, the possible transformations of closed quantum
systems are commonly divided up into two types: the deterministic unitary
evolution, described by the Schrödinger equation, and the non-deterministic
measurement process, described by projective measurement operators. More
generally, one may choose to focus on an open system’s effective dynamics
induced by the unitary evolution of a system coupled to an ancilla and
measurements of the latter. Such an effective dynamics is given, according to
Stinespring’s and Naimark’s dilation theorems [22], by a completely positive
trace decreasing (CPTD) map that admits the Choi-Kraus representation [23]

$\rho\mapsto\tilde{\Phi}(\rho):=\sum_{m=1}^{N}G_{m}\rho G_{m}^{\dagger},$ (1)

where $\rho$ is a reduced density matrix of the subsystem of interest and
$G_{m}$ are the so-called Kraus operators fulfilling
$\sum_{m=1}^{N}G_{m}^{\dagger}G_{m}\leq\mathds{1}$ [24].

For a composite system initialized in a product state with a pure ancilla
state, $\rho_{0}\otimes\outerproduct{\phi_{\textrm{A}}}{\phi_{\textrm{A}}}$,
the above CPTD map reduces to the action of a single Kraus operator. Such
protocol is known as an _effective measurement_ [25]. Equation (1) then gives
a state conditioned on the outcome of that selective measurement, which is to
be contrasted with trace-preserving non-selective measurements [26].

One can then pick another ancilla system in its pure state and iterate the
procedure. In the simplest case, the old ancilla is being reused since the
projective measurement collapses its density matrix to a pure state. The limit
in which the time of the intermediate unitary evolution goes to zero is known
as a _continuous measurement_. The time evolution of the conditioned state can
then be described by a map of the form
$\rho\mapsto\tilde{\Phi}_{t}(\rho):=G(t)\rho G^{\dagger}(t)$. Having chosen an
initial state $\rho_{0}$, we will talk about its trajectory
$\rho(t):=\tilde{\Phi}_{t}(\rho_{0})$. Given that $G(t)$ is differentiable and
invertible, the dynamical map $\tilde{\Phi}_{t}$ can be associated with the
following dynamics

$\dot{\rho}(t)=-i\left(K(t)\rho(t)-\rho(t)K^{\dagger}(t)\right),$ (2)

where $K(t)=i\dot{G}(t)G^{-1}(t)$ acts as a generator of the evolution
operator $G(t)$ and need not be Hermitian. Physically, we can think of the map
$\tilde{\Phi}_{t}$ as a channel describing a continuous information flow from
the system to the observer [27, 28].

###  2.2 States with fixed rank: manifold $\mathcal{Q}_{r}$

The invertibility condition above implies that any state has a non-zero
probability of passing through the channel $\tilde{\Phi}_{t}$. But since the
map $\tilde{\Phi}_{t}$ is trace decreasing, the state passes through the
channel only with probability $\Tr(\tilde{\Phi}_{t}(\rho))$. Consequently, the
output of the map $\tilde{\Phi}_{t}$ does not need to be a physical density
matrix. The remedy to this problem is to normalize the output.

Formally, let us denote by $\mathcal{Q}\subset\mathsf{L}(\mathcal{H}_{n})$ the
set of all density operators acting on a $n$-dimensional Hilbert space
$\mathcal{H}_{n}$. The non-Hermitian evolution leading to a normalized state,
described above, takes the general form

$\Phi_{t}:\mathcal{Q}\to\mathcal{Q},\quad\Phi_{t}(\rho):=\frac{G(t)\rho
G^{\dagger}(t)}{\Tr(G(t)\rho G^{\dagger}(t))},$ (3)

where, as before, $G(t)$ is a smooth function of $t$ valued in invertible
operators, i.e., elements of the general linear group
$\mathrm{GL}(n,\mathbb{C})$. Note that $G(t)$ can be unitary, but we do not
impose it. Since $G(t)$ is necessarily of full rank, when acting on density
matrices $\rho$, it preserves their rank,111Notice that (3) is also the most
general form of a normalized CPTD map that preserves the rank of any positive
operator. i.e., if $\mathrm{rk}\left(\rho\right)=r$ then
$\mathrm{rk}\left(\Phi_{t}(\rho)\right)=r$.222Although the rank of the
evolving state does not change, its trace distance to states with a different
rank can decrease rapidly, but never reach zero in a finite time. Hence, to
study the evolution of a density operator initially of rank $r$, we may
restrict our attention to the subset $\mathcal{Q}_{r}\subset\mathcal{Q}$ of
density operators with the same rank.

This observation allows us to introduce a differential structure. Indeed, the
set $\mathcal{Q}_{r}$ is naturally a $(2nr-r^{2}-1)$-dimensional manifold with
a differential structure inherited from the space of $n\times n$ complex
matrices, isomorphic to $\mathbb{R}^{2n^{2}}$, in which it can be smoothly
embedded [29]. This is not the case for $\mathcal{Q}$, which is not a manifold
in the usual sense (see Fig. 3). However, it can also be endowed with a
generalized differential structure, on which we comment in the conclusion.

Figure 1: (a) The surface and the interior of the Bloch’s ball respectively
form $2-$ and $3-$dimensional manifolds. (b) The Bloch’s ball itself does not
form a standard manifold as open sets touching its surface cannot be mapped to
open sets in $\mathbb{R}^{d}$ for any $d$.33footnotemark: 3 22footnotetext:
The Bloch’s ball forms a manifold with boundaries. Defining a differential
structure on such manifolds does not pose difficulties. However, higher
dimensional systems can no longer be described as such, and extending the
presented results to them is not straightforward.

###  2.3 Velocities on $\mathcal{Q}_{r}$: tangent space
$\mathsf{T}_{\rho}\mathcal{Q}_{r}$

Consider any curve describing the evolution of a quantum state $\rho(t)$,
governed by the map (3) and passing through the state $\rho$ at some time that
we take to be zero. The tangent to this curve at $\rho$ can be associated with
its time derivative $\dot{\rho}\equiv\dot{\rho}(0)$. Importantly, the set of
all such tangents forms the tangent space $\mathsf{T}_{\rho}\mathcal{Q}_{r}$,
illustrated in Fig. 2.

Figure 2: The manifold $\mathcal{Q}_{r}$ (white/grey surface). We illustrate
two evolution curves on this manifold, $\rho_{1}(t)$ and $\rho_{2}(t)$, that
pass through $\rho$; the arrows in magenta show the corresponding tangent
vectors at this point, $\dot{\rho}_{1}$ and $\dot{\rho}_{2}$. These tangent
vectors are elements of the tangent space at $\rho$, denoted by
$\mathsf{T}_{\rho}\mathcal{Q}_{r}$, and depicted as the cyan plane.

The tangent vector can be expressed in different ways. First, considering that
any two states in $\mathcal{Q}_{r}$ can be connected by the group action

$\begin{split}&\Phi:\mathcal{Q}_{r}\times\mathrm{GL}(n,\mathbb{C})\to\mathcal{Q}_{r},\\\
&\rho\mapsto\Phi(\rho,G):=\frac{G\rho G^{\dagger}}{\Tr(G\rho
G^{\dagger})}.\end{split}$ (4)

Any tangent vector at $\rho$ ($\dot{\rho}\in\mathsf{T}_{\rho}\mathcal{Q}_{r}$)
follows as

$\dot{\rho}=-i(K\rho-\rho K^{\dagger})+i\Tr((K-K^{\dagger})\rho)\rho,$ (5)

where $K\in\mathfrak{gl}(n,\mathbb{C})$ is an element of the Lie algebra
associated with $\mathrm{GL}(n,\mathbb{C})$, consisting of all $n\times n$
matrices [30].

Alternatively, one can decompose the generator of the dynamics into its
Hermitian and anti-Hermitian parts, namely $K=H-i\Gamma$ with
$H=(K+K^{\dagger})/2$ and $\Gamma=(K^{\dagger}-K)/(2i)$. The tangent vector
then takes the form [31]

$\dot{\rho}=-i\left[H,\rho\right]-\\{\Gamma,\rho\\}+2\Tr(\Gamma\rho)\rho.$ (6)

This equation arises in continuous quantum measurements in the absence of
quantum jumps, i.e., under null-measurement conditioning [3]. It also occurs
in spectral filtering, which modifies the dynamics and enhances the dynamical
manifestations of quantum chaos [32, 33, 34].

The non-linear term appearing in Eqs. (5) and (6) is due to the
renormalization of the state to preserve its norm. Note that a master equation
similar to Eq. (6) has also been considered to describe classical dissipative
systems [35]. We discuss this ”dynamical decomposition” further in Section
3.4, but first focus on a different ”static decomposition” of the tangent
vectors, which refers to the geometry of the tangent space itself.

##  3 Geometry of non-Hermitian trajectories

###  3.1 Coherent and incoherent directions: spaces
$\mathsf{U}_{\rho}\mathcal{Q}_{r}$ and $\mathsf{C}_{\rho}\mathcal{Q}_{r}$

Using the spectral decomposition of the density matrix
$\rho=\sum_{k}\lambda_{k}\Pi_{k}$, where $\Pi_{k}$ are spectral projections
and $\lambda_{k}$ are distinct eigenvalues, any vector tangent to
$\mathcal{Q}_{r}$ at this point can be decomposed into
$\dot{\rho}=\dot{\rho}_{\textrm{u}}+\dot{\rho}_{\textrm{c}}$, with

$\displaystyle\dot{\rho}_{\textrm{u}}=\sum_{k\neq m}\Pi_{k}\dot{\rho}\Pi_{m},$
(7a) $\displaystyle\dot{\rho}_{\textrm{c}}=\sum_{k}\Pi_{k}\dot{\rho}\Pi_{k}.$
(7b)

This decomposition, illustrated in Fig. 3, has been used in the context of
differential geometry [36, 37, 38, 39, 40] and was used in Refs. [41, 42, 43]
for bounding the speed of the evolution, and in [44] for the identification of
work and heat in quantum thermodynamics. It can be generalized to arbitrary
observables [43].

It follows that the tangent space $\mathsf{T}_{\rho}\mathcal{Q}_{r}$ can be
decomposed into a direct sum of unitary and commutative subspaces [7, 45, 40]
respectively defined as

$\displaystyle\\!\\!\mathsf{U}_{\rho}\mathcal{Q}_{r}$
$\displaystyle=\\{\dot{\rho}\in\mathsf{T}_{\rho}\mathcal{Q}_{r}|\dot{\rho}=-i[H,\rho],H=H^{\dagger}{}\\},$
(8a) $\displaystyle\\!\\!\mathsf{C}_{\rho}\mathcal{Q}_{r}$
$\displaystyle=\\{\dot{\rho}\in\mathsf{T}_{\rho}\mathcal{Q}_{r}|[\dot{\rho},\rho]=0\\}.$
(8b)

It is easy to check that
$\dot{\rho}_{\textrm{u}}\in\mathsf{U}_{\rho}\mathcal{Q}_{r}$ and
$\dot{\rho}_{\textrm{c}}\in\mathsf{C}_{\rho}\mathcal{Q}_{r}$, and that these
spaces are linearly independent.

Remember that elements of $\mathsf{T}_{\rho}\mathcal{Q}_{r}$ can be thought of
as tangents at $\rho$ to all the smooth curves in $\mathcal{Q}_{r}$ passing
through this point at time zero. The set of allowed curves can be restricted
to those that form smooth submanifolds, i.e., curves without cusps. It is then
possible to define an eigen-decomposition along each such path
$\rho(t)=\sum_{k=1}^{r}\lambda_{k}(t)\outerproduct{\lambda_{k}(t)}{\lambda_{k}(t)}$,
where the one-dimensional projectors
$\outerproduct{\lambda_{k}(t)}{\lambda_{k}(t)}$ depend smoothly on
time.333Here the eigenvalues $\lambda_{k}$ are no longer assumed to be
distinct. Higher dimensional projectors may be discontinuous at times when the
degeneracy changes. The unitary and commutative components of
$\dot{\rho}=\dot{\rho}(0)$ take then a simple form

$\displaystyle\dot{\rho}_{\textrm{u}}$
$\displaystyle=\sum_{k}\lambda_{k}(\outerproduct*{\dot{\lambda}_{k}}{\lambda_{k}}+\outerproduct*{\lambda_{k}}{\dot{\lambda}_{k}}),$
(9a) $\displaystyle\dot{\rho}_{\textrm{c}}$
$\displaystyle=\sum_{k}\dot{\lambda}_{k}\outerproduct{\lambda_{k}}{\lambda_{k}}.$
(9b)

See also Fig. 3.

Physically, $\mathsf{U}_{\rho}\mathcal{Q}_{r}$ and
$\mathsf{C}_{\rho}\mathcal{Q}_{r}$ can be thought of as ”coherent” and
”incoherent” directions respectively and we will henceforth refer to them as
such [43, 42]. The evolution along the former does not change populations in
the instantaneous eigenbasis of the state but generates coherence by contrast
to the evolution along the latter.

Note that pure states ($r=1$) have an empty commutative subspace,
$\mathsf{C}_{\rho}\mathcal{Q}_{1}=\emptyset$, which agrees with the fact that
the only quantum operations preserving purity are unitaries.

Figure 3: At every point of the trajectory in the interior of the Bloch’s
ball, the tangent to it (black arrow) can be decomposed into commutative
(magenta) and unitary (cyan) components.

###  3.2 Classical direction: subspace $\mathsf{Class}_{\rho}\mathcal{Q}_{r}$
and complementary $\mathsf{L}_{\rho}\mathcal{Q}_{r}$

The commutative subspace can be decomposed further when the spectrum of the
state is degenerate [40]. Indeed, the incoherent component
$\dot{\rho}_{\textrm{c}}\in\mathsf{C}_{\rho}\mathcal{Q}_{r}$ can be expressed
as $\dot{\rho}_{\textrm{c}}=\dot{\rho}_{\textrm{cl}}+\dot{\rho}_{\textrm{l}}$,
with

$\displaystyle\dot{\rho}_{\textrm{cl}}$
$\displaystyle=\sum_{k}\frac{\Tr(\Pi_{k}\dot{\rho})}{\Tr\Pi_{k}}\Pi_{k},$
(10a) $\displaystyle\dot{\rho}_{\textrm{l}}$
$\displaystyle=\sum_{k}\left(\Pi_{k}\dot{\rho}\Pi_{k}-\frac{\Tr(\Pi_{k}\dot{\rho})}{\Tr\Pi_{k}}\Pi_{k}\right).$
(10b)

These directions form what we call the classical
$\mathsf{Class}_{\rho}\mathcal{Q}_{r}$ and the lifting
$\mathsf{L}_{\rho}\mathcal{Q}_{r}$ subspaces correspondingly.

To sum up, the tangent vector can be decomposed as

$\dot{\rho}=\dot{\rho}_{\textrm{u}}+\dot{\rho}_{\textrm{cl}}+\dot{\rho}_{\textrm{l}}.$
(11)

The lifting direction $\dot{\rho}_{\textrm{l}}$ is responsible for lifting the
degeneracy of the state and is, for example, relevant in symmetry breaking in
adiabatic processes. Evolution along the unitary and lifting directions can
generally lead to a change of eigenbasis. As such, they can be viewed as non-
classical. By contrast, the classical direction $\dot{\rho}_{\textrm{cl}}$ can
be interpreted as classical information processing since we can, in a
consistent way, map $\rho$ to a probability vector and consider the evolution
along $\dot{\rho}_{\textrm{cl}}$ as arising through stochastic
transformations. Lastly, it is worth noting that $\dot{\rho}_{\textrm{u}}$
preserves the von Neumann entropy of the state while $\dot{\rho}_{\textrm{l}}$
strictly decreases it.

###  3.3 Maximal distinguishability of the coherent and incoherent directions

Let us, for a moment, restrict our attention to the manifold of full-rank
states $\mathcal{Q}_{n}$. It turns out that the coherent and incoherent
directions are maximally distinguishable, or orthogonal, with respect to any
monotone Riemannian metric on $\mathcal{Q}_{n}$. These are the metrics whose
geodesic distance between any two density operators in $\mathcal{Q}_{n}$
either decreases or stays the same under completely positive trace preserving
(CPTP) transformations. It can be argued that any meaningful notion of
statistical distance on $\mathcal{Q}_{n}$ should have this property.
Intuitively, two density matrices should not become more distinguishable after
post-processing the data they describe.

Any monotone metric on the manifold $\mathcal{Q}_{n}$ is an inner product
defined locally and associated with the following norm [46, 47]

$\begin{split}&\norm{\cdot}:\mathsf{T}_{\rho}\mathcal{Q}_{n}\to\mathbb{R},\\\
&\norm{\dot{\rho}}=\sqrt{\sum_{j,k}|\dot{\rho}_{jk}|^{2}c(\lambda_{j},\lambda_{k})},\end{split}$
(12)

where $\dot{\rho}_{jk}=\bra{\lambda_{j}}\\!\dot{\rho}\\!\ket{\lambda_{k}}$.
The function $c(\lambda,\mu)$ is a symmetric, homogeneous, and of order $-1$,
i.e., $c(\alpha\lambda,\alpha\mu)=\alpha^{-1}c(\lambda,\mu)$. Its choice
defines the metric. For example, choosing
$c(\lambda,\mu)=\frac{2}{\lambda+\mu}$ leads to the Bures metric, widely used
in quantum theory [23].

With the definition of the monotone metric together with the polarization
identity, it becomes apparent that the coherent and incoherent directions are
mutually orthogonal since
$\dot{\rho}_{\textrm{c}}\in\mathsf{C}_{\rho}\mathcal{Q}_{n}$ is purely
diagonal and $\dot{\rho}_{\textrm{u}}\in\mathsf{U}_{\rho}\mathcal{Q}_{n}$
purely off-diagonal when expressed in the eigenbasis of $\rho$ – see Appendix
B.1 for a detailed proof. This maximal distinguishability can be visualized on
the Bloch ball for a qubit, as seen in Fig. 3. Similarly, the lifting and
classical subspaces can also be shown to be orthogonal with respect to one
another (Appendix B.1). These results reveal a hidden rigidity condition that
all monotone metrics needs to respect.

The orthogonality of coherent and incoherent directions with respect to all
monotone metrics can be proven as above for $\mathcal{Q}_{n}$ only, as only on
the manifold of full rank states are the norms associated with these metrics
known to be of the form $\eqref{eq: monotone metric}$. Moreover, the
monotonicity of the metric is relevant for such states only, as information is
indisputably gained under rank preserving dynamics (3) in open systems, due to
effective measurements described above.444The set of CPTP maps will in general
only preserve the rank for full rank states. However, the Bures metric can be
extended to the manifold of states with rank $r<n$ [48]. Specifically, on that
manifold, the Bures metric
$(\cdot,\cdot)_{\textsc{b}}:\mathsf{T}\mathcal{Q}_{r}\times\mathsf{T}\mathcal{Q}_{r}\to\mathbb{R}$
takes the form

$(\dot{\rho},\dot{\rho}^{\prime})_{\textsc{b}}=\frac{1}{2}\Tr(L\dot{\rho}^{\prime}),$
(13)

where $L$ is known as a symmetric logarithmic derivative (SLD) and is any
solution to the equation $L\rho+\rho L=\dot{\rho}$ [49, 48].555Note that we do
not include any factor of two in this definition. The coherent and incoherent
directions are also orthogonal in $\mathcal{Q}_{r}$ with respect to this
extended Bures metric.

###  3.4 Hamiltonian and gradient vector fields

Consider now a vector field representing a dynamics generated by a non-
Hermitian operator $K=H-i\Gamma$. Equation (6) shows a decomposition of the
vector field into two parts where the first term comes from the Hermitian part
of $K$

$\dot{\rho}_{H}=-i[H,\rho],$ (14)

and the other two terms from the anti-Hermitian part

$\dot{\rho}_{\Gamma}=-\\{\tilde{\Gamma}_{\rho},\rho\\},$ (15)

where we introduced the state-dependent operator
$\tilde{\Gamma}_{\rho}=\Gamma-\Tr(\Gamma\rho)$. The latter can be seen as a
gradient of the function
$\expectationvalue{\Gamma}:\mathcal{Q}_{r}\to\mathbb{R},\quad\expectationvalue{\Gamma}_{\rho}:=\Tr(\Gamma\rho)$
with respect to the Bures metric [30].

We recall that the gradient of any scalar field
$f:\mathcal{Q}_{r}\to\mathbb{R}$ is a vector field $\nabla
f:\mathcal{Q}_{r}\to\mathsf{T}\mathcal{Q}_{r}$ satisfying

$\forall\rho\in\mathcal{Q}_{r},\forall\dot{\rho}\in\mathsf{T}_{\rho}\mathcal{Q}_{r}:\,(\nabla
f(\rho),\dot{\rho})_{\textsc{b}}=\mathrm{d}f(\dot{\rho}),$ (16)

where $\mathrm{d}f$ is the differential of $f$. In the case when $f$ is the
expectation value of an observable $A$,
$f(\rho)=\textrm{Tr}(A\rho)=\expectationvalue{A}_{\rho}$, then
$\mathrm{d}\expectationvalue{A}_{\rho}=\Tr(A\dot{\rho})$. Interestingly, the
fields $-i[A,\rho]$ and $\\{A,\rho\\}$ are everywhere orthogonal with respect
to the Bures metric. The first one generates a rotation that preserves the
expectation value of $A$, while the latter generates a flow that moves along
the direction of maximal change and cuts each level hypersurface666A level
hypersurface is a set on which the function takes a constant value.
orthogonally; see Fig. 4. For $A$ with a non-degenerate spectrum, the latter
gives rise to $1$ stable, $1$ unstable, and $n-2$ hyperbolic fixed points
corresponding to the observable’s eigenstates. More generally, the vector
fields $-i[H,\rho]$ and $\\{\tilde{\Gamma}_{\rho},\rho\\}$ are everywhere
orthogonal when $[H,\Gamma]=0$.

Note that while the decomposition into coherent and incoherent directions
purely refers to the differential structure of $\mathcal{Q}_{r}$, the
decomposition into Hermitian and anti-Hermitian directions is dynamical, as it
depends on the chosen generator of the dynamics. Moreover, while
$\dot{\rho}_{H}\in\mathsf{U}_{\rho}\mathcal{Q}_{r}$, in general
$\dot{\rho}_{\Gamma}\in\mathsf{U}_{\rho}\mathcal{Q}_{r}\oplus\mathsf{C}_{\rho}\mathcal{Q}_{r}$.

Figure 4: Vector fields $-i[A,\rho]$ (on the left) and $\\{A,\rho\\}$ (on the
right) on the Bloch ball for a qubit, with $A=\sigma_{z}$. Colors represent
the speed with respect to the Bures metric. The dynamics $-i[A,\rho]$
preserves the expectation values of $A$ as it rotates the sphere while
$\\{A,\rho\\}$ flows from the south pole to the north pole along the direction
of maximal increase.

###  3.5 Maximal success rate

When the non-Hermitian generator arises from the continuous measurements (see
Sec. 2.1), the Hermitian operator $\Gamma$ is necessarily positive-semi
definite due to the monotonic decrease of the state’s norm. The norm decreases
at a rate $\gamma=2\Tr(\Gamma\rho)$, which is the non-linear contribution in
(6).

At each time step, the anti-Hermitian part of the generator naturally splits
into two parts $\Gamma=\Gamma_{\textrm{u}}+\Gamma_{\textrm{c}}$, where
$\Gamma_{\textrm{u}}=\sum_{j\neq k}\Pi_{j}\Gamma\Pi_{k}$ induces motion in the
coherent directions and $\Gamma_{\textrm{c}}=\sum_{k}\Pi_{k}\Gamma\Pi_{k}$ in
the incoherent directions. Since $\Gamma_{\textrm{u}}$ generates the dynamics
along coherent directions, by definition, there exists a (state-dependent)
Hamiltonian generating the same dynamics. Therefore, by exchanging
$\Gamma_{\textrm{u}}$ for this Hamiltonian, one can mitigate the decrease of
norm along the given trajectory, as there is no probability decay for purely
unitary dynamics. This leads to finding a maximal success rate, as we explain
below.

Let us first observe that after correcting for the change in norm (6), the
dynamics is independent of the joint shifts of the eigenvalues of $\Gamma$.
However, the decay rate is not independent of these shifts and, under the
constraint $\Gamma\geq 0$, is minimized when the smallest eigenvalue of
$\Gamma$ equals zero. We will view this as trivial optimization and compare
rates for operators already shifted in this manner.

Let $\mu_{\textrm{c}}^{\textrm{(min)}}$ and $\mu^{\textrm{(min)}}$ be the
smallest eigenvalues of $\Gamma_{\textrm{c}}$ and $\Gamma$ respectively. In
the light of the above comment, we take $\mu^{\textrm{(min)}}=0$. For any
eigenvector $\ket{\mu_{\textrm{c}}}$ of $\Gamma_{\textrm{c}}$, we have
$\bra{\mu_{\textrm{c}}}\\!\Gamma\\!\ket{\mu_{\textrm{c}}}=\bra{\mu_{\textrm{c}}}\\!\Gamma_{\textrm{c}}\\!\ket{\mu_{\textrm{c}}}$,
so $\mu_{\textrm{c}}^{\textrm{(min)}}\geq 0$. The shifted operator
$\Gamma_{\textrm{opt}}=\Gamma_{\textrm{c}}-\mu_{\textrm{c}}^{\textrm{(min)}}\mathds{1}$
is positive ($\Gamma_{\textrm{opt}}\geq 0$) and satisfies

$\Tr(\Gamma_{\textrm{opt}}\>\rho)=\Tr(\Gamma\rho)-\mu_{\textrm{c}}^{\textrm{(min)}}\leq\Tr(\Gamma\rho).$
(17)

This inequality is saturated iff the kernel of $\Gamma_{\textrm{c}}$ is
contained within the kernel of $\Gamma$777The kernel of a linear operator $A$
on a vector space $\mathcal{V}$ is the subspace
$\textrm{ker}A\subset\mathcal{V}$ defined by
$\textrm{ker}A=\\{v\in\mathcal{V}\,|\,Av=0\\}$.—something that never happens
for full-rank qubit states when $\Gamma_{\textrm{u}}\neq 0$.

The Hermitian operator $H_{\textrm{opt}}=H-i\sum_{j\neq
k}\frac{\lambda_{j}+\lambda_{k}}{\lambda_{j}-\lambda_{k}}\Pi_{k}\Gamma\Pi_{j}$
generates, in the coherent directions, the same evolution as
$H-i\Gamma_{\textrm{u}}$. Consequently,
$K_{\textrm{opt}}=H_{\textrm{opt}}-i\Gamma_{\textrm{opt}}$ generates the same
evolution as $K$, but with a smaller decrease of the state’s norm. In this
sense, the non-Hermitian dynamics that maximizes the success rate for
obtaining the desired state-trajectory is achieved when the anti-Hermitian
part of the generator commutes with the instantaneous state.

###  3.6 Shortcuts to adiabaticity

The geometrical interpretation presented in the preceding sections may be
applied to STAs generalizing counterdiabatic driving to open systems as
described in [5]. Counterdiabatic driving provides an alternative to adiabatic
protocols. Under slow driving, the adiabatic theorem guarantees that the
adiabatic trajectory describes the evolution of a reference system [50, 51].
Under counterdiabatic driving, the same adiabatic trajectory describes exactly
the evolution, without the requirement of slow driving, provided that the
dynamics is assisted by auxiliary control terms. The original formulation is
restricted to unitary evolution, preserving the von Neumann entropy of the
evolving state. The auxiliary terms admit then a Hamiltonian form [16, 17, 18,
19]. For nonunitary evolution, different generalizations have been put forward
[20, 5], that require a counterdiabatic Liouvillian involving Hamiltonian
terms and dissipators.

Let us consider the trajectory

$\rho(t)=e^{-\beta H_{0}(t)}/Z(t),$ (18)

governed by the Hermitian time-dependent Hamiltonian $H_{0}(t)$, with inverse
temperature $\beta>0$, and $Z(t)=\textrm{Tr}(e^{-\beta H_{0}(t)})$ the
partition function. Note that any $\rho(t)$ can be written in this form. Given
the finite temperature, the rank of the state equals that of $H_{0}(t)$. The
velocity of (18) is given by
$\dot{\rho}(t)=\dot{\rho}_{\textrm{u}}(t)+\dot{\rho}_{\textrm{c}}(t)$, where

$\displaystyle\dot{\rho}_{\textrm{u}}(t)$
$\displaystyle=-i[H_{{}_{\textrm{CD}}}(t),\rho(t)],$ (19a)
$\displaystyle\dot{\rho}_{\textrm{c}}(t)$
$\displaystyle=-\frac{\dot{Z}(t)}{Z(t)^{2}}\rho(t)-\frac{\beta}{Z(t)}\sum_{k=1}^{n}\dot{E}_{k}e^{-\beta
E_{k}}\outerproduct{E_{k}}{E_{k}}$ (19b)

with the eigendecomposition $(H_{0}(t)-E_{k})\ket{E_{k}}=0$. The operator
$H_{{}_{\rm CD}}(t)=H_{0}(t)+H_{1}(t)$ is the counterdiabatic Hamiltonian with
$H_{1}(t)=i\sum_{k=1}^{n}(\outerproduct*{\dot{E}_{k}}{E_{k}}+\innerproduct*{\dot{E}_{k}}{E_{k}}\outerproduct{E_{k}}{E_{k}})$.
The subscripts of the two components were chosen deliberately as it is easy to
check that $\dot{\rho}_{\textrm{u}}(t)\in\mathsf{U}_{\rho(t)}\mathcal{Q}_{r}$
and $\dot{\rho}_{\textrm{c}}(t)\in\mathsf{C}_{\rho(t)}\mathcal{Q}_{r}$.

The same trajectory can be generated by the non-Hermitian generator
$K(t)=H_{{}_{\textrm{CD}}}(t)-i\Gamma(t)$, where
$\Gamma(t)=-\frac{\beta}{2}\sum_{k=1}^{n}\dot{E}_{k}\outerproduct{E_{k}}{E_{k}}-\frac{1}{2}\derivative{t}\log{Z}(t)$.
Since $[\Gamma(t),\rho(t)]=0$, this implementation can optimize the success
rate after shifting the spectrum of $\Gamma$ such that its smallest eigenvalue
equals zero.

##  4 Geodesics and speed limit

Let us now consider two points $\rho_{1},\rho_{2}\in\mathcal{Q}_{r}$ and the
trajectories connecting them. The shortest geodesic is the shortest trajectory
with a constant speed. To compute it, it is useful to go into a purification
space that facilitates a geometric analysis through a fiber bundle
construction. Let us first recall some notions of purification and fiber
bundle.

###  4.1 Purifications, fiber bundle, and geodesics

Any density matrix $\rho\in\mathcal{Q}_{r}$ acting on a Hilbert space
$\mathcal{H}_{\textrm{S}}$ with dimension $n$ can be purified using an ancilla
modelled on $\mathcal{H}_{\textrm{A}}$ with
$\dim(\mathcal{H}_{\textrm{A}})\geq r$. Specifically, there always exists a
normalized vector
$\ket{W}\in\mathcal{H}_{\textrm{S}}\otimes\mathcal{H}_{\textrm{A}}$ such that
$\rho=\Tr_{\textrm{A}}{\outerproduct{W}{W}}$. We call such a vector a
purification of the state $\rho$. It can be explicitly expressed in terms of
the spectral decomposition as

$\ket{W}=\sum_{k=1}^{r}\sqrt{\lambda_{k}}\ket{\lambda_{k}}\otimes\ket{a_{k}},$
(20)

where $\\{\ket{a_{k}}\\}_{k=1}^{r}$ forms an orthonormal set in
$\mathcal{H}_{\textrm{A}}$. We consider minimal purifications by fixing
$\dim(\mathcal{H}_{\textrm{A}})=r$. Even with this restriction, purifications
are not unique because of possible local unitary transformations on the
ancilla.

For convenience, we write any, not necessarily normalized, vector
$\ket{W}\in\mathcal{H}_{\textrm{S}}\otimes\mathcal{H}_{\textrm{A}}$ in the
form of a matrix, namely

$\ket{W}\\!\\!=\\!\\!\sum_{j,k=1}^{r}w_{jk}\ket{S_{j}}\otimes\ket{A_{k}}\mapsto\\!\\!\sum_{j,k=1}^{r}\\!\\!w_{jk}\outerproduct{S_{j}}{A_{k}}=W,$
(21)

where $w_{jk}\in\mathbb{C}$, and $\ket{S_{j}}$ and $\ket{A_{k}}$ span the
basis of $\mathcal{H}_{\textrm{S}}$ and $\mathcal{H}_{\textrm{A}}$,
respectively. This defines an isomorphism between
$\mathcal{H}_{\textrm{S}}\otimes\mathcal{H}_{\textrm{A}}$ and the space of
linear maps
$\mathcal{B}_{r}:=B(\mathcal{H}_{\textrm{A}},\mathcal{H}_{\textrm{S}})$. Such
a matrix representation considerably simplifies computation, as we will see
below. In particular we may now write
$\rho=\Tr_{\textrm{A}}{\outerproduct{W}{W}}=WW^{\dagger}$.

Figure 5: Schematic illustration of the bundle structure (not specific to any
$n,r$). To each state $\rho=WW^{\dagger}$, a fiber $\pi^{-1}(\rho)$ is
attached (cyan circle). The unitary group action transforms the purification
$W$ to a different point on the fiber $WU$. The vertical space
$\mathsf{V}_{W}\mathcal{W}_{r}$ forms the tangent space to the fiber while
$\mathsf{H}_{W}\mathcal{W}_{r}$ forms the orthogonal compliment.

We restrict the set $\mathcal{B}_{r}$ to matrices representing purifications
of density matrices of rank $r$ which defines the set

$\\!\\!\\!\\!\mathcal{W}_{r}=\\{W\in\mathcal{B}_{r}|\textrm{Tr}(WW^{\dagger}){=}1\textrm{
and }\exists(W^{\dagger}W)^{-1}\\},\\!$ (22)

where the first condition corresponds to the normalization and the second is
equivalent to having $WW^{\dagger}$ with a fixed rank $r$. In what follows, we
refer to the elements of $\mathcal{W}_{r}$ simply as purifications.

At each point $\rho\in\mathcal{Q}_{r}$, we can think of ‘attaching’ the set of
all purifications satisfying $\rho=WW^{\dagger}$. The triple
$(\mathcal{Q}_{r},\mathcal{W}_{r},\pi)$ forms a _fiber bundle_ , with $\pi$
being a projection map

$\pi:\mathcal{W}_{r}\to\mathcal{Q}_{r},\quad\pi(W)=WW^{\dagger}.$ (23)

The pre-image $\pi^{-1}(\rho)$ is known as the fiber over $\rho$ and are
precisely those purifications for which $WW^{\dagger}=\rho$. More precisely,
it is an example of a _principal fiber bundle_ as one can define a unitary
group action on the fibers which account for the unitary freedom on the
ancilla system

$\Xi:\mathcal{W}^{r}\times\textrm{U}(r)\to\mathcal{W}^{r},\quad\Xi(W,U)=WU.$
(24)

See also Appendix A. The projection map, in turn, induces a mapping between
the tangent spaces of $\mathcal{W}_{r}$ and $\mathcal{Q}_{r}$, namely

$\begin{split}&\mathrm{d}\pi_{{}_{W}}:\mathsf{T}_{{}_{W}}\mathcal{W}_{r}\rightarrow\mathsf{T}_{{}_{WW^{\dagger}}}\mathcal{Q}_{r},\\\
&\mathrm{d}\pi_{{}_{W}}(\dot{W})=\dot{W}W^{\dagger}+W\dot{W}^{\dagger}.\end{split}$
(25)

The tangent vectors in $\mathsf{T}_{{}_{W}}\mathcal{W}_{r}$ that are tangent
to the fiber form the vertical space, defined as
$\mathsf{V}_{{}_{W}}\mathcal{W}_{r}=\ker\mathrm{d}\pi_{{}_{W}}$. It may be
complemented to form the whole tangent space
$\mathsf{T}_{{}_{WW^{\dagger}}}\mathcal{W}_{r}$. The compliment is known as
the horizontal space, denoted $\mathsf{H}_{{}_{W}}\mathcal{W}_{r}$. We choose
this compliment to be orthogonal with respect to the metric locally defined as

$\begin{split}&(\cdot,\cdot):\mathsf{T}_{{}_{W}}\mathcal{W}_{r}\times\mathsf{T}_{{}_{W}}\mathcal{W}_{r}\to\mathbb{R},\\\
&(\dot{W},\dot{W}^{\prime})=\frac{1}{2}\Tr(\dot{W}^{\dagger}\dot{W}^{\prime}+\dot{W}^{\prime\dagger}\dot{W}),\end{split}$
(26)

where $\dot{W}$ and $\dot{W}^{\prime}$ are two tangent vectors at the point
$W$. This is a restriction of the Euclidean metric on
$\mathsf{T}_{{}_{W}}\mathcal{B}_{r}$. The horizontal space can then be thought
of as orthogonal to the fiber itself, see Fig. 5. An important property of
this metric is that it is invariant with respect to the unitary group action,
$(\dot{W}U,\dot{W}^{\prime}U)=(\dot{W},\dot{W}^{\prime})$. Such a metric is
called right-invariant and will allow us to connect the metric on
$\mathcal{W}_{r}$ to a metric on $\mathcal{Q}_{r}$ as seen below.

Any tangent vector $\dot{W}\in\mathcal{W}_{r}$ can be now decomposed into a
vertical and horizontal component
$\dot{W}=\dot{W}_{\textrm{v}}+\dot{W}_{\textrm{h}}$, with

$\displaystyle\dot{W}_{\textrm{v}}=WA,$ (27a)
$\displaystyle\dot{W}_{\textrm{h}}=LW,$ (27b)

where $A^{\dagger}=-A$, $L=L^{\dagger}$ and $\textrm{Tr}(LWW^{\dagger})=0$,
see for example [52].

In addition, the space $\mathsf{H}_{{}_{W}}\mathcal{W}_{r}$ is isomorphic with
the tangent space $\mathsf{T}_{{}_{WW^{\dagger}}}\mathcal{Q}_{r}$ [53]. This
in combination with (LABEL:euclidean_metric) being right-invariant allows for
$\mathcal{Q}_{r}$ to inherit the metric from $\mathcal{W}_{r}$. The induced
metric turns out to be exactly the Bures metric (13), namely

$(\dot{W}_{\textrm{h}},\dot{W}_{\textrm{h}}^{\prime})=(\dot{\rho},\dot{\rho}^{\prime})_{B},$
(28)

where
$\dot{W}_{\textrm{h}},\dot{W}_{\textrm{h}}^{\prime}\in\mathsf{H}_{{}_{W}}\mathcal{W}_{r}$,
$W\in\pi^{-1}(\rho)$, $\mathrm{d}\pi_{{}_{W}}\dot{W}_{\textrm{h}}=\dot{\rho}$
and $\mathrm{d}\pi_{{}_{W}}\dot{W}_{\textrm{h}}^{\prime}=\dot{\rho}^{\prime}$.
Importantly, the right-invariance of (LABEL:euclidean_metric) implies that
(28) will not depend on which purification $W$ we choose from the fiber.

Since $\mathcal{B}_{r}$ may be seen as a real vector space, it can be equipped
with an inner product. We pick the inner product compatible with
(LABEL:euclidean_metric), which takes the same form but with elements from
$\mathcal{B}_{r}$ instead of $\mathsf{T}\mathcal{W}_{r}$. This inner product
allows interpreting $\textrm{Tr}(WW^{\dagger})=1$ in (22) as a unit sphere.

The shortest geodesic is the shortest path between two given points
parameterized by its arc length. On a sphere, they are the great arcs. The set
$\mathcal{W}_{r}$ is a homogeneous space and forms an open and dense subset of
the unit sphere in $\mathcal{B}_{r}$. As a consequence, the geodesics on
$\mathcal{W}_{r}$ are also great arcs. The geodesic distance between two
purifications $W_{1},W_{2}$ is thus the arc length $\arccos(W_{1},W_{2})$.
Each geodesic in the base space $\mathcal{Q}_{r}$ can in turn be lifted to a
great arc in the purifications space, i.e., there exists a great arc in
$\mathcal{W}_{r}$ that projects down to the geodesic in $\mathcal{Q}_{r}$. In
particular, the shortest geodesic between $\rho_{1}$ and $\rho_{2}$ is a
projection of a shortest great arc between the corresponding fibers, i.e., a
geodesic between two purifications $W_{1}\in\pi^{-1}(\rho_{1})$,
$W_{2}\in\pi^{-1}(\rho_{2})$ minimizing $\arccos(W_{1},W_{2})$ [52]. This is
equivalent to requiring

$W_{1}^{\dagger}W_{2}\geq 0,$ (29)

as first shown in [54]; see also Appendix B.2. The corresponding geodesic
distance on $\mathcal{Q}_{r}$ equals
$\arccos\Tr\sqrt{W_{1}^{\dagger}W_{2}W_{2}^{\dagger}W_{1}}$. It may be
expressed independently of the purifications as

$\operatorname{dist}(\rho_{1},\rho_{2})=\arccos\sqrt{F(\rho_{1},\rho_{2})},$
(30)

where we recover the Uhlmann fidelity
$F(\rho_{1},\rho_{2})=\left(\Tr\sqrt{\sqrt{\rho_{1}}\rho_{2}\sqrt{\rho_{1}}}\right)^{2}$
[54]; see also Appendix B.2 for a self-contained derivation of this fact.

Figure 6: Both paths between $W_{1}$ and $W_{2}$, and between $W_{1}$ and
$W_{2}^{\prime}$ are geodesic on $\mathcal{W}_{r}$. The first one however is
the shortest geodesic between fibers $\pi^{-1}(\rho_{1})$ and
$\pi^{-1}(\rho_{2})$ and hence its projection onto $\mathcal{Q}_{r}$ is the
shortest geodesic between $\rho_{1}$ and $\rho_{2}$.

###  4.2 Speed limit

The Bures metric gives a natural speed limit of the evolution in terms of the
geodesic distance and the time-averaged speed of the trajectory [55, 56, 57].
Here, we will describe the speed in terms of a non-Hermitian generator of the
evolution and analyze the tightness of the speed limit for different upper
bounds of this speed.

A non-Hermitian evolution satisfying (6) can be lifted to an evolution in the
purification space satisfying

$\dot{W}=-iHW-\tilde{\Gamma}_{\rho}W,$ (31)

where $\tilde{\Gamma}_{\rho}$ can be written as $\Gamma-\textrm{Tr}(\Gamma
WW^{\dagger})$. Indeed the tangent
$\dot{\rho}=\mathrm{d}\pi_{{}_{W}}(\dot{W})$ can be found using Eq.
(LABEL:differential) and the fact that $\rho=WW^{\dagger}$. From Eq. (27a), it
follows that $\tilde{\Gamma}_{\rho}W\in H_{{}_{W}}\mathcal{W}_{r}$, while
$iHW$ in general contains both vertical and horizontal contributions. In fact,
(31) is horizontal iff $\Pi H\Pi=0$, where $\Pi=\sum_{k:\lambda_{k}\neq
0}\Pi_{k}$ is the projection onto the support of $\rho$ [52]. Consequently,
when moving horizontally within the support of the state, the term
$\tilde{\Gamma}_{\rho}W$ is non-zero, implying that the system is necessarily
open. This will be especially relevant for shortest geodesics, as we will see
below. Note that, in the special case of a full-rank state, the unitary flow
has to be zero, i.e., $H=0$, for evolution to be completely horizontal.

The horizontal part of (31) is obtained by decomposing $-iHW$ according to Eq.
(27a),

$\dot{W}_{\textrm{h}}=LW-\tilde{\Gamma}_{\rho}W,$ (32)

where $L$ is an SLD fulfilling $\dot{\rho}_{H}=L\rho+\rho L$. A solution to
this equation can be found by first applying the projection
$d\pi_{{}_{W}}(-iHW)=d\pi_{{}_{W}}(LW)$ which yields, using
(LABEL:differential), $-i[H,\rho]=\\{L,\rho\\}$. By using the spectral
decomposition of $\rho$ and projecting both sides of the equation on
$\Pi_{j}\bullet\Pi_{k}$, we find

$L=i\sum_{j,k}\frac{\lambda_{j}-\lambda_{k}}{\lambda_{j}+\lambda_{k}}\Pi_{j}H\Pi_{k}.$
(33)

As the metric (28) on $\mathsf{T}\mathcal{Q}_{r}$ is defined from the
horizontal components of the purifications, we find

$(\dot{\rho},\dot{\rho})_{\textsc{b}}=\textrm{Tr}(\dot{W}_{\textrm{h}}^{\dagger}\dot{W}_{\textrm{h}})=\Tr((L-\tilde{\Gamma}_{\rho})^{2}\rho).$
(34)

The form of $L$ given in (33) can help to recognize
$\Tr(L^{2}\rho)=\frac{1}{4}\mathcal{F}(H,\rho)$, where $\mathcal{F}(H,\rho)$
is the quantum Fisher information of the state with respect to $H$ [58]. As a
result

$(\dot{\rho},\dot{\rho})_{\textsc{b}}=\frac{1}{4}\mathcal{F}(H,\rho)+\Delta^{2}\Gamma+\frac{1}{4}\Tr(\\{L,\Gamma\\}\rho),$
(35)

where $\Delta^{2}\Gamma=\Tr(\Gamma^{2}\rho)-\Tr(\Gamma\rho)^{2}$.

Combining (35) and (30) results in the lower bound of the evolution time

$\begin{split}\tau\geq\frac{\arccos{\sqrt{F(\rho_{0},\rho_{\tau})}}}{\frac{1}{\tau}\int_{0}^{\tau}\sqrt{(\dot{\rho},\dot{\rho})_{\textsc{b}}}dt}.\end{split}$
(36)

We can yet define a weaker speed limit by upper bounding the speed, and obtain

$\begin{split}\tau\geq\frac{\arccos{\sqrt{F(\rho_{0},\rho_{\tau})}}}{\frac{1}{\tau}\int_{0}^{\tau}\sqrt{\Delta^{2}H+\Delta^{2}\Gamma-i\Tr([H,\Gamma]\rho)}dt}.\end{split}$
(37)

To see this, note that
$\dot{W}^{\prime}=-i\tilde{H}_{\rho}W-\tilde{\Gamma}_{\rho}W$ is also a lift
of $\dot{\rho}$, i.e., $\mathrm{d}\pi_{{}_{W}}(\dot{W}^{\prime})=\dot{\rho}$.
As before, the tilde with subscript $\rho$ indicates a shift of the operator
such that the expectation value with respect to $\rho$ equals zero. The
inequality then follows from
$(\dot{\rho},\dot{\rho})_{\textsc{b}}=(\dot{W}_{\textrm{h}}^{\prime},\dot{W}_{\textrm{h}}^{\prime})\leq(\dot{W}^{\prime},\dot{W}^{\prime})$.
This is saturated iff $\Pi\tilde{H}_{\rho}\Pi=0$, which is equivalent to $\Pi
H\Pi=\alpha\Pi$, $\alpha\in\mathbb{R}$. It trivially holds for pure states but
not for general mixed states [52].

A similar speed limit to (36) has been proposed in [59]. However, their
derivation relies on using the metric (LABEL:euclidean_metric), which is
dependent on the choice of purification, in contrast to (28). Consequently,
they consider the distance between two arbitrarily chosen purifications
$W_{0}$ and $W_{\tau}$ in the fibers of $\rho_{0}$ and $\rho_{\tau}$ which in
general does not amount to the distance between the fibers. This means that
their proposed bound can be violated whenever $W^{\dagger}_{0}W_{\tau}$ does
not satisfy Eq. (29).

###  4.3 Generator of geodesic

We move now to deriving an explicit expression for the shortest geodesics in
$\mathcal{Q}_{r}$ and describing a non-Hermitian dynamics that can generate
it. This section is based on [60, 61] and extends the results to states with
rank $r<n$.

The shortest geodesic in $\mathcal{Q}_{r}$ between two states $\rho_{1}$ and
$\rho_{2}$ can be lifted to the shortest geodesic between the fibers in
$\mathcal{W}_{r}$ over these states, i.e., to a geodesic between $W_{1},W_{2}$
satisfying condition (29). As an arc of a great circle, it takes the form

$W(\tau)=\frac{1}{\sin\theta}\left(\sin(\theta-\tau)W_{1}+\sin\tau
W_{2}\right),$ (38)

where $\theta=\arccos{(W_{1},W_{2})}$ is the geodesic distance. The geodesic
on $\mathcal{Q}_{r}$ is then given by
$\rho_{g}(\tau)=W(\tau)W^{\dagger}(\tau)$.

We now ask what evolution follows such a path. More precisely, we look for a
geodesic evolution operator $G_{\textrm{g}}(\tau)$ such that
$W(\tau)=G_{\textrm{g}}(\tau)W_{1}$, or equivalently
$\rho_{g}(\tau)=G_{\textrm{g}}(\tau)\rho_{1}G_{\textrm{g}}^{\dagger}(\tau)$.
To do so, we introduce the operator $M$ mapping the initial purification onto
the final one, $W_{2}=MW_{1}$. This operator is defined uniquely only for
full-rank states. For $r<n$, we have some freedom and pick

$M:=W_{2}W_{1}^{+}+iR\\\
+(\mathds{1}{-}\Pi_{1})W_{2}W_{1}^{+}+(W_{2}W_{1}^{+})^{\dagger}(\mathds{1}{-}\Pi_{1}),$
(39)

where $\Pi_{1}=W_{1}W^{+}_{1}$ is the projection onto the support of $W_{1}$,
defined from the Moore-Penrose inverse $W_{1}^{+}$. We choose $R\geq 0$ such
that $R=(\mathds{1}-\Pi_{1})R\,(\mathds{1}-\Pi_{1})$ to ensure invertibility
of the geodesic evolution operator $G_{\textrm{g}}(\tau)$, and detail this
choice in Appendix B.4.

Any point $W(\tau)$ on the shortest trajectory can thus be obtained from the
initial state as $W(\tau)=G_{\textrm{g}}(\tau)W_{1}$, where

$G_{\textrm{g}}(\tau)=\frac{1}{\sin\theta}\left(\sin(\theta-\tau)\mathds{1}+\sin\tau\,M\right).$
(40)

Trivially, conjugation with such an operator yields the evolution on
$\mathcal{Q}_{r}$.

Note that such dynamics preserves the trace along the chosen geodesic, but not
necessarily along another trajectory. We therefore define the map

$\Phi_{\tau}^{\textrm{g}}:\mathcal{Q}_{r}\to\mathcal{Q}_{r},\quad\Phi_{\tau}^{\textrm{g}}(\rho):=\frac{G_{\textrm{g}}(\tau)\rho
G_{\textrm{g}}^{\dagger}(\tau)}{\Tr(G_{\textrm{g}}(\tau)\rho
G_{\textrm{g}}^{\dagger}(\tau))},$ (41)

which is of the form (3). As described in Sec. 2, the generator of this
evolution is given by

$K_{\textrm{g}}(\tau):=i\derivative{\tau}G_{\textrm{g}}(\tau)\,G_{\textrm{g}}^{-1}(\tau).$
(42)

Note that, by construction, the trajectory in the purification space is
horizontal, i.e., $\dot{W}(\tau)\in H_{W(\tau)}\mathcal{W}_{r}$. For full-rank
states, this implies that $K_{\textrm{g}}(\tau)$ is anti-Hermitian at all
times and corresponds to the unique solution of
$K_{\textrm{g}}(\tau)\rho(\tau)+\rho(\tau)K_{\textrm{g}}(\tau)=i\derivative{\tau}\rho(\tau)$.

In general, the generator (42) is time-dependent and not unique. It prescribes
the motion along the shortest path between two states with constant speed. A
natural question is whether it can be made time-independent, by possibly
loosening the restriction on the speed being constant. In the qubit case, it
is always possible to do so. The resulting generator is necessarily anti-
Hermitian for the full rank state, as illustrated in the application below. In
the general case, however, a time-independent generator does not always exist.
This claim is supported by the counter-example of a qutrit, which we detail in
Appendix B.5.

Figure 7: (a) State norm $\Tr(\rho)$ along the evolution between $\rho_{1}$
and $\rho_{2}$ generated by (42) (red dashed) and its optimization (solid
blue). Inset: relative difference of the norms. (b) Decay rate $\gamma$ for
the generator (42) (red dashed) and $\gamma_{\textrm{opt}}$ for its
optimization (solid blue) along the evolution between $\rho_{1}$ and
$\rho_{2}$. Inset: difference of the rates.

##  5 Application: Non-Hermitian control in a qubit

To illustrate our findings, we consider the shortest geodesic between the
qubit states $\rho_{1}=\frac{1}{2}(\mathds{1}+\frac{1}{5}\sigma_{z})$ and
$\rho_{2}=\frac{1}{2}(\mathds{1}-\frac{1}{2}\sigma_{x}-\frac{1}{2}\sigma_{z})$.
This trajectory is generated using (42) and then compared with the
optimization described in Sec. 3.5. For a fair comparison, the spectrum of the
anti-Hermitian part is shifted so that the smallest eigenvalue equals zero for
both generators. Figure 7 shows how the state norm and its decay rate change
over time for the two trajectories. As expected, we see that the optimization
leads to a slower decay rate of the norm. Notably, the relative difference of
the norms, which quantifies how many more times the experiment has to be re-
run for the trajectory $\rho$ in order to reach the same success rate as the
optimized trajectory $\rho_{\textrm{opt}}$, gets relatively close to one. The
decay rate for the optimized dynamics $\gamma_{\textrm{opt}}$ drops to zero
and becomes non-analytic at the time when the state passes the minor axis of
an ellipse associated with the geodesic, as discussed below. At that point,
such an ellipse is tangent to a concentric sphere, meaning that the velocity
at this point has no commutative component and can be fully expressed in terms
of Hamiltonian vector field. That is exactly what the optimization procedure
does. The non-analyticity there is caused by the level crossing in
$\Gamma_{\textrm{c}}$.

In order to construct these generators, we first took the trivial
purifications of the states, respectively $W_{1}=\sqrt{\rho_{1}}$ and
$W_{2}^{\prime}=\sqrt{\rho_{2}}$, which in general do not satisfy condition
(29). Keeping $W_{1}$ fixed and using the unitary freedom we then found a
purification $W_{2}$ satisfying (29) by picking
$W_{2}=W_{2}^{\prime}U^{\dagger}$, where $W_{1}^{\dagger}W_{2}^{\prime}=PU$ is
the polar decomposition with $P\geq 0$ and $U$ unitary. We defined $W(\tau)$
according to (38) and found $G_{g}(\tau)=W(\tau)W_{1}^{-1}$ (notice the
simplification thanks to the existence of the inverse $W_{1}^{-1}$). Finally,
we defined $K_{g}(\tau)$ according to (42).

Interestingly, the image of any shortest geodesic in a qubit can be generated
by time-independent anti-Hermitian Hamiltonians (see Appendix B.6 for details)
being a rescaled generator (42). Consequently, each shortest geodesic will
form a segment of an ellipse that has its centre at the fully mixed state and
the points contained along its major axis being the two eigenstates of (42),
as illustrated in Fig. 8.

Figure 8: The figure depicts the plane (cyan) containing the maximal mixed
state ($\textrm{p}_{3}$), and the initial and final states ($\rho_{1}$ and
$\rho_{2}$). The intersection with the Bloch sphere forms a great circle and
the ellipse (magenta) for which the shortest geodesic is a segment of is the
unique ellipse intersecting $\rho_{1}$ and $\rho_{2}$ and with its two points
on its major axis ($\textrm{p}_{1}$ and $\textrm{p}_{2}$) lying on the great
circle. The point $\textrm{p}_{4}$ lies on the minor axis of the ellipse and
is always an interior point for full-rank states.

##  6 Conclusion

Starting from the basic description of quantum evolution, we have shown how a
trace-preserving non-Hermitian evolution preserves the rank of the density
matrix. This allowed us to develop a geometrical understanding of such
dynamics. Specifically, the tangent vector (master equations (5) and (6)) can
be decomposed in the instantaneous state eigenbasis into a direction that
generates coherence and another that changes population only. These directions
are mutually orthogonal. The incoherent part can further be decomposed into a
component that lifts degeneracy and decreases the von Neumann entropy, and a
part that can be viewed as classical since it does not change the eigenbasis.
By decomposition of the anti-Hermitian part of the generator into the coherent
and incoherent direction, we showed that the non-Hermitian dynamics that
maximizes the success rate of a given trajectory is one whose anti-Hermitian
part commutes with the instantaneous state.

We then considered the shortest path linking two states. Using techniques of
purification, we found the condition (29) to obtain the shortest path. We
derived a quantum speed limit (37) and constructed the generator of the
geodesic (40), that is, the non-Hermitian evolution that saturates the speed
limit.

It is natural to ask whether the above geometrical interpretation could be
extended to rank-changing dynamics, as resulting from a Lindblad master
equation, including the jump terms. Even though the set of all quantum states
$\mathcal{Q}$ is no longer a manifold, it forms a Whitney stratified space,
i.e., a collection of manifolds glued together, and as such, can be endowed
with a generalized differential structure [62]. It is then possible to extend
the notion of tangent spaces to $\mathcal{Q}$ and distinguish commutative and
unitary directions. The precise treatment of this problem exceeds the scope of
this paper.

We hope that the insights presented in this manuscript and the intuition that
can be gained from our results will help develop control protocols in many-
body systems.

##  7 Acknowledgements

The authors thank Dan Allan for early contributions, and Ole Sönnerborn and
Pablo Martínez-Azcona for providing valuable feedback on the manuscript. This
research was funded by the Luxembourg National Research Fund (FNR, Attract
grant 15382998 and grant 17132054).

##  Appendix A Geometrical preliminaries

This Appendix provides a brief overview of the concepts from differential
geometry used in the main text.

###  A.1 Differentiable manifolds

Let us start with reviewing the notion of manifold. A real $d$-dimensional
manifold $\mathcal{M}$ is a topological space locally homeomorphic to open
subsets of $\mathbb{R}^{d}$. For example, a sphere is a $2$-dimensional
manifold. However, a sphere with an attached string is not because the string
itself is $1$-dimensional. If these local homeomorphisms, called charts, ”glue
together” in a differentiable way, the local similarity to real space allows
for introducing the notion of differentiability, otherwise not defined on the
topological spaces. We call the collection of such charts covering the whole
$\mathcal{M}$ a _differentiable atlas_ and the manifold itself a
_differentiable manifold_.

One may ask if such a differentiable manifold could not be embedded in
$\mathbb{R}^{n}$ with $n\geq d$ and inherit the notion of differentiability
from there instead of having it defined locally in the charts. It turns out
that it is indeed possible, as made precise by Whitney embedding theorem, and
that the two notions are compatible (in the sense that every embedding induces
a differentiable atlas and every differentiable atlas defines an embedding).
Throughout this paper, we use the later picture as a natural embedding of our
manifolds exists in higher dimensional spaces. However, all notions discussed
here can be seen as internal to the manifold without the need for any external
space, which is exactly where the true power of the geometrical picture lies.

###  A.2 Tangent spaces and bundles

The tangent space is one of the most important and used notions in
differential geometry. Let us consider a differentiable path
$\gamma:[0,1]\to\mathcal{M}$ on a differentiable $d$-dimensional manifold
$\mathcal{M}$ (embedded in $\mathbb{R}^{n}$) passing through a point $p$.
Trivially, the velocity of $\gamma$ at $p$ is given by its derivative at $p$
(in the embedding) and is tangent to $\gamma$. The set of all possible
velocities of all differentiable paths passing through $p$ is the _tangent
space_ $\mathsf{T}_{p}\mathcal{M}$. Indeed, $\mathsf{T}_{p}\mathcal{M}$ can be
thought of as a $d$-dimensional hyperplane tangential to $\mathcal{M}$ at
point $p$.

The velocity of a curve at each of its points belongs to different tangent
spaces. It would be useful to have a unified description of this velocity
along the whole curve. The proper way to do so is to introduce the notion of a
tangent bundle as a locally trivial fiber bundle.

A locally trivial fiber bundle, later called just fiber bundle, is a triple
$(\mathcal{E},\pi,\mathcal{M})$ consisting of a total space $\mathcal{E}$ and
a base space $\mathcal{M}$, both being differentiable manifolds, and a
surjective map $\pi:\mathcal{E}\to\mathcal{M}$, called projection, satisfying
the _local triviality condition_. Namely, $\mathcal{E}$ has to look locally
like a product space $\mathcal{M}\times\mathcal{F}$ for some manifold
$\mathcal{F}$ called the _fiber_. More formally, we require that there exist a
smooth manifold $\mathcal{F}$, such that for any $p\in\mathcal{M}$ there
exists a neighborhood $U$ and a smooth homeomorphism $\phi$ such that the
following diagram commutes

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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(43)

The _tangent bundle_ $\mathsf{T}\mathcal{M}$ is then an example of a fiber
bundle over $\mathcal{M}$ with the total space being a disjoint union of all
tangent spaces
$\mathcal{E}=\bigsqcup_{p\in\mathcal{M}}\mathsf{T}_{p}\mathcal{M}$ and a
natural projection $\mathsf{T}\mathcal{M}\supset\mathsf{T}_{p}\mathcal{M}\ni
X\mapsto\pi(X)=p$. A velocity of $\gamma$ is then a path in
$\mathsf{T}\mathcal{M}$. However, we are still unable to talk about its change
along the path as we did not specify how to relate different fibers
$\mathsf{T}_{p}\mathcal{M}\subset\mathsf{T}\mathcal{M}$ and take derivatives.
We are going to fill this gap in the next subsection.

We can further enrich the fiber bundle by adding a smooth right action of a
Lie group $G$ on the total space $\mathcal{E}$, which preserves the fibers and
acts freely and transitively on them. Such a fiber bundle is then called a
principal fiber bundle.

###  A.3 Connections, vertical and horizontal spaces

To connect different fibers, we need to add yet another structure, a
_connection_. A connection is a smooth way of splitting all tangent spaces to
the total space $\mathsf{T}_{X}\mathcal{E}$ into two complementary subspaces:
a _vertical space_ $\mathsf{V}_{X}\mathcal{E}$ consisting of directions along
the fiber $\mathcal{F}$, and a _horizontal space_ $\mathsf{H}_{X}\mathcal{E}$.
Their disjoint unions together with the bundle projection
$\pi^{\prime}:\mathsf{T}\mathcal{E}\to\mathcal{E}$ form vertical
$\mathsf{V}\mathcal{E}$ and horizontal $\mathsf{H}\mathcal{E}$ bundles
respectively.

Note that specifically, to talk about a change in the velocity of a curve or
acceleration, we have to define connection on a tangent bundle with a tangent
bundle as its base space $\mathsf{T}\mathsf{T}\mathcal{M}$.

###  A.4 Riemannian manifolds

So far, we have defined a differentiable manifold and a tangent bundle over
it. It is easy to check that every $\mathsf{T}_{p}\mathcal{M}$ is a vector
space. However, we cannot yet quantify how different are the vectors, i.e.,
there is no scalar product. It is tempting just to inherit the Euclidean inner
product from the embedding (this is what we indeed do in the main text, as the
considered embeddings will be, in some sense, natural). In general, however, a
different choice may be more natural (although Nash’s theorem states that for
any such choice, there exists an embedding inducing the same local inner
product). A smooth assignment of scalar products to each tangent space is
called a Riemannian metric, and a differentiable manifold now endowed with
such a metric becomes a Riemannian manifold.

As an additional side note, let us mention that there also exist complex
manifolds that locally look like $\mathbb{C}^{d}$ and not $\mathbb{R}^{d}$.
For example, a physical Hilbert space is, in a natural way, such a complex
manifold. Complex manifolds can, however, be seen as real manifolds of double
the dimension with an additional structure, a particular endomorphism of the
tangent bundle, which amounts to specifying the multiplication by an imaginary
unit. We use this fact to introduce a natural Riemannian structure on Hilbert
spaces.

##  Appendix B Fixed rank dynamics

###  B.1 Orthogonality between coherent and incoherent directions

The inner product $\langle\cdot,\cdot\rangle$ corresponding to the monotone
metric (12) can be computed through the polarization identity

$\langle\dot{\rho},\dot{\rho}^{\prime}\rangle=\frac{1}{4}\big{(}\norm{\dot{\rho}+\dot{\rho}^{\prime}}^{2}-\norm{\dot{\rho}-\dot{\rho}^{\prime}}^{2}\big{)}.$
(44)

For any pair of tangent vectors
$\dot{\rho}_{\textrm{u}}\in\mathsf{U}_{\rho}\,\mathcal{Q}_{n}$ and
$\dot{\rho}_{\textrm{c}}\in\mathsf{C}_{\rho}\,\mathcal{Q}_{n}$, we can choose
an eigenbasis common to $\rho$ and $\dot{\rho}_{\textrm{c}}$ such that
$(\dot{\rho}_{\textrm{c}})_{jk}=0$ $\forall j\neq k$ and
$(\dot{\rho}_{\textrm{u}})_{jj}=0$ $\forall j$. From (12), we have

$\begin{split}&\norm{\dot{\rho}_{\textrm{c}}\pm\dot{\rho}_{\textrm{u}}}^{2}=\sum_{j,k}\absolutevalue{(\dot{\rho}_{\textrm{c}}\pm\dot{\rho}_{\textrm{u}})_{jk}}^{2}c(\lambda_{j},\lambda_{k})\\\
&=\sum_{j}\absolutevalue{(\dot{\rho}_{\textrm{c}})_{jj}}^{2}c(\lambda_{j},\lambda_{j})+\sum_{j\neq
k}\absolutevalue{(\dot{\rho}_{\textrm{u}})_{jk}}^{2}c(\lambda_{j},\lambda_{k})\\\
&=\norm{\dot{\rho}_{\textrm{c}}}^{2}+\norm{\dot{\rho}_{\textrm{u}}}^{2}.\end{split}$
(45)

It then follows from (44) that
$\langle\dot{\rho}_{\textrm{u}},\dot{\rho}_{\textrm{c}}\rangle=0$.

To show orthogonality between two tangent vectors
$\dot{\rho}_{\textrm{cl}}\in\mathsf{Class}_{\rho}\,\mathcal{Q}_{n}$ and
$\dot{\rho}_{\textrm{l}}\in\mathsf{L}_{\rho}\,\mathcal{Q}_{n}$, consider a
common eigenbasis between $\rho$, $\dot{\rho}_{\textrm{cl}}$ and
$\dot{\rho}_{\textrm{l}}$; $\\{\ket{\lambda_{jk}}\\}$, where the first index
runs over the number of distinct eigenvalues and $k$ over the multiplicities
$m_{j}$. We let $\lambda_{j}$ denote the distinct eigenvalues so that
$\lambda_{jk}=\lambda_{j}$ $\forall k\in\\{1,2,\dots m_{j}\\}$. We can then
write
$\dot{\rho}_{\textrm{cl}}=\sum_{j}\sum_{k=1}^{m_{j}}\mu_{j}\outerproduct{\lambda_{jk}}{\lambda_{jk}}$
and
$\dot{\rho}_{\textrm{l}}=\sum_{j}\sum_{k=1}^{m_{j}}\nu_{jk}\outerproduct{\lambda_{jk}}{\lambda_{jk}}$,
where
$\sum_{k=1}^{m_{j}}\nu_{jk}=0\impliedby\Tr(\dot{\rho}_{\textrm{l}}\Pi_{j})=0$.
We then have

$\begin{split}&\norm{\dot{\rho}_{\textrm{cl}}\pm\dot{\rho}_{\textrm{l}}}^{2}=\sum_{j}\sum_{k=1}^{m_{j}}\absolutevalue{(\dot{\rho}_{\textrm{cl}}\pm\dot{\rho}_{\textrm{l}})_{jk,jk}}^{2}c(\lambda_{jk},\lambda_{jk})\\\
&=\sum_{j}c(\lambda_{j},\lambda_{j})\sum_{k=1}^{m_{j}}(\mu_{j}\pm\nu_{jk})^{2}\\\
&=\sum_{j}c(\lambda_{j},\lambda_{j})\sum_{k=1}^{m_{j}}(\mu_{j}^{2}+\nu_{jk}^{2}).\end{split}$
(46)

Together with (44), it follows that
$\langle\dot{\rho}_{\textrm{cl}},\dot{\rho}_{\textrm{l}}\rangle=0$.

In conclusion, we have shown that the subspaces
$\mathsf{U}_{\rho}\mathcal{Q}_{n}$, $\mathsf{Class}_{\rho}\mathcal{Q}_{n}$ and
$\mathsf{L}_{\rho}\mathcal{Q}_{n}$ are mutually orthogonal to one another with
respect to any monotone metric.

###  B.2 The Bures angle

The Bures angle between two states $\rho_{1}$ and $\rho_{2}$ equals the length
of the shortest spherical arc connecting the respective fibers and reads

$\Theta(\rho_{1},\rho_{2})=\inf_{W_{1},W_{2}}\arccos(W_{1},W_{2}),$ (47)

where $(W_{1},W_{2})=\real\Tr(W_{1}^{\dagger}W_{2})$ is the euclidean inner
product. The minimization over the purification can be done by fixing
purifications $W_{1}$, $W_{2}^{\prime}$ of $\rho_{1}$, $\rho_{2}$
respectively, and then considering all possible gauge transformations
$W_{2}=W_{2}^{\prime}U$. The angle is minimized when the inner product between
$W_{1}$ and $W_{2}$ is maximized.

Let $A$ be an operator and let $\\{\ket{k}\\}$ be the set of the eigenvectors
of $AA^{\dagger}$. We then have $\real(\Tr
A)\leq\sum_{k}\absolutevalue{\bra{k}\\!A\\!\ket{k}}=\sum_{k}\sqrt{\bra{k}\\!A\\!\ket{k}\\!\bra{k}\\!A^{\dagger}\\!\ket{k}}\leq\sum_{k}\sqrt{\bra{k}\\!AA^{\dagger}\ket{k}}=\Tr\sqrt{AA^{\dagger}}$.
The inequalities are saturated iff $A$ is positive. If we now take
$A=W^{\dagger}_{1}W_{2}$ we see that the upper bound is independent of $U$ and
the angle is minimized iff $W^{\dagger}_{1}W_{2}\geq 0$.

Note that $A\geq 0\iff
A=\sqrt{AA^{\dagger}}=\sqrt{W_{1}^{\dagger}W_{2}W_{2}^{\dagger}W_{1}}=\sqrt{\sqrt{\rho_{1}}\rho_{2}\sqrt{\rho_{1}}}$
where we fixed $W_{1}=\sqrt{\rho_{1}}$ without loss of generality.
Consequently

$\Theta(\rho_{1},\rho_{2})=\arccos{\Tr\sqrt{\sqrt{\rho_{1}}\rho_{2}\sqrt{\rho_{1}}}},$
(48)

which recovers the Uhlmann fidelity.

###  B.3 Rank of the sum of operators

Consider two positive operators $A\geq 0$ and $B\geq 0$. A vector belongs to
the kernel of a positive operator iff its expectation value equals zero.
Moreover, we have

$\bra{\psi}\\!(A+B)\\!\ket{\psi}=0\iff\bra{\psi}\\!A\\!\ket{\psi}=\bra{\psi}\\!B\\!\ket{\psi}=0.$
(49)

Hence $\textrm{ker}(A+B)=\textrm{ker}(A)\cap\textrm{ker}(B)$. If $d$ is the
dimension of the Hilbert space, we have that

$\begin{split}\textrm{rank}(A+B)&=d-\textrm{dim}\big{(}\textrm{ker}(A+B)\big{)}\\\
&\geq d-\textrm{dim}\big{(}\textrm{ker}(A)\big{)}\\\
&=\textrm{rank}(A).\end{split}$ (50)

In other words, the rank of the sum of two positive operators either increases
or stays the same.

###  B.4 Invertibility of the geodesic evolution operator

We show that the evolution operator $G_{\textrm{g}}(\tau)$ (40) has a trivial
kernel and is thus invertible for all $\tau\in[0,\theta]$.

It is enough to ensure that there are no negative or zero eigenvalues in the
spectrum of $M$. Indeed, the condition
$\forall\ket{\psi}\in\mathcal{H}:M\ket{\psi}\neq-\frac{\sin(\theta-\tau)}{\sin{\tau}}\ket{\psi}$
is equivalent with the kernel of $G_{\textrm{g}}(\tau)$ being trivial.

Using the projectors $W_{1}W_{1}^{+}$ and $(\mathds{1}-W_{1}W_{1}^{+})$, the
operator $M$ can be expressed as the symmetric block matrix

$M=\begin{pmatrix}A&C^{\dagger}\\\ C&B\end{pmatrix},\quad A\geq 0,$ (51)

where dim$(A)=r$ and dim$(B)=n-r$. We let this matrix act on any vector

$\begin{pmatrix}A&C^{\dagger}\\\ C&B\end{pmatrix}\begin{pmatrix}a\\\
b\end{pmatrix}=\begin{pmatrix}Aa+C^{\dagger}b\\\ Ca+Bb\end{pmatrix},$ (52)

and compute the inner product between the vector and its image with respect to
$M$. It reads $(a,b)_{M}:=(a,Aa)+(b,Bb)+2\real(b,Ca).$ Now, choosing $B$ such
that $B=\pm iR$ where $R>0$ is sufficient for $(a,b)_{M}$ to never be a
negative number. This implies that no negative numbers are included in the
spectrum of $M$, and thus ensures $G_{g}(\tau)$ has a trivial kernel.

###  B.5 Time-independent geodesic generator – a counterexample

In the main text, we derive the generator of the geodesic (42), which, in
general, is not unique and is time-dependent. We asked whether it could be
made time-independent by loosening the constant speed requirement. Here, we
show an example in a qutrit for which this is not possible. Consider the
initial and target states

$\rho_{1}=\begin{pmatrix}\frac{1}{5}&0&0\\\ 0&\frac{2}{5}&0\\\
0&0&\frac{2}{5}\end{pmatrix},\quad\rho_{2}=\begin{pmatrix}\frac{1}{5}&0&0\\\
0&\frac{3}{10}&0\\\ 0&0&\frac{1}{2}\end{pmatrix}.$ (53)

In this case, choosing the purifications $W_{1}=\sqrt{\rho_{1}}$ and
$W_{2}=\sqrt{\rho_{2}}$ satisfies (29) and, according to (39)-(40) and (42),
the generator of the evolution at initial time reads

$K_{\textrm{g}}(0)=i\begin{pmatrix}\frac{1-\cos{\theta}}{\sin{\theta}}&0&0\\\
0&\frac{\frac{\sqrt{3}}{2}-\cos{\theta}}{\sin{\theta}}&0\\\
0&0&\frac{\frac{\sqrt{5}}{2}-\cos{\theta}}{\sin{\theta}}\end{pmatrix}.$ (54)

Because $\rho_{1}$ and $\rho_{2}$ are both full rank and diagonal, the
geodesic $\rho(t)$ is diagonal at all times. This means that it has no
velocity in the coherent directions, and hence, any time-independent generator
$K$, if it exists, can be made anti-Hermitian. Such a generator can thus be
assumed to be of the form $K=\alpha K_{\textrm{g}}(0)+i\mu\mathds{1}$,
$\alpha\in\mathbb{R}_{+},\mu\in\mathbb{R}$. We further show in Appendix B.6
that if the generator of the shortest path is time-independent and anti-
Hermitian, then it necessarily has only two distinct eigenvalues. Since this
is not the case for such $K$, we must conclude that a time-independent
generator that generates the shortest path does not always exist.

###  B.6 Time-independent geodesic generators on $\mathcal{Q}_{n}$

Consider a full-rank state evolution generated by a non-Hermitian time-
independent operator $K=H-i\Gamma$. One possible lift of this dynamics to the
purification space is described by the Schrödinger equation
$\dot{W}(t)=-iHW(t)-(\Gamma-\langle\Gamma\rangle)W(t)$. Due to the fact that
the images of geodesics are great arcs, we may say that if the image of $W(t)$
is an image of a geodesic, then it lies in the real span formed by the initial
state $W(0)$ and the initial tangent $\dot{W}(0)$. Consequently, by Taylor
expanding the evolution over some open interval and using the fact that $W$ is
invertible because of the full-rank condition, we get
$K^{n}\in\text{sp}\\{\mathds{1},K\\}$ for any power $n\in\mathbb{N}$. In this
case, we can always subtract a scaling of the identity from $K$ to ensure that
$K^{2}=k\mathds{1}$, where $k$ is a real number888Indeed, since
$K^{2}=a\mathds{1}+ibK$ holds for some pair of real numbers $a,b$, we can look
at $K^{\prime}=K-c\mathds{1}$ for which $K^{\prime
2}=K^{2}-2cK+c^{2}\mathds{1}=(a+c^{2})\mathds{1}+(ib-2c)K$. Taking $c=ib/2$
and $k=a-b^{2}/4$, we get $K^{\prime 2}=k\mathds{1}$.. So up to rescaling, we
get three cases: $K^{2}=\pm\mathds{1}$ and $K^{2}=0$; thus the generator is
either diagonalizable with eigenvalues $\\{1,-1\\}$, $\\{i,-i\\}$, or
nilpotent of degree two.

For the geodesic to be a shortest one between the fibers, it must be
horizontal. This is the case iff $W^{\dagger}(K^{\dagger}+K)W=0$, which for
invertible $W$ is equivalent to $K^{\dagger}+K=0$. We hence have that the
generator of the shortest geodesic on $\mathcal{Q}_{n}$ has to have its
spectrum contained in $\\{i,-i\\}$, up to a global shift and rescaling.

###  B.7 Qubit geodesics contained on an ellipse

Let $H=i\sigma_{z}$ and consider a general qubit state which, without loss of
generality, is assumed to be
$\rho=\frac{1}{2}(\mathds{1}+x\sigma_{x}+z\sigma_{z})$. The (normalized)
evolution is then given by

$\rho\mapsto\frac{1}{2(\cosh{2t}+z\sinh{2t})}(e^{2\sigma_{z}t}+x\sigma_{x}+ze^{2\sigma_{z}t}\sigma_{z}).$
(55)

The Bloch ball coordinates $(X,Z)$ are thus expressed in terms of
$X=\frac{x}{\cosh{2t}+z\sinh{2t}}$ and
$Z=\frac{z\cosh{2t}+\sinh{2t}}{\cosh{2t}+z\sinh{2t}}$. Now, note that

$\frac{X^{2}}{a}+Z^{2}=1\iff a=\frac{X^{2}}{1-Z^{2}}=\frac{x^{2}}{1-z^{2}}.$
(56)

The fact that $a$ is positive and time-independent proves that the state moves
along an ellipse spanned by the $zx$-plane. We also see that the major axis is
equal to one and thus touches two antipodal pure states, which are the
eigenstates of $H$.

## References

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