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Isotropy and control of dissipative quantum
dynamics
Benjamin Dive1, Daniel Burgarth2and Florian Mintert1
1Department of Physics, Imperial College, SW7 2AZ London, UK
2Institute of Mathematics, Physics and Computer Science, Aberystwyth University,
SY23 3FL Aberystwyth, UK
(Dated: 25 September 2015)
Abstract. We investigate the problem of what evolutions an open quantum system
with Hamiltonian controls can undergo. A series of no-go theorems which exclude
channels from being reachable, and an expression for the required evolution time, are
given by considering noisy dynamics as an anisotropic flow in state space. As well as
studying examples of the strength of these criteria in control theory, we explore their
relation with existing approaches and links with quantum thermodynamics.
Contents
1 Introduction 1
2 Set up 3
3 Theoretical Results 5
3.1 Discussion................................... 7
3.2 Proofs ..................................... 9
4 Numerical Results 13
5 Conclusion 15
1. Introduction
The ability to coherently control quantum dynamics has received considerable interest
in the last few decades, both for its potential application in technology [1,2] and the
insight it provides to fundamental science [3,4,5]. It is therefore surprising that the
question of what dynamics can be reached with such controls has not been answered
for open systems, where interactions with the environment cannot be neglected [6,7].
A thorough understanding of this would allow the optimisation of both the design and
operation of devices for quantum computation, communication and sensing.
arXiv:1509.07163v1 [quant-ph] 23 Sep 2015
CONTENTS 2
The corresponding question for finite dimensional noiseless systems has been
answered by bilinear control theory using algebraic tools from Lie theory [8,9]. Attempts
to generalise these methods to open systems have met considerable mathematical
difficulties and have only been attempted for Markovian systems, which are memoryless
and governed by Lindbladians [10]. The two principle results of this approach is an
accessibility criterion [11,12,13], which describes which directions can be explored
for short times; and Lie wedges [14,15], which provide a partial characterisation of
the geometry of the reachable set but in general cannot be calculated exactly. Other
approaches focus on the related question of state-controllability, where the interest is in
the ability to map one state to another [16,17,18,19].
Figure 1: The figure illustrates the principle ideas of this paper by showing a cross section
of state space at two different time, with the small arrows indicating the direction of
flow induced by a Lindbladian. The large arrows correspond to the available controls.
We see that different parts of the space are contracting at different rates; by rotating
the system in time with Hamiltonian controls some of these decay rates can be averaged
together.
In this paper we look at operator controllability by investigating another geometric
approach and, rather than attempting to characterise what can be done, show that
there are broad ranges of evolutions that an open system with local coherent control
cannot reach. Similar ideas were introduced in [20] in the case of Markovian, time-
independent and unital quantum systems with a special focus on qubits. In order to get
an intuitive feel for the principle idea behind our work, it is useful to consider the action
of a Lindbladian as a flow in state space, as illustrated in figure 1. A Lindbladian can
act in a variety of ways on the state space, including rotating and shrinking the space
(corresponding to decay) in a potentially anisotropic way. Hamiltonian controls allow
us to impose additional rotations on the system such that different parts of the state
CONTENTS 3
space feel different contraction rates at different times. This results in the ability to
mix the decay rates together and leads to the overall evolution obeying some averaged
rates. These cause the final state space, which represents the total evolution, to be more
isotropic than in the absence of controls.
Our main results relate directly to this, and state that a noisy quantum operation
cannot be replicated with another source of noise if the former has a more ordered
structure than the latter. After introducing the relevant mathematics in section 2,
we define the decay rates of an open system with this structure in mind in section 3,
and show that the action of any coherent controls is to make these more uniform. We
demonstrate that the sum of these rates is unaffected by control and thus obtain a
strict condition for the times at which a target evolution can be reached. In the case
of Markovian noise this fully determines a single time when the target may be reached.
In addition, we obtain bounds on how the purifying power of noise can be enhanced by
control. The strength of these criteria is tested numerically in section 4for common
examples of noise, and we show that the no-go theorems are strong and tight enough
to provide a major restriction on what evolutions are possible. Both the language and
the mathematics used to describe these processes are reminiscent of thermodynamics, a
relation which we explore in section 5, where we also compare the methods of this paper
with prior work on Lie wedges.
2. Set up
The aim of this paper is to formalise the intuition described above and apply it to more
complex processes which are time-dependent, non-unital, and non-Markovian in order
to obtain some very general rules for which operations can or cannot be performed on an
open quantum system by varying controls. To do this, we first state the control problem
formally. We consider Hamiltonian controls on a finite dimensional system interacting
with an environment, with the requirement that the reduced system obeys a master
equation:
d
dtρ=Gt(ρ) = G0
t(ρ)−i[Ht, ρ]
ρT=MT(ρ0)≡ T eRT
0dtGt(·)ρ0,(1)
where ρis a quantum state, Gtis the generator for the motion, and MTis the resulting
dynamical map, the set of which (varying over total times and controls) we aim to
characterise. The generator is divided into an uncontrollable drift G0
tand a controllable
Hamiltonian term Ht. The latter is a time-dependent control Hamiltonian chosen so
as to generate the desired dynamics, and we assume no restrictions on it beyond being
Hermitian. G0
trepresents the intrinsic part of the dynamics, such as an internal energy
splitting or an interaction with the environment, and is unaffected by the controls. If
we restrict it to be a (time-dependent) Lindblad operator, which we will denote by Lt,
then the allowed solutions are (time-dependent) Markovian, completely-positive trace-
preserving maps. Although we will sometimes make this restriction in order to gain
CONTENTS 4
more physical insight, the key results of this paper do not rely on this and hold for a
more general generator.
It is worth noting that there is an implicit assumption in describing the system in the
form of Eq.(1): that the local Hamiltonian controls do not affect the interaction with the
environment. While there are cases where this holds exactly, this is not always the case;
tracing out the interaction with the environment and then adding a local Hamiltonian
yields different results than performing the trace after the controls have been added
[21,22,23]. This assumption should therefore be treated as an approximation, one
which is commonly made as it simplifies the problem such that the full environment
does not need to be modelled explicitly.
In order to make the picture introduced in figure 1rigorous, and derive our results,
it is necessary to introduce some mathematical concepts and notation. The starting
point is to work in the generalised Bloch representation [24], where a quantum state
ρis represented as the real vector |ρi= (x0, x1, x2, ..., xd2−1)T, where xi= Tr[σiρ] are
the expectation values over an orthonormal set of traceless Hermitian matrices, with
the exception of σ0=1
√d1, and where dis the dimension of the underlying Hilbert
space. In this representation super-operators acting on states become matrices. We will
denote dynamical maps and generators in this representation by Mand Grespectively to
distinguish them from their super-operator form. The spectral properties (eigenvalues,
singular values, trace and determinant) of the super-operators are given by those of their
matrix representation. The matrix representation of the duals of the super-operators
(denoted by †) are the Hermitian conjugate of the matrix representation of the original
super-operator.
Writing out the explicit form of Mhighlights some of its properties. If the
dynamical map is trace-preserving then it can easily be shown that Mis of the form
1 0 0 ...
v1˜
M11 ˜
M12 ...
v2˜
M21 ˜
M22 ...
... ... ... ...
(2)
where the top row is fixed and all the elements are real if Mis Hermiticity preserving.
This form explicitly separates the unital and non-unital part of the dynamics. Unitality
refers to leaving the maximally mixed state unchanged; as this is the only state left
invariant by all Hamiltonians and it is the centre of rotations, this is an important
property for control. The left hand column consisting of the elements vifully describe
the non-unitality of the map and quantifies how much the maximally mixed state
is shifted by the dynamical map. For unital operations, such as unitary evolution,
these vanish and the dynamical map reduces to M= 1 ⊕˜
M. The unital dynamics
described by ˜
Mare partially decoupled from the non-unital dynamics in the sense that
^
MBMA=˜
MB˜
MA, meaning that the total unital dynamics of a concatenation is given
by the unital part of the individual super-operators. This can easily be seen by noting
that the concatenation of super-operators is given by their matrix product. It is also
CONTENTS 5
useful to have a measure of how unital or non-unital a dynamical map is; a convenient
one is the purity of the maximally mixed state after having been acted on by the map,
ν(M) = Tr M(1
d1)2=1
d(1 + Pv2
i). This value is maximised at one if the maximally
mixed state is mapped to a pure state and is minimised to 1
dif it is unital.
The matrix form of a trace-preserving generator Gis identical, expect that in this
case the entire top row is 0. Its unital and non-unital parts can be separated in a similar
way and the same results on concatenation holds. Because of this, we have that f
eG=e˜
G
and, hence, that if Mis the dynamical map generated by Gtthen ˜
Mis the one given
by ˜
Gt. The meaning of this is that inside every non-unital problem is buried a unital
one with a dimension 1 smaller, and any solution to the control problem of generating
Mmust also solve ˜
M.
As they are particularly important, we note the form that the super-operators of
closed dynamics take in this representation. Unitary propagators, M(·) = U·U†, become
matrices in the defining representation of the rotation group 1⊕SO(d2−1). Hermitian
generators, G(·) = −i[H, ·], are in the corresponding Lie algebra, 0 ⊕so(d2−1), which
consist of real antisymmetric matrices. In the case of d= 2, unitary propagators form
all such matrices (and like-wise for Hamiltonians), but in higher dimension they only
form a subgroup. This property is one of the fundamental reasons why the results of
this paper are no-go theorems rather than a complete characterisation of the allowed
dynamics.
The intuitive picture described in the introduction relies on a notion of averaging a
list (the decay rates) to obtain another. This process is described by the majorization
relation [25] which tests if one real list is more uniformly distributed (or less ordered)
than an other. A list a
a
ais majorized by another list b
b
b, written as a
a
a≺b
b
b, if:
a↓
1≤b↓
1
a↓
1+a↓
2≤b↓
1+b↓
2(3)
...
Xa↓
i=Xb↓
i,
where a↓signifies that the elements of a
a
aare sorted in decreasing order. Another way
of stating this is that a
a
ais majorized by b
b
bif and only if a
a
ais a convex combination of
permutations of b
b
b. It is this property which makes it suitable to describe an averaging
procedure. A point to note is that, unlike standard inequalities on the reals, majorization
provides only a partial order.
3. Theoretical Results
With this formalism, we are now in a position to refine the intuition developed in figure
1. The picture was of a Lindbladian (or a more general generator) acting on the state
space such that it flowed from one shape to another. By coherently controlling the
system, the space can be rotated so that some of the decay rates are averaged together.
CONTENTS 6
For a purely dissipative Markovian two-level system these decay rates are the eigenvalues
of the Lindbladian, which are always non-positive. In higher dimensions however these
eigenvalues can be complex and it is far less evident which quantities can be averaged by
rotations. The situation is further complicated in the non-unital case where the centre
of rotations moves over time in a way which depends on the controls, and is made even
more difficult in the case of non-Markovian dynamics where the generators are not of
Lindblad form and less can be said about their eigenvalues.
The correct decay rates to consider are the eigenvalues of the generator plus its dual,
or equivalently, of the Hermitian part of G. This arises naturally in the mathematics
and make physical sense for two of reasons. Firstly, they are real, so the majorization
condition is meaningful. Secondly, they are independent of any rotational part of the
dynamics (whether it corresponds to a Hamiltonian degree of freedom or not), and hence
capture only the decay/growth component of the flow. Similarly, it is possible to see
that the anisotropy of the dynamical map is described by its singular values (as they
are rotationally invariant), which loosely correspond to the characteristic lengths of the
final state space.
This, together with majorization as described above, allows us to write the
conditions which must all be satisfied for a dynamical map Mto be reachable by a
system with drift G0
t. These are: an expression for the evolution time to a fixed state
space volume,
det (M) = eRT
0Tr(G0
t)dt,(4)
a constraint on the anisotropy of the dynamical map,
log [σ
σ
σ(M)] ≺ZT
0
λ
λ
λ G0
t+G0†
t
2!dt, (5)
a unital version of this condition (for trace-preserving G0
t),
log [σ
σ
σ(˜
M)] ≺ZT
0
λ
λ
λ ˜
G0
t+˜
G0†
t
2!dt, (6)
which was shown in [20] for a time-independent Lindblad drift, and a bound on the
maximal non-unitality that can be reached,
ν(M)≤sup
ρ,t
Tr[ρ2] where Tr ρ G0
t(ρ)= 0.(7)
λ
λ
λand σ
σ
σrefer to the list of eigenvalues and singular values of the super-operator
respectively, and the log acts element-wise on the lists. In order to make the equations
look more similar to each other, and stress that it is the Hermitian part of the drift that
matters, G0
tcan be replaced by G0
t+G0†
t
2in Eqs.(4) and (7).
Instead of seeing if the system can reach a target map, we can instead ask if it can
effectively simulate another generator G0. In this case we we obtain the condition
λ
λ
λG0+G0†≺λ
λ
λG0+G0†,(8)
CONTENTS 7
where we have picked the relative time scales such that Tr[G0] = Tr[G0]. The unital
version of this relation,
λ
λ
λ˜
G0+˜
G0†≺λ
λ
λ˜
G0+˜
G0†,(9)
also holds provided both generators are trace-preserving.
We now proceed to give a detailed discussion of these criteria, followed by their
proofs.
3.1. Discussion
Evolution time — The first criterion, Eq.(4), is an equality while the others are
inequalities, and it is interesting to investigate its meaning in more detail. The modulus
of the determinant of Mis the volume occupied by its image, while the trace of Gtis the
rate at which this state space is growing (this will be non-positive, unless the system is
non-Markovian, leading to a contraction of the space). One interpretation of this result
is thus that the total rate at which volume is lost in state space is independent of the
Hamiltonian controls. More physical insight can be gained if we restrict the drift; if the
generator is Hermiticity preserving (a weaker condition than it being positive), then its
trace is always real. This means that such a system can only reach maps with positive
determinant. As this is not the case for every completely-positive trace-preserving map,
this condition allows us to immediately rule out large sections of the space as unreachable
for a broad class of dynamical systems [26,14].
If we further restrict the drift to be a Lindbladian at all times (with a non-vanishing
dissipative part), then the trace is always negative, signifying that a target map can
only be reached at a single instant in time (if at all), and that this time can be easily
calculated. This is in stark contrast to the case of closed systems, where in general little
is known about the time required to reach a target. In the case of a unitary target
map, which has determinant one, the longer the controls take the more the map reached
will deviate from the target. In a non-Markovian system, where the interplay between
memory effects and controls has received much recent attention [27,28], the trace of the
drift can be positive for certain times, leading to revivals in the determinant. Indeed, this
has already been suggested as an indicator of non-Markovianity [29]. For our purposes,
this particular feature leads to the possibility of there being several solutions to Eq.(4)
for a given target map. Our result thus gives a clear interpretation of one of the benefits
of non-Markovianity in control: it provides more flexibility in when a target map can
be reached.
Dynamical map anisotropy — The conditions of Eqs.(5) and (6) are a refinement of
the intuition that controlling the system allows us to mix the decay rates of the drift
together. This is most easily seen by looking at the unital case. The left hand side
of the relation are the singular values of the dynamical map which, in the same way
as the determinant is the volume in state space, are the characteristic lengths of the
CONTENTS 8
final state space. Thus, while Eq.(4) determines the volume reached, Eq.(6) provides
a constraint on the anisotropy of the dynamical maps that can be reached. The non-
unital majorization relation has broadly the same interpretation, although the overall
shift caused by the non-unitality manifests itself in the decay rates and singular values in
a complex way. Indeed, one of the eigenvalues of G0
t+G0†
twill typically be positive in the
non-unital case, even if the drift is Markovian. Thus, although the two conditions look
very similar, in practice they give very different results and there are many dynamical
maps that pass one test but fail the other for a given drift (as we will later see in figure
2). It is also worth noting that Eq.(4) is recovered, up to a modulus sign, by the last
term in the majorization relations.
Maximal non-unitality — This last condition on dynamical maps, Eq.(7), is
conceptually very different from the others. Rather than restricting the shape of the
dynamical map, it is a constraint on how much it can move the maximally mixed
state which corresponds to where the centre of the image of the dynamical map lies
in state space. Although the right hand side of Eq.(7) is independent of controls, the
maximisation over all states (and over all tif the drift is time-dependent) makes this
criterion somewhat harder to evaluate in higher dimensions. It is worth noting that
the constraint is less strict than G0(ρ) = 0, which means that the non-unitality is not
bounded by the fixed points of the drift and can go beyond it. The interpretation of this
criterion is therefore that it is possible to increase the ability of noise to purify states
by using controls, but only up to the limit given.
A physical example of this is a three-level system in a Λ configuration, with the
excited state decaying into the two low level states. In the absence of controls, the
system has some non-unitality as the maximally mixed state over the three levels will
decay to a mixed state over two levels. With the presence of controls, however, we
can coherently transfer the population back from one of the two ground states to the
excited state where it will once again decay. Doing this many times results in the
total population being transferred to the other ground state and the total action of
the dynamics is to map everything to a single pure state. Thus, this dynamical map
induced by a specific set of controls has maximal non-unitality. This is the principle
behind optical pumping and shows that non-unitality can be increased with controls. If
the two lower levels had some decay between them, however, this scheme may not work
perfectly; Eq.(7) provides a bound for this.
Generator anisotropy — Instead of seeing if the system can reach a target map, we
can instead ask if it can effectively simulate another generator, such that it gives rise to
the same dynamical map for all times. In this case we take G0
tto be time-independent
and see if it can reach M=eG0tfor all t; from which we obtain Eq.(8). This should
be thought of as a condition for the simulation of some noisy dynamics with another
source of noise in a time continuous fashion, rather than a criterion to evolve to a single
point in time. This criterion is therefore stricter than the anisotropy conditions on
CONTENTS 9
dynamical maps; it imposes not only a target map but the whole trajectory in time
to the map. This result is at first hand surprising, as the only generators which can
be reached exactly from a given drift are given precisely by the drift plus all possible
controls. However, by quickly rotating the system it is possible to get arbitrarily close to
the desired trajectory by winding tightly around the desired path without ever moving
exactly along it.
In the case of the generator G0being a unital qubit Lindbladian, Eq.(8) can be
simplified further. This is because as L0is symmetric (up to Hamiltonian components
which can be cancelled out with controls) [30], the previous equation collapses to
λ
λ
λ(L0)≺λ
λ
λL0(10)
and is sufficient, an observation that was made in [20]. Furthermore the eigenvalues
of such unital qubit Lindbladians, λ
λ
λ(L), are constrained by complete positivity [31].
From this it is straightforward to show that λ
λ
λ(L) = −(1
2,1
2,0) majorizes all others.
Hence, a Lindbladian with such a spectrum is universal - it can simulate all other
unital qubits Lindbladians. An example of such a Lindbladian is given by dephasing,
L(·) = −[σz,[σz,·] ]. Conversely, the completely depolarising channel with eigenvalues
λ
λ
λ(L) = −(1
3,1
3,1
3) is majorized by all other Lindbladians and is therefore at the bottom
of the hierarchy defined by majorization.
3.2. Proofs
Evolution time — To derive Eq.(4), we begin by noting that the formal solution for M
is given in terms of time-ordered matrix exponential, which can be expressed according
to the Magnus expansion
M=TeRT
0G(t)dt =eRT
0G(t1)dt1+1
2Rt
0dt1Rt1
0dt2[G(t1),G(t2)]+....(11)
As the determinant of a matrix exponential is the exponential of the trace we can rewrite
this as
det (M) = eTrhRT
0G(t1)dt1+1
2Rt
0dt1Rt1
0dt2[G(t1),G(t2)]+...i=eTr[RT
0G(t)dt](12)
where we have used the fact that commutators are traceless. Finally, as discussed
previously, the controls are also traceless. As the trace and determinant of Mand G
are identical to those of Mand G, this gives the desired expression
det (M) = eRT
0Tr[G0
t]dt.(13)
In the case that the Magnus expansion does not converge (which may happen if
RT
0||G(t)||2dt > π), then the time-ordered exponential in Eq.(11) can be split into
several terms. As the determinant of a product is the product of the determinants,
the proof follows through in the same way, and the integrals over the trace can be
recombined into a single term at the end.
CONTENTS 10
Dynamical map anisotropy — The proof of the dynamical map anisotropy conditions
arise from two observations. Firstly, that the evolution can be decomposed into
infinitesimal time-steps in a Trotter-like way, alternating between coherent and
incoherent evolution. Secondly, the controls only affect the coherent steps which are
all rotation matrices and so do not contribute to the singular values. To prove (5) we
expand the time-ordered exponential in terms of short time steps
M=TeRT
0G(t)dt,
= lim
δt→0eG0(T)δt eH(T)δt ... eG0(0)δt eH(0)δt .(14)
We now look at the singular values of both sides of the equation, denoted by the operator
σ
σ
σ. Specifically, we use the result in [25] for the majorization relation between the singular
values of matrices and their products:
log σ
σ
σ(AB)≺log σ
σ
σ(A) + log σ
σ
σ(B) (15)
where the log is understood as acting on each element in the list, and the sum on the
right hand size acts on the ordered lists. This is straightforward to generalise to the
case of products of many matrices. Applying this to Eq.(14) we obtain:
log σ
σ
σ(M)≺lim
δt→0nlog σ
σ
σeG0(T)δteH(T)δt+... + log σ
σ
σeG0(0)δteH(0)δto.
As mentioned above, the coherent steps corresponds to rotation matrices do not affect
the singular values. This allows the expression for the singular values to be simplified
to
log σ
σ
σ(M)≺lim
δt→0nlog[σ
σ
σ(eG0(T)δt)] + ... + log[σ
σ
σ(eG0(0)δt)]o.(16)
We recall that singular values are defined according to σ
σ
σ(A) = λ
λ
λ√AA†, where λ
λ
λ
signifies the eigenvalues. From this, each term in the previous equation can be expressed
as
lim
δt→0log[σ
σ
σ(eG0(t)δt)] = lim
δt→0log "λ
λ
λeG0(t)δt+G0†(t)δt 1
2#,
= lim
δt→0λ
λ
λ G0(t) + G†
0(t)
2!δt. (17)
Substituting this result into Eq.(16), yields:
log [σ
σ
σ(M)] ≺ZT
0
λ
λ
λG0(t) + G0†(t)
2dt, (18)
which is independent of the representation used and so holds for the super-operators
themselves. The proof for condition (6), the unital version of this for trace-preserving
generators, follows immediately from the fact that M=Texp{RT
0G(t)dt}implies
˜
M=Texp{RT
0˜
G(t)dt}, as was discussed in section 2. A different proof of this latter
unital result was shown in [20] for time-dependent Lindbladians only, and relied on
similar mathematical ideas.
CONTENTS 11
Maximal non-unitality — To prove the non-unitality bound, Eq.(7), we begin with the
formal expression and then derive an easy to calculate bound for it. We also make the
additional assumption that Gtis continuous. The maximal non-unitality of a channel
is, by construction,
ν(M) = sup
t,Hτ
Tr[ρ2] where ρ=TeRt
0dτGτ(·)1
dI,(19)
where the supremum is over all possible evolution times and all possible controls. This
appears as difficult to calculate as solving the initial problem, and is therefore of limited
use. However an upper bound for it can be found more readily. To do this we consider
the maximisation as happening in two steps. Firstly by maximising over τ∈[0, t] for a
given set of controls, and then over all possible controls. The first part corresponds to
finding the maximum of a curve, and the second as finding which curve has the highest
maxima. For continuous Gτthe first maximisation must obey one of following three
conditions: the maximum is reached at t= 0, or d
dt Tr[ρ2] = 0, or the maximum is
reached at t→ ∞. The first case can be ruled out immediately by noting that at t= 0
the purity is in fact a minimum, due to the initial condition of Eq.(19). As the purity
is bounded from above (and is smooth for continuous generators) if the maximum is
reached at t→ ∞, the gradient must also tend to 0 in that limit. Hence, we have that
a necessary condition for the first step in the maximisation of Eq.(19), which must also
be obeyed by the overall supremum, is:
d
dt Tr[ρ2] = 0,
Tr[ρ G0
t(ρ)] −iTr [ρ[Ht, ρ]] = 0,
Tr[ρ G0
t(ρ)] = 0.(20)
We note that the gradient as calculated above depends solely on G0
tand not on
the controls, hence we can relax the condition on Gtbeing continuous to G0
tbeing
continuous. Instead of calculating the supremum over all controls, we can compute it
over all states which satisfy this condition. This allows us to place an easier to calculate
bound on the maximum non-unitality reachable by a dynamic system as desired:
ν(M)≤sup
ρ,t
Tr[ρ2] where Tr ρ G0
t(ρ)= 0.(21)
Generator anisotropy — The derivation of the criterion for generator anisotropy begins
with the result for dynamical map majorization and the additional constraint that it
holds for all time. To do this, we replace Mby eG0tand consider the system drift to be
time-independent. If this condition is satisfied for infinitesimal tthen, by concatenation,
it holds for all time and we say that G0can effectively simulate G0. Under these
conditions Eq.(5) simplifies to
log [σ
σ
σ(eG0δt)] = log [λ
λ
λ(eG0+G0† δt
2)] = λ
λ
λG0+G0†δt
2(22)
=⇒λ
λ
λG0+G0†≺λ
λ
λG0+G0†,
CONTENTS 12
where we have picked the relative time scales such that Tr[G0] = Tr[G0]. The unital
version of this relation also holds provided both generators are trace-preserving for the
reasons discussed in section 2.
To prove the sufficiency of this criterion for the case of unital qubit Lindbladians,
Eq.(10), we provide an explicit way to reach ˜
L0using a drift ˜
L0and unrestricted
Hamiltonian controls. As we are dealing with solely unital operators, we drop the
tilde notation for this section and pick the time scale such that Tr[L0] = Tr[L0] = 1. By
using the singular value decomposition, the target can be expressed as:
M=eL0t=UDV =U
e−ν1t0 0
0e−ν2t0
0 0 e−ν3t
V(23)
where −νare the eigenvalues of L0, and Uand Vare members of O(3). Furthermore,
as Mhas positive determinant and the diagonal block is positive, we can pick Uand V
to have determinant +1, thereby restricting them to SO(3). In a similar way, we can
express the free evolution of the system for time tas
eL0t=W F (t)W†=W
e−σ1t0 0
0e−σ2t0
0 0 e−σ3t
W†(24)
where the −σiare the eigenvalues of L0. Using the same argument as above, Wcan
be constraint to be in SO(3). Controls on the system allow the implementation of
any R=eHwhich, as we noted previously, corresponds to any matrix in SO(3). The
control scheme to reach the target map corresponds to alternating free evolution and
instantaneous controls as
UDV =R1W F (t1)W†R2....W F (tn)W†Rn+1 (25)
We pick R1=U W †,Rn+1 =W V and relabel W†RjW=R0
j. This can always be done
due to the group structure. Next we pick the R0to be permutation matrices (which all
lie in SO(3)) such that Eq.(25) consists solely of diagonal matrices where each of the
term is an exponential. This lets us express it in the simple form
ν1
ν2
ν3
=
σ1
σ2
σ3
t1+
σ1
σ3
σ2
t2+... +
σ3
σ2
σ1
t6,(26)
where, we recall from the way we picked the scale of L0, that Pti= 1. This control
scheme thus allows us to reach any M=eL0twhere the eigenvalues of L0are a convex
combination of those of L0; which is equivalent to saying that they are majorized by
them [25]. This means that λ
λ
λ(L0)≺λ
λ
λ(L0) is a sufficient, as well as necessary, condition
for reachability in unital qubit systems with unconstrained Hamiltonian control.
CONTENTS 13
4. Numerical Results
The conditions detailed above are all necessary but not sufficient, so the question of
how strong they are is important. We carried out extensive numerical simulations to
quantify this in two ways: firstly by how often the tests forbid a target from being
reached, and secondly by how often a target can be reached when it is not excluded.
Figure 2: The figures characterise the strength of the different no-go theorems on a
qubit system for a family of drifts. The top graph shows what fraction of channels are
excluded by each test as a function of the σzexpectation value of the steady state of
the drift, itself a function of temperature. The bottom graph shows the same channels
and what fraction satisfied all the tests and, out of those, how many could be reached
with a numerical optimisation package [32] to within a distinguishability (given by the
diamond norm) of at least 0.1%.
We investigate these two measures by first looking at a family of qubit systems.
The drifts we consider are generalised amplitude damping Lindbladians (corresponding
to spontaneous emission to a bath at finite temperature [33]), the targets are randomly
generated time-dependent Markovian maps [34], and the evolution time is chosen
according to the determinant condition. Which maps are excluded by the no-go theorems
and which are reached are plotted in figure 2. Taken together the criteria cut out over
90% of the space as unreachable for all the drifts considered, and show that at least
CONTENTS 14
10% of those not ruled out can be reached. It is also interesting to note that the relative
importance of the different conditions varies with temperature: when the bath is at
T= 0 (corresponding to hzi=1) the unitality test provides no information and the
unital majorization test is the most restrictive, while at high temperatures (low hzi)
their importance is reversed.
The second example is a qutrit in a non-symmetric Λ configuration where the top
level decays to the two lower levels and we focus on the skew in the system - the ratio of
the two decay rates. We investigated how close such a drift could get to maps generated
by a similar drift but with a different skew. Figure 3shows that those which were as or
more symmetric as the drift could be reached with a very high fidelity, and increasingly
poorly those which were less symmetric; in very good agreement with the majorization
criteria as plotted. The tightness of the no-go theorems in this scenario show how useful
they are in cases where there is a clear measure of non-uniformity, demonstrating that
we can go from a highly ordered evolution to a less ordered one, but not the other way
around.
Figure 3: The graph shows how the dynamical map anisotropy conditions relate to
the controllability of a non-symmectric Λ system. The system drift is an amplitude-
decay Lindbladian fully characterised by a skew of 10, the ratio of the decay rates
from the top level to each bottom level [35]. Plotted is the minimal distance (given
by the diamond norm) to targets with different skews that could be reached with
a numerical optimisation package [32]. The thick line at 10.3 is the non-reachable
boundary given by the majorization criteria - everything to the right of it is excluded
- in excellent agreement with the numerical results. The slight bump just below 10 is
due to the difficulty of numerically finding solutions which require strong, sharp control
Hamiltonians.
This example was also discussed in section 3.1 on maximal non-unitality in
the context of optical pumping, although the conclusions may at first hand appear
contradictory. There we said that a qutrit in a Λ configuration could have a pure fixed
CONTENTS 15
point regardless of the ratio of the decay rates; while here we stress that the skew
cannot be increased. The resolution to this problem is that although the fixed point of
the dynamics can be chosen independent of the skew, this only determines the evolution
at t→ ∞, at all other times the state space occupies a finite volume. The shape of this
volume is what is constrained by the skew.
5. Conclusion
The importance of the eigenvalues of the drift plus its dual have been stressed throughout
this paper, and is most clearly illustrated in the generator anisotropy condition, showing
that they are critically important and physically relevant quantities to consider when
controlling open systems. Further evidence that this is the case can be found by relating
them to the entropy production rate, given in unital channels by the rate of change of
entropy of the reduced system [36]. An expression for this in terms of the spectral
properties of the channel is given in [37] which can be easily modified for Markovian
channels, using Eq.(5), to give:
d
dtS[ρ(t)] ≥λ1
2||ρ(t)−1
d||2
2(27)
where Sis the von-Neumann entropy, ρ(t) is a state evolving under a unital Lindbladian,
||·||2is the L2norm, and λ1is the smallest (in magnitude) eigenvalue of the unital part
of the Lindbladian plus its dual. Not only does this relate the decay rates to a physical
quantity, it also means that this lower bound on the entropy production rate can only
be increased with the addition of controls.
It is interesting to relate these results on majorization of generators, and especially
of Lindbladians, to existing work based on Lie wedges [14,15]. The Lie wedge provides
a sufficient but not necessary condition for controllability (as the semigroup closure still
needs to be taken), while the criteria of this paper are necessary but not sufficient.
Taken together, they allow us to approximate the reachable set from both sides. Our
approach gives results which are easier to use and can be calculated numerically, while in
many cases there are no known methods to determine the exact Lie wedge, especially in
the non-unital case. It also has the considerable advantage of allowing drifts which are
non-Markovian and time-dependent. However, the method used here does not enable us
to see what effect reducing the allowed set of controls has. In the simplest case of unital
qubit Lindbladians, we showed that majorization was sufficient if we had unrestricted
Hamiltonian controls; a result which can also be obtained from Lie wedges, showing the
consistency of the two methods.
As well as addressing issues of controllability, the results can also be understood
in a wider context. In particular, Eq.(8) demonstrates that performing local operations
on a system can modify its effective interaction with an environment. As majorization
is a one way relation, this means that a series of reversible operations on a system
can change its dynamics with an environment in a non-reversible way. This is at first
CONTENTS 16
hand surprising, and suggests a potential new avenue of research to understand how
irreversibility arises from unitary dynamics. The relation to thermodynamics can also
be seen in Eq.(27), where the bound on entropy production rate could be used to study
what can be achieved with algorithmic cooling [38] in the presence of continuous noise.
Finally, we note the importance that majorization plays both here and, in a slightly
modified form, in thermodynamics [39]. There it acts as a relation between initial and
final states, while in this paper it is elevated to the level of super-operators. The normal
interpretation of majorization is that of the Second Law - systems go from being well
ordered to less well ordered. Our results hint that this could be applied to super-
operators too; that local reversible control leads to an evolution operator which is less
well ordered, independent of the state it acts on.
Acknowledgement — We would like to thank the QuTIP project [32] for providing the
numerical packages used, and especially to Alex Pitchford for developing the pulse opti-
misation package for it. We are grateful to HPC Wales for providing the computational
time required. Fruitful discussions were had with David Jennings, Kamil Korzekwa and
Matteo Lostaglio on the links with thermodynamics, for which we are thankful. This
work was supported by EPSRC through the Quantum Controlled Dynamics Centre for
Doctoral Training, the EPSRC grant EP/M01634X/1, and the ERC project ODYC-
QUENT.
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