Completely positive master equation for arbitrary driving and
small level spacing
Evgeny Mozgunov
1
and Daniel Lidar
1,2,3,4
1
Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA
2
Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, California 90089, USA
3
Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA
4
Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA
Markovian master equations are a ubiquitous tool in the study of open quantum systems,
but deriving them from first principles involves a series of compromises. On the one hand, the
Redfield equation is valid for fast environments (whose correlation function decays much faster
than the system relaxation time) regardless of the relative strength of the coupling to the system
Hamiltonian, but is notoriously non-completely-positive. On the other hand, the Davies equation
preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with
a finite level spacing, which makes it incompatible with arbitrarily fast time-dependent driving.
Here we show that a recently derived Markovian coarse-grained master equation (CGME), already
known to be completely positive, has a much expanded range of applicability compared to the
Davies equation, and moreover, is locally generated and can be generalized to accommodate
arbitrarily fast driving. This generalization, which we refer to as the time-dependent CGME,
is thus suitable for the analysis of fast operations in gate-model quantum computing, such as
quantum error correction and dynamical decoupling. Our derivation proceeds directly from the
Redfield equation and allows us to place rigorous error bounds on all three equations: Redfield,
Davies, and coarse-grained. Our main result is thus a completely positive Markovian master
equation that is a controlled approximation to the true evolution for any time-dependence of the
system Hamiltonian, and works for systems with arbitrarily small level spacing. We illustrate
this with an analysis showing that dynamical decoupling can extend coherence times even in a
strictly Markovian setting.
1 Introduction
Modeling experiments requires taking into account that physical systems are open, i.e., not ideally isolated
from their environments. Usually an environment contains many more degrees of freedom than the system
itself, but is not in any interesting phase of matter where the detailed modeling of each of those degrees of
freedom is required. This makes the problem of modeling open systems tractable. More precisely, the problem
is tractable under a list of assumptions on the parameters of the bath (environment) and its interaction with
the system [13].
One natural approach is to consider the case when this interaction is weak compared to all other energy
scales in the problem. In this limit, Davies showed [4] that the dynamics of the system are given by a
Markovian master equation with what is now called Davies generators. Rigorous bounds on the error between
the solution of this equation and the true evolution can be obtained [presented here in Eq. (264)]. The largest
contribution to the error comes from making the rotating wave approximation (RWA).
1
After the system
evolves for a time set by a relevant relaxation timescale, the error in its density matrix is given by:
system-bath coupling strength ×
q
bath correlation time/system level-spacing . (1)
1
This approximation is stated explicitly in Ref. [3], page 87, around Eq. (3.6.67).
Accepted in Quantum 2020-01-19, click title to verify. Published under CC-BY 4.0. 1
arXiv:1908.01095v2 [quant-ph] 30 Jan 2020
This quantity, in general, becomes exponentially large in the system size due its denominator, so that the
coupling strength or the bath correlation time need to be sufficiently small for the use of Davies generators to
be justified, up to some upper system size limit. In practice, for convenience and also because it may apply
in a range that is larger than the rigorous but conservative bounds imply, the Lindblad master equation [5]
with Davies generators is often used outside its formal range of applicability in modeling of experiments
(e.g., Ref. [69]). In such cases it should be understood as a semi-empirical model of open system dynamics,
that possesses all the physical properties of the dynamics without reproducing them exactly. Unfortunately,
since we naturally wish to apply modeling tools to problems for which we do not know a priori what the
correct answer is, this approach cannot provide correctness guarantees. However, other methods are even
more susceptible to this problem, e.g., a path-integral (non-master equation) approach based on integrating
out the bath [1012]: this involves integration over a very high-dimensional space, so methods [13] that bring
it back into the realm of numerical feasibility usually involve uncontrollable approximations.
One of the important requirements of open system dynamics is complete positivity [14], under which
density matrices whose eigenvalues are by definition non-negative are mapped to other such matrices, even
when applied to a subsystem of a larger system.
2
This property is equivalent to the Markovian master
equation being in Lindblad canonical form [5, 16]. There are numerous other master equations (e.g., Refs. [17
39]), some in Lindblad form and some not, whose domains of validity and ranges of applicability were not
always studied in detail. One promising candidate is a coarse-grained master equation (CGME) [40], which
is in Lindblad form and thus automatically completely positive (CP). In this work we show that a suitably
generalized variant of this equation, that includes an arbitrarily time-dependent system Hamiltonian (hence
called the “time-dependent CGME”), has a much milder error than (1):
system-bath coupling strength ×bath correlation time , (2)
which does not diverge with the system size n for local observables, and has only a polynomial [O(n
α
) with
1/2 α 1] prefactor for nonlocal observables. We note that the full form of the error estimate (2) is not
uniform over the evolution time, unlike some of the error bounds developed elsewhere [39, 41]. We also show
that this new master equation is an improvement over the Lindblad equation with Davies generators in that
it has a local structure, compatible with approximate simulation methods with asymptotically smaller costs
[O(n) vs. O(exp(n))], and that the error estimate (2) is valid for an arbitrary time-dependence of the system
Hamiltonian. In particular, the time-dependent CGME is compatible with the assumption of arbitrarily fast
gates, often made in the circuit model of quantum computing, e.g., in the analysis of fault-tolerant quantum
error correction [42]. This assumption was shown to be incompatible with the derivation of the Davies master
equation [43].
This paper is structured as follows. We summarize our main results in Sec. 2, where we present the main
master equations involved in this work: Redfield and Davies-Lindblad (well known), and coarse-grained, both
time-independent and time-dependent (new). We also present bounds that provide the range of applicability
of each type of master equation (new). We show that these bounds can be expressed entirely in terms of
only two timescales, namely the fastest system decoherence timescale, and the characteristic timescale of the
decay of the bath correlation function. The reader interested only in the results and not in the details of
derivations can safely read only this section and then skip to the conclusions in Sec. 7. Derivations begin in
Sec. 3, where we present a simple, new derivation of the time-independent CGME. We express the equation
in CP (Lindblad) form, state the range of validity for different approximations made in the derivation, and
thus for the equation itself. We also compare the range of applicability with other master equations, and note
the spatial locality of the Lindblad generators of the CGME. At this point we are ready to address the case
of time-dependent Hamiltonians. The derivation of the equation for this case (which happens to result in
exactly the same form) is given in Sec. 4. In particular, this equation is well suited for the simulation of open
system adiabatic quantum computation, but also for dynamical decoupling, which involves the opposite limit
of very fast system dynamics. We note that while we will sometimes refer to qubits, none of the results we
discuss in this work are limited to qubits, and in fact any finite multi-level system, or interacting set of such
systems, is captured by the formalism. We study some applications and examples in Sec. 5, including error
suppression using a dynamical decoupling protocol, and numerical results of the comparison of our master
2
It should be noted that complete positivity can be relaxed without losing physicality; see, e.g., Ref. [15].
Accepted in Quantum 2020-01-19, click title to verify. Published under CC-BY 4.0. 2
equation with the Redfield and Lindblad-Davies master equations. In the remainder of the paper we derive
the range of validity of the various master equations we study. We give a detailed treatment of the Born
approximation and the other approximations involved, in terms of rigorous bounds presented in Sec. 6, where
we also discuss the generalization to multiple coupling terms and analyze the scaling of the error bounds for
large system sizes. Conclusions are presented in Sec. 7. Various additional technical details are given in the
Appendix.
2 Three master equations
This section provides a summary of our main results. We first present a brief background to define basic
concepts and notation, followed by the definition and properties of the two main timescales we use later to
provide bounds and ranges of applicability. We then summarize the three master equations we focus on in
this work, followed by upper bounds on the distance of their solutions from the true state.
2.1 Background
Consider a system interacting with its environment as described by the total Hamiltonian
H
tot
= H I
b
+ A B + I H
b
. (3)
Here H is the time-independent system Hamiltonian (we treat the time-dependent case in Sec. 4), H
b
is the
bath Hamiltonian, and A and B are, respectively, Hermitian system and bath operators. The more general
situation, with V = A B replaced by
P
i
A
i
B
i
, is easily treated as well (see Sec. 3.7), but we avoid it
here to simplify the notation. We choose A to be dimensionless with kAk = 1 [the operator norm is defined
in Eq. (36)] and B to have energy units, but we deliberately do not set kBk = 1, since we wish to account
for baths (such as harmonic oscillator baths) for which kBk can be infinite. We also set ~ 1 throughout,
so that energy and frequency have identical units.
Let the eigenstates of H be {|ni} , and the corresponding eigenvalues {E
n
}. Equivalently, H =
P
n
E
n
Π
n
,
where Π
n
is a projector onto the eigenspace with eigenvalue E
n
, and Π
m
Π
n
= δ
mn
Π
n
. Note that in the
interaction picture A(t) = U
(t)AU(t) =
P
nm
A
nm
e
iE
mn
t
|nihm| where E
mn
= E
m
E
n
, A
nm
= hn|A |mi,
and U is the solution of
˙
U = iHU. Let
A
ω
=
X
mn:E
mn
=ω
A
nm
|nihm| =
X
mn:E
mn
=ω
Π
n
AΠ
m
= A
ω
, (4)
so that
3
A(t) =
X
ω
A
ω
e
t
. (5)
Here ω runs over all the energy differences (Bohr frequencies) between the eigenvalues of H.
We assume henceforth that the bath is stationary [ρ
b
, H
b
] = 0 where ρ
b
is the bath state and
introduce the bath correlation function C(t) and its Fourier transform γ(ω), known as the bath spectral
density:
C(t) = Tr[ρ
b
B(t)B] = C
(t) =
1
2π
Z
−∞
e
t
γ(ω) (6a)
γ(ω) =
Z
−∞
C(t)e
t
dt = f(ω) + f
(ω) , (6b)
where B(t) = e
iH
b
t
Be
iH
b
t
and
f(ω) =
Z
0
C(t)e
t
dt =
1
2
γ(ω) + iS(ω) , S(ω) =
1
2i
[f(ω) f
(ω)] = S
(ω) . (7)
3
The choice of the sign for this notation, as well as other notation choices, follow the textbook [2], p.133-134.
Accepted in Quantum 2020-01-19, click title to verify. Published under CC-BY 4.0. 3
Note that the real-valued functions S(ω) and γ(ω) are related via a Kramers-Kronig transform
S(ω) =
1
2π
Z
−∞
P
1
ω ω
0
[γ(ω
0
)]
0
, (8)
where P
1
x
[f] = lim
0
R
f(x)
x
dx is the Cauchy principal value.
4
When A B is replaced by
P
i
A
i
B
i
,
we note the positivity of γ
ij
(ω) =
R
dte
t
Tr[ρ
b
B
i
(t)B
j
] as a matrix in the indices i, j (of course this reduces
to γ(ω) 0 in the scalar case). This is a non-trivial fact that is usually associated with Bochner’s theorem,
as e.g., in the textbook [2]. In Appendix A we give two different proofs, one using Bochner’s theorem and
another that is direct. If we assume not only that the bath state is stationary, but that it is also in thermal
equilibrium at inverse temperature β, i.e., ρ
b
= e
βH
b
/Z, Z = Tre
βH
b
, then it follows that the correlation
function satisfies the time-domain Kubo-Martin-Schwinger (KMS) condition [2, 44]:
C(t) = Tr[ρ
b
B(0)B(t + )] . (9)
If in addition the correlation function is analytic in the strip between t = and t = 0, then it follows that
the Fourier transform of the bath correlation function satisfies the frequency domain KMS condition [2, 44]:
γ(ω) = e
βω
γ(ω) ω 0 . (10)
We note that the thermal equilibrium assumption is not necessary for the results we derive in this work, and
we never use it in our proofs. We mention it here for completeness, and also since we use it in one of our
dynamical decoupling examples later on. Finally, note that C(t) has units of frequency squared.
2.2 The bath correlation time τ
B
and the fastest system decoherence timescale τ
SB
We define the two key quantities
1
τ
SB
=
Z
0
|C(t)|dt (11a)
τ
B
=
R
T
0
t|C(t)|dt
R
0
|C(t)|dt
(11b)
Here T is the total evolution time, used as a cutoff which can often be taken as . We show later that in the
interaction picture k ˙ρk
1
4kρk
1
SB
, where ρ is the system density matrix and the trace norm defined in
Eq. (35), so that we can interpret τ
SB
as the fastest system decoherence timescale, or timescale over which ρ
changes due to the coupling to the bath, in the interaction picture (i.e., every other system decay timescale,
including the standard T
1
and T
2
relaxation and dephasing times, must be longer). The quantity τ
B
is the
characteristic timescale of the decay of C(t). We return in Sec. 6 to why we define these timescales in this
manner, but note that Eq. (11b) becomes an identity if we choose |C(t)| e
t/τ
B
and take the limit T .
We further note that τ
SB
and τ
B
are the only two parameters relevant for determining the range of
applicability of the various master equations; in particular, nothing about the norm or time-dependence of
H
tot
affects the accuracy of our time-dependent CGME, given below. In particular, H
tot
can be arbitrarily
large or small in the operator norm (strong coupling, unbounded bath), or have an arbitrarily large derivative
(non-adiabatic regime). This remark is important in light of previous approaches, so we expand on it some
more.
In previous work it was common to extract a dimensionful coupling parameter g, i.e., to replace B by
g
˜
B, where k
˜
Bk = 1. One then defines τ
B
=
R
0
|
˜
C(t)|dt in terms of a dimensionless correlation function
˜
C(t) = C(t)/g
2
[31]. One problem with this approach is the arbitrariness of distributing the numerical
factors between g and
˜
B. Another is that it precludes unbounded baths, such as oscillator baths, for which
kBk = . In contrast, the formalism we present here is applicable even when kBk diverges. Furthermore,
kBk contains an extra scale that does not carry any new information about the range of applicability of the
master equations we discuss and derive. By not introducing this extra scale we highlight that there are only
two free parameters in our analysis of the error: τ
B
and τ
SB
, and no other information about the bath is
4
This follows from the identity
R
0
e
ixτ
= πδ(x) + iP
1
x
.
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needed. I.e., even though different baths will lead to different equations, our results are universal for any
bath with the same values of these two parameters (in fact only their dimensionless ratio matters).
Note that we can relate τ
B
and τ
SB
to the spectral density. First, using γ(ω) > 0 and C(t) = C
(t):
γ(ω) = |γ(ω)|
Z
−∞
|C(t)|dt = 2
Z
0
|C(t)|dt =
2
τ
SB
. (12)
The dephasing time T
2
and relaxation time T
1
for a single qubit are usually defined as 1(0) and 1(ω)
respectively, where ω is the qubit operating frequency (e.g., see Ref. [45]), so we see that τ
SB
. T
1
, T
2
, which
illustrates our earlier comment that τ
SB
is the fastest system decoherence time-scale. Second, γ
0
(ω) =
i
R
−∞
tC(t)e
t
dt, so that
|γ
0
(ω)|
Z
−∞
|t||C(t)|dt = 2
Z
0
t|C(t)|dt
T →∞
= 2
τ
B
τ
SB
, (13)
where the limit is taken in Eq. (11b). And lastly,
(ω)
=
(ω)
|
ω
, while differentiating the KMS condition
yields
(ω)
= βe
βω
γ(ω) + e
βω
(ω)
, so that:
γ
0
(0) =
1
2
βγ(0) . (14)
This implies that γ
0
(0) > 0, so that under the KMS condition (10) we can replace Eq. (13) by
βγ(0) 4
τ
B
τ
SB
. (15)
The upper bounds on the approximation errors of all the master equations we present below involves the
ratio τ
B
SB
[see, e.g., Eqs. (26)-(28)]. The smallness of this ratio is sufficient for the CGME to be useful,
while additional assumptions are required for the Davies-Lindblad equation and some versions of the Redfield
equation. It follows from Eq. (15) that for our bound to be small it is necessary that the temperature β
1
is not too low and/or the spectral density at ω = 0 (roughly the same as the dephasing rate T
1
2
) is not too
large. The simple condition βγ(0) 1 already rules out diverging spectral densities such as the case of 1/f
noise, though this can be ameliorated by introducing a low-frequency cutoff.
We now present the three master equations, in the Schrödinger picture.
2.3 Redfield master equation
For the derivation see Sec. 3.2. This equation was known earlier than the Davies-Lindblad equation [19], and
is often used in quantum chemistry. It is not CP and hence cannot be put in Lindblad form:
˙ρ
R
(t) = i[H, ρ
R
(t)] + (
R
A
f
ρ
R
A
f
A + h.c.) , (16)
where we defined the “filtered" operator:
A
f
=
Z
0
C(t)A(t)dt =
X
ω
A
ω
Z
0
C(t)e
t
dt =
X
ω
f
(ω)A
ω
, (17)
where f(ω) is defined in Eq. (7), and we used the subscript R to denote that the density matrix satisfies the
Redfield equation (we use a similar subscript notation below to distinguish the solutions of different master
equations). There is no natural way to separate the Lamb shift. One of the benefits of the Redfield equation
is that there is only one generator A
f
per interaction with the environment A B. We will show that as long
as it preserves positivity, the Redfield equation is more accurate than the Davies-Lindblad master equation.
2.4 Davies-Lindblad master equation
For the derivation see Sec. 3.3. This is the familiar result [4]
˙ρ
D
(t) = i[H + H
LS
, ρ
D
] +
X
ω
γ(ω)(A
ω
ρ
D
A
ω
1
2
{ρ
D
, A
ω
A
ω
}) , (18a)
H
LS
=
X
ω
S(ω)A
ω
A
ω
, S(ω) =
1
2i
Z
−∞
sgn(t)C(t)e
t
dt , (18b)
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where H
LS
is the Lamb shift. We note that ad hoc forms of the Lindblad equation are often written down and
used without justification, e.g., with just one Lindblad term per qubit (e.g., σ
or I σ
z
[46, 47]). In reality
the Davies generators are derived from first principles, i.e., from the description of the total Hamiltonian of
the system and bath. The number of Davies generators is unfortunately exponential in the number of qubits
n:
P
ω
is over all 4
n
energy differences.
2.5 Coarse-grained master equation (CGME) for time-independent or time-dependent system Hamilto-
nians
This is our main new set of results, generalizing Ref. [40] to the time-dependent case.
2.5.1 The time-independent case
For the derivation see Sec. 3.4. First, considering time-independent system Hamiltonians, we obtain:
˙ρ
C
(t) = i[H + H
LS
, ρ
C
(t)] +
Z
−∞
d
A
ρ
C
(t)A
1
2
n
ρ
C
(t), A
A
o
, (19)
where the Lindblad operators are
A
=
s
γ()
2πT
a
Z
T
a
/2
T
a
/2
e
it
A(t)dt (20a)
=
X
ω
f(, ω)A
ω
, f(, ω) =
s
γ()T
a
2π
sinc[T
a
( ω)/2] , (20b)
with sinc(x) sin(x)/x, and where T
a
is the averaging, or coarse-graining time, discussed below. The
continuous family of terms labeled by is an unusual form of the Lindblad equation, but is manifestly CP.
The more standard form in terms of a discrete sum over transition frequencies is equivalent to this one, and
is given in Eq. (71) below. The range of applicability of the CGME is only slightly smaller than that of
the Redfield ME, as we discuss in Sec. 3.5. Much like for Davies generators, there are exponentially many
frequency differences, but unlike the Davies case the discretization of the continuous integral
R
d makes it
possible to keep the number of generators constant in the system size for each A B term. These results are
presented in Sec. 3.6.
The same applies for the Lamb shift, which is
H
LS
=
X
ωω
0
F
ωω
0
A
ω
0
A
ω
(21a)
F
ωω
0
=
1
2T
a
ω
+
Re
Z
T
a
0
e
i(ωθT
a
ω
+
)
e
i(ω
0
θ