Markovian to non-Markovian phase transition in the operator dynamics of a
mobile impurity

D. Gribben1, J. Marino1, S. P. Kelly1,2*

1 Institute for Physics, Johannes Gutenberg University of Mainz, D-55099
Mainz, Germany

2 Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and
Astronomy, University of California at Los Angeles, Los Angeles, CA 90095

*<EMAIL_ADDRESS>
## Abstract

We study a random unitary circuit model of an impurity moving through a
chaotic medium. By varying the velocity of the impurity, $v_{d}$, relative to
the speed of information propagation within the medium, $v_{B}$, we control
the exchange of information between the medium and impurity. Above supersonic
velocities, $v_{d}>v_{B}$, information cannot flow back to the impurity after
it has moved into the medium, and the resulting dynamics are Markovian. Below
supersonic velocities, $v_{d}<v_{B}$, the dynamics of the impurity and medium
are non-Markovian, and information is able to flow back onto the impurity. We
show the two regimes are separated by a continuous phase transition with
exponents directly related to the diffusive spreading of operators in the
medium. This is demonstrated by monitoring an out-of-time-order correlator
(OTOC) in a scenario where the impurity is substituted at an intermediate
time. During the Markovian phase, information from the medium cannot transfer
onto the replaced impurity, manifesting in no significant operator
development. Conversely, in the non-Markovian phase, we observe that operators
acquire support on the newly introduced impurity. We also characterize the
dynamics using the coherent information and provide two decoders which can
efficiently probe the transition between Markovian and non-Markovian
information flow. Our work demonstrates that Markovian and non-Markovian
dynamics can be separated by a phase transition, and we propose an efficient
protocol for observing this transition.

###### Contents

  1. 1 Introduction
  2. 2 Model
  3. 3 Capturing operator dynamics and backflow
    1. 3.1 Operator backflow
  4. 4 Operator dynamics
    1. 4.1 One-dimensional medium
    2. 4.2 Two-dimensional medium
  5. 5 Information transition
    1. 5.1 Echo protocol
      1. 5.1.1 Echo decoder
    2. 5.2 Teleportation protocol
      1. 5.2.1 Teleportation decoder
  6. 6 Conclusion
  7. A Alternate 1D circuit
  8. B Derivation of Eq. (10)
  9. C Details of HP fidelity and derivation of Eq. (14)

##  1 Introduction

The scrambling of quantum information is a concept crucial to a number of
fields [1, 2, 3, 4, 5, 6, 7, 8, 9]. Its study helps quantify the complexity of
quantum processes and can be related to the local recoverability of
information [1, 10]. Quantum experiments carried out in reality are
inextricable from their environments which often hinder information tasks when
quantum information in the system is irretrievably lost to the environment
[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. However, there is always a
finite probability that this information can flow back into the system and for
certain physical processes this backflow is crucial. For example, in closed
system thermalization the bulk of the system acts as an environment to, and
becomes entangled with, local subsystems [22, 23, 24]. Furthermore, this
information back flow, or non-Markovanity, may be a usefully resource for
quantum information processing [25, 26].

Recently, phase transitions in information dynamics have been demonstrated in
a variety of settings ranging from PT-symmetric non-Hermitian systems [27, 28,
29, 30, 31, 32, 33] to systems monitored by projective measurements [34, 35,
36, 37, 38, 39, 40, 41, 42, 43], to more general open systems involving qubits
arranged in a variety of space-time geometries [44, 45, 46, 47, 48, 49, 50,
51, 52, 31, 53, 21, 54, 55, 56, 57, 58, 59, 14]. It is therefore natural to
wonder if open system dynamics undergo a phase transition in information flow
as they are tuned between Markovian and non-Markovian limits. In this light,
the authors of Refs. [60, 61] investigated non-Markovian effects of monitored
systems with a non-Markovian environment but did not find a transition in non-
Markovanity. In another context, Refs. [62, 63] found the non-Markovianity of
an impurity is sensitive to a topological phase transition occurring within
the environment. However, it is not clear if the two phases are distinguished
by Markovian and non-Markovian dynamics. Furthermore, the environment in both
systems are non-scrambling and Gaussian which is not characteristic of generic
quantum systems.

Thus, in this work, we investigate the possibility of a phase transition in
non-Markovianity of an impurity coupled to a quantum chaotic environment. We
present a model in which the system can be taken between regions of zero and
non-zero information backflow by only varying the form of the coupling between
the system and environment. Distinct from previous results [62, 63, 60], we
find that these regions are separated by a continuous phase transition, and
identify a critical point characterized by scale-free temporal correlations.

Figure 1:  Cartoon of the model we consider: an impurity (blue circle) moving
through a medium. The circles within the medium correspond to the unitary
brickwork which drives the evolution of the medium. Initially, the medium
contains no information about the impurity (clear circles). Then at a random
time, the impurity interacts with the medium and leaks information into it at
that particular point in spacetime. The information is rapidly scrambled away
from this point (green circles) at a characteristic velocity. In this work, we
investigate the behaviour for subsonic (left) and supersonic (right) impurity
velocities.

The model we consider describes an impurity moving through a chaotic medium.
By varying the velocity of the impurity relative to the information scrambling
velocity within the medium we can tune the capacity at which information can
leave the impurity and re-enter at a later time. We simulate the chaotic
dynamics of the model using random unitary circuits and capture the flow of
information by computing the out-of-time-order correlator (OTOC) [64, 65, 66].
The strength of the OTOC on the impurity at late times results from a
combination of information retained by the impurity over time and information
backflow from the medium. To isolate the contribution from backflow we enact a
protocol in which we remove the impurity at an intermediate time and replace
it with a fresh impurity. Any non-trivial operator support on the fresh
impurity must be a result of feedback from the medium. Beginning with a single
qubit impurity moving through a 1D medium, we find that on varying the
velocity of the impurity a phase transition occurs between zero and non-zero
backflow. Using a mapping between operator growth and a random walk [65, 66],
we show that the criticality is a result of the diffusive growth of the
operator within the medium.

We then extend our model to that describing a 1D impurity moving through a 2D
medium to study the effect of the backflow transition on scrambling within the
impurity. In particular we focus on how the presence of backflow affects a
distinct phase transition known to occur in the Markovian limit of this
extended model [14]. This transition is between phases of persistent and
vanishing scrambling within the impurity and we shall refer to it here as the
_scrambling transition_. We find that the scrambling transition is only
possible when there is no operator backflow.

As in the scrambling transition, the backflow phase transition can also be
observed in a particular channel capacity. The relevant channel in this case
captures the information about the initial state contained within the medium
and discarded impurity at late times. We show that a party with access to only
these degrees of freedom can recover the initial impurity state with perfect
fidelity when the channel capacity is maximal. This is achieved via a protocol
where a fresh impurity is coupled to the medium and the dynamics are reversed.
Additionally, we show that the backflow transition can be observed in a
complementary channel. This channel characterizes the ability to decode
information on the initial impurity state given access to its final state
along with the initial states of the medium and the fresh impurity inserted in
the reset step. The decoder for the complementary channel is reminiscent of
that proposed in [10], for the Hayden-Preskill protocol. We associate these
two quantum channels with distinct decoding protocols and in each case derive
relations between the decoder fidelity and the channel capacity. In one case
the decoder fidelity is optimal in the limit of zero backflow whereas for the
other decoding it is enhanced in the opposite limit where non-Markovianity is
maximal.

The paper is organized as follows. In Section 2 we introduce the model and
give the details of the circuit implementation for both the 1D and 2D cases.
We continue, in Section 3, to detail how we capture the operator dynamics and
in particular the protocol we implement to capture operator backflow. First,
we consider a single qubit impurity coupled to a 1D medium, and discuss the
OTOC dynamics of this model in Section 4.1. Here we compute the degree of
information backflow as a function of velocity and determine finite-time
scaling exponents. In Section 4.2 we take a 1D impurity moving through a 2D
medium and investigate the impact of the backflow transition on scrambling
within the impurity. Finally, in Section 5, we investigate how the backflow
transition is manifest in a pair of complementary channel capacities. We
detail decoding protocols whose fidelity is directly related to these channel
capacities and can be used to observe the transition in small, near term
quantum computers.

##  2 Model

Below we construct a random circuit model aimed at capturing the information
dynamics of an impurity moving through a chaotic medium. This could describe,
for instance, a mobile impurity immersed in a Bose gas, or equivalently a
static impurity in a flowing gas [67, 68, 69, 70]. In such a system, the
general expectation is that information transferred from impurity to medium is
rapidly scrambled away from the location of the interaction in a
characteristic lightcone [65, 66]. This scrambling lightcone captures the
velocity at which information is scrambled through the medium. By tuning the
impurity velocity relative to the speed of information scrambling within the
medium, one may expect the information dynamics to undergo a transition. When
the impurity moves slower than the information scrambling in the medium, i.e.
at _subsonic_ velocities, there is potential for information to flow back into
the impurity and hence the reduced dynamics of the impurity can be non-
Markovian. On the other hand, if we increase the velocity to be _supersonic_ ,
such that the trajectory falls outside the light cone, information backflow
becomes impossible and the medium acts as a Markovian environment for the
impurity. The general model and these two cases are depicted in Figure 1.

To capture the information dynamics, we will consider the full unitary
evolution of the total system of impurity and medium. By partitioning a closed
system into subsystems labelled system and environment, we are then able to
apply open systems concepts such as Markovianity. In this paper we always
consider Markovianity of the impurity as a system with the medium as its
environment. The main result is that the change from non-Markovian to
Markovian operator dynamics on the impurity corresponds to a continuous phase
transition.

Figure 2:  Circuit diagram of the model implementation. Here the medium qubits
are all initialized in an infinite temperature state, $\rho_{\infty}$, and the
impurity is initially in state $\rho_{I}(0)$. In this realization the impurity
initially drifts two sites away from its origin and then interacts with the
medium, after a further drift of one site it once again swaps qubits with the
medium. The green gates indicate the lightcone associated with the information
transferred into the medium by the first swap. The second swap still within
this scrambling lightcone and thus could lead to backflow of information from
medium to impurity.

We first model this dynamics by considering a single impurity qubit moving
over a 1D chain of qubits describing the medium. The chaotic evolution of the
medium is captured by a random circuit and the impurity-medium interaction
consists of swap gates applied at a fixed rate, $p$. The circuit
implementation of this evolution is depicted in Figure 2. The medium is
initialized in an infinite temperature state and evolves under a brickwork of
random unitaries drawn from the Clifford group. The Clifford group is a
unitary 3-design meaning that it exactly reproduces the first, second and
third order moments of the Haar distribution [71]. Below, we consider the
dynamics of OTOCs, which are second order functions of the brickwork
unitaries, and hence are equivalent to those that would be generated by Haar
random circuits.

Between every two layers of unitaries, the impurity qubit is swapped with the
medium qubit at position $x\in\mathbb{Z}$ with probability $p$. Initially we
set $x=0$ but after each interaction step we shift $x\to x+d$ where the drift,
$d\in\mathbb{Z}$, is drawn from the following binomial distribution:

$p(d)=\binom{d_{max}}{d}p_{D}^{d}(1-p_{D})^{d_{max}-d}.$ (1)

The probability of a single shift, $p_{D}$, is set to give an average velocity
$p_{D}d_{\text{max}}=v_{d}\tau=v_{d}$ where $\tau$ is the time between shifts,
and we work in units of $\tau\equiv 1$. For all results in this paper we fix
$d_{\text{max}}=20$. We apply two layers of unitaries before shifting in order
to allow for more scrambling to occur in the medium relative to the impurity
velocity i.e. to broaden the lightcone. This broadening is stochastic but we
can associate a butterfly velocity, $v_{B}$, with the typical rate at which it
occurs. The average increase in the number of sites on which an operator has
support after a time $t$ is then given by $2v_{B}t$ [66]. By tuning $v_{d}$
relative to $v_{B}$ we can control the degree of feedback from the medium. The
exact Markovian limit, where the probability of feedback is zero, corresponds
to $v_{d}>c$ where $c$ is the maximum velocity of information in the medium;
in our case the two layers of unitaries are applied per time step such that
$c=2$. In this limit the medium is effectively reset with respect to the
dynamics of the impurity and there is no possibility of feedback. In the
Appendix A, we consider an alternate circuit where the scrambling within the
medium only occurs up to the position of the impurity.

Figure 3:  Illustration of the model implementation for a 1D impurity moving
through a 2D medium. The left diagram shows the circuit which evolves the
impurity through time with swaps occurring between each site and its
independent strip of the medium. The right-hand cartoon shows how the impurity
moves through the medium. Each green strip of the medium evolves independently
via the circuit displayed in Figure 2.

Motivated by Ref. [14] we also consider a chain of $N$ impurity qubits moving
through a 2D lattice of $N\times M$ qubits. The 2D medium evolves as $N$
decoupled chains of $M$ qubits evolving under the brickwork circuit in Figure
2. The 1D impurity chain takes steps through this medium in the $x$-direction
with the step length being drawn from the distribution in Eq. (1). We allow
for interactions to occur between the individual impurity sites via a random
unitary brickwork equivalent to that occurring in the medium. As the chain
drifts and scrambles each qubit within is swapped independently into the
medium with probability $p$.

The 2D model is depicted in Figure 3; another perspective is gained by
considering Figure 2 as a cross-section. Here we must account for the
scrambling in the medium, the scrambling on the impurity, and the interaction
between them. However, this interaction vanishes when taking the limit of
Markovian impurity dynamics $v_{d}>c$, and this circuit is equivalent to that
studied in Ref. [14]. There, the medium was a stationary reservoir of infinite
temperature qubits. In the extended model, on taking $v_{d}>c$ the impurity
drifts beyond the scrambling lightcone and any non-trivial dynamics occurring
within the medium; the impurity only ever interacts with infinite temperature
qubits. In the Markovian model, the swaps occurring between the impurity and
the medium act solely as a loss process on the impurity. This loss acts to
suppress scrambling of the operator within the impurity, with the butterfly
velocity of the scrambling lightcone decreasing as $p$ is increased. On
continued increase of $p$ through a critical swap rate, $p_{c}$, the impurity
undergoes a phase transition from a steady state with non-zero OTOC within the
impurity to one with the entire operator having flowed into the environment.
This scrambling transition was identified to be part of the directed
percolation universality class [14].

Outside of the Markovian limit, $v_{d}<c$, we are again in a regime where
operator backflow is possible. This backflow occurs locally where, in between
layers of impurity scrambling, each site along with its independent medium
evolve one step according to the 1D setup of the previous section. The $N$
media are treated as independent but evolve under equivalent parameters i.e.
$p$ and $v_{d}$ are global properties. Although treating the environments as
independent is artificial it presents a simple limit in which we can explore
the effect of backflow on the scrambling transition within the impurity.

##  3 Capturing operator dynamics and backflow

In this section we start with the 1D problem. We initialize an operator
localized on the impurity, after some time on the order of $p^{-1}$ it is
swapped into to the medium and then scrambled. To capture how the operator is
scrambled within the medium we compute the out-of-time-order correlator (OTOC)
defined by

$C^{(M)}(x,t)=\frac{1}{4}\mathrm{tr}\left\\{\rho_{0}^{IM}[X^{(I)}(t),Y^{(M)}_{x}]^{\dagger}[X^{(I)}(t),Y^{(M)}_{x}]\right\\},$
(2)

where $X^{(I)}$ is the Pauli-X operator acting on the impurity and
$X^{(I)}(t)=U^{\dagger}_{t}X^{(I)}U_{t}$ is this operator evolved to time $t$
in the Heisenberg picture. $Y^{(M)}_{x}$ is the Pauli-Y operator acting on the
medium qubit at site $x$. OTOCs quantify where in Hilbert space a time-evolved
operator has support. In addition to the operator weight in the medium, we
capture the operator weight remaining on the impurity with the OTOC given by

$C^{(I)}(t)=\frac{1}{4}\mathrm{tr}\left\\{\rho_{0}^{IM}[X^{(I)}(t),Y^{(I)}]^{\dagger}[X^{(I)}(t),Y^{(I)}]\right\\},$
(3)

where $Y^{(I)}$ is the Pauli-Y operator acting on the impurity. The choice of
Pauli operators in the above expressions is arbitrary in our case; on
averaging over Clifford circuit realizations, any two distinct non-identity
Pauli operators for the initial and time-evolved operators would yield
identical results.

If $C^{(I)}(t)$ vanishes at some time $t$ then the entire information content
of the initial operator has flowed into the environment. In the case of
Markovian environments this information is irretrievable unless one can access
the environment. However, for a generic environment, information can flow back
into the system at later times leading to non-Markovian revivals of the OTOC
on the system. But this non-Markovianity is distinct from that typically
considered in the field of open quantum systems which concerns the evolution
and/or correlations of a state rather than an operator [72]. We will now
outline a protocol in which we can measure the degree of this backflow.

###  3.1 Operator backflow

Figure 4: Diagram of the protocol used to measure the operator backflow
quantified by
$B=\frac{1}{4}\mathrm{tr}\left\\{\rho_{0}^{IM}[X^{(I)}(t),Y^{(I)}]^{\dagger}[X^{(I)}(t),Y^{(I)}]\right\\}$.
The unitaries, $U_{1}$ and $U_{2}$, are given by the circuit in Figure 2.
Alice, Bob and Charlie refer to the parties in the information games outlined
in Section 5.

We capture operator backflow with a protocol analogous to the causal break of
the process tensor formalism [73, 74, 75, 76, 77, 78, 79]; this protocol is
depicted in Figure 4. We first evolve the model for a time $t_{1}$, then reset
the impurity by swapping the qubit with one in an infinite temperature state.
This fresh impurity is then evolved for a further time $t_{2}$ to give a total
evolution time of $T\equiv t_{1}+t_{2}$. After these steps we again track the
operator weight on the impurity via an OTOC, $B(v_{d},T)$, whose definition is
equivalent to that of $C^{(I)}(T)$, but with the operators evolved under the
backflow. We enforce that $t_{1},t_{2}\gg p^{-1}$ such that the impurity and
medium have significant time to interact before the reset step and subsequent
OTOC readout.

In the case of random unitary evolution the environment strongly scrambles any
information that flows into it and the system evolution is Markovian as
characterized by a process tensor which factorizes between timesteps. However,
it has recently been shown that although the state evolution may be Markovian
or near-Markovian, the OTOC can still display non-Markovian correlations [80].
These correlations arise due to the Heisenberg evolution of the operator; in
our model this causes the support of the operator to first spread onto the
medium and then, after the reset step, onto the fresh impurity. Although we
treat the full system unitarily, by considering the impurity sector as the
system and the medium as the environment, we can consistently refer to the
Markovianity of the reduced operator dynamics on the impurity.

##  4 Operator dynamics

In this section, we present the OTOC dynamics for both the 1D and 2D models
introduced in Section 2. First, we investigate the dynamics of the backflow
protocol for the 1D system and perform finite-time scaling on the late-time
dynamics. For the 2D system, we consider the steady state of the OTOC without
applying the reset step and investigate the effect of operator backflow on the
scrambling of operators within the 1D impurity.

###  4.1 One-dimensional medium

We shall focus exclusively on $B(v_{d},T)$, the weight of the OTOC on the
impurity at time $T$ during the reset protocol depicted in Figure 4. In Figure
5 we plot $B(v_{d},T)$ at a late time as a function of the drift velocity
$v_{d}$. A transition occurs between at a critical velocity of $v^{*}\approx
1.2$, in the vicinity of the long-time butterfly velocity $v_{B}=1.2$. In
Figure 6 we show that the backflow satisfies a finite-time scaling of the
following form:

$B(v_{d},T)=f((v_{d}-v^{*})\sqrt{T}),$ (4)

where $f$ is a universal scaling function.

The origin of this exponent can be understood by analyzing the operator
dynamics within the medium. Given that $t_{2}$ is much larger than the inverse
swap rate $1/p$, we expect that, at $t_{2}$, the impurity has equilibrated
with the local medium and will follow the dynamics of the nearest medium
qubit. We are therefore interested in the weight of the OTOC in the medium at
the position of the impurity; after disorder averaging that position is
$\overline{x_{I}}=v_{d}t$.

$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$$0$$0.2$$0.4$$0.6$$0.8$$v_{d}$$B$
Figure 5: Backflow order parameter on variation of impurity drift velocity.
Here we set $t_{1}=100$, $t_{2}=1000$ and $p=0.1$. This is the result of an
average over $10^{4}$ disorder realizations.

To determine the OTOC at this position, we use the approach detailed in Ref.
[66]. In that approach, the average OTOC is considered by choosing the unitary
bricks from the Clifford group and using the fact that the Clifford group is a
3-design [71]. This ensures the average over the Clifford group is the same as
the average over the Haar group for the OTOC. Clifford circuits map Pauli
strings to Pauli strings, and since we consider an initial Pauli $X$ operator,
this operator remains a Pauli string for all time. From the definition of the
OTOC we have $B(v_{d},t)=1$ if the operator $X^{(I)}(t)$ has character of $X$
or $Z$ on the impurity and otherwise equals zero. On disorder averaging if the
operator has non-trivial (i.e. non-identity) support on a particular site then
it has equal probability of having $X$, $Y$ or $Z$ character. Therefore we
have that $\overline{B(v_{d},t)}=\frac{2}{3}\overline{n_{I}(v_{d},t)}$, where
$n_{I}(v_{d},t)$ is a density that is equal to one if $X^{(I)}(t)$ has non-
trivial support on the impurity, and zero otherwise. We can equivalently
express the disorder-averaged OTOC within the medium at position $x$ as
$\overline{B^{(M)}(x,t)}=\frac{2}{3}\overline{n_{M}(x,t)}$, where $B^{(M)}$ is
the OTOC within the medium during the backflow protocol. The late-time
dynamics of $\overline{n_{M}(x,t)}$ is characterized by regions of zero and
non-zero density separated by diffusively propagating boundaries. The dynamics
of these boundaries can be well captured by a hydrodynamic description where
the probability distribution of the boundary position evolves according to a
Fokker-Planck equation [66]. The OTOC weight at the position of the impurity
is given by

$P(v_{d},t)=\overline{n_{M}(\overline{x_{I}},t)}=\frac{1}{2}\left[1-\mathrm{erf}\left(\frac{v\sqrt{t}}{2\sqrt{D}}\right)\right],$
(5)

where $v=v_{d}-v_{B}$ is the difference between the impurity velocity, $v_{d}$
and operator butterfly velocity $v_{B}$, and $D$ is a constant. This satisfies
the diffusion-like scaling relation:

$P(v,t)=F(v\sqrt{t}).$ (6)

The backflow transition satisfies precisely the same scaling relation as that
followed by the OTOC weight within the medium at the position of the impurity.
This occurs because at late times the impurity matches the free dynamics of
the OTOC within the medium without affecting it, the diffusive nature of this
dynamics gives us the numerically observed scaling relations shown in Fig. 6.
Moreover, the dynamics described by Eq. (5) are scale-free when $v=0$. Away
from this point the dynamics instead follow an error function which at long
times corresponds to exponential relaxation on a timescale given by
$\frac{v^{2}}{4D}$.

$1$$1.1$$1.2$$1.3$$1.4$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$v_{d}$$B$$-6$$-4$$-2$$0$$2$$4$$6$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$(v_{d}-v^{*})\sqrt{T}$$B$$t_{2}$1252505001000
Figure 6: Left: Expanded view of the backflow phase transition near the
critical point. Right: Data collapse under a finite-time rescaling. Here we
set $t_{1}=100$ and $p=0.1$. This is the result of an average over $10^{4}$
disorder realizations.

###  4.2 Two-dimensional medium

For the 2D problem the quantity of interest is the density of the OTOC over
the entire impurity. We first extend the definition of the OTOC in Eq. 3 to
have spatial dependence:

$C^{(I)}(y,t)=\frac{1}{4}\mathrm{tr}\left\\{\rho_{0}^{IM}[X^{(I)}_{0}(t),Y^{(I)}_{y}]^{\dagger}[X^{(I)}_{0}(t),Y^{(I)}_{y}]\right\\},$
(7)

where $y$ has been introduced to index the position on the impurity. From this
we can analogously define the, now site-dependent, density:
$\overline{C^{(I)}(y,t)}=\frac{2}{3}\overline{n_{I}(y,t)}$. The site-
dependence of the operator weight on the impurity is irrelevant to the
question of backflow we study here. Instead, the quantity we compute is
$\overline{n_{I}(y,t)}$ integrated over $y$ i.e.

$\overline{N_{I}(t)}=\frac{1}{N}\sum_{y}\overline{n_{I}(y,t)},$ (8)

where $N$ is the number of qubits within the impurity. This is the order
parameter of the phase transition in scrambling observed in the Markovian
limit of this model [14]; the transition is characterised by the steady state
value of $\overline{N_{I}}$ vanishing as the swap rate, $p$, is increased
through a critical value, $p_{c}$. This was shown to belong to the directed
percolation universality class, where bond formation is promoted by the
unitary gates and suppressed by the swap operations [14]. In the language of
directed percolation, operator backflow can be associated with the formation
of long-range bonds in the temporal direction. We now consider the effect of
these long-range bonds on the percolation within the impurity.

AbsorbingPercolating$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$0$$0.2$$0.4$$0.6$$0.8$$1$$v_{d}$$p$$p_{c}$$v^{*}$
Figure 7:  Phase diagram for the two-dimensional model as a function of swap
rate, $p$, and impurity drift velocity, $v_{d}$. The absorbing phase
corresponds to a steady state with $\overline{N_{I}}=0$ the the percolating
phase when $\overline{N_{I}}>0$ in the steady state. The vertical line in the
plot indicates the position of the critical velocity, $v^{*}$, of the backflow
transition in the one-dimensional model. The horizontal line indicates the
critical swap rate, $p_{c}$, of the scrambling phase transition in the
Markovian model as reported in Ref. [14].

Figure 7 shows the phase diagram we observe on varying the $p$ as we take
$v_{d}$ across the backflow transition. There are four relevant regimes:
subsonic velocities $v_{d}<v^{*}$, supersonic yet subluminal velocities
$v^{*}<v_{d}\leq c$, superluminal velocities $v_{d}>c$, and a critical
velocity $v_{d}=v^{*}$. In the subsonic regime, $v_{d}<v^{*}$, the operator
flow is in non-Markovian phase of the 1D model meaning operators flow back
into the system and operator support on the impurity is always maintained;
this region is purely scrambling. Conversely, in the superluminal regime,
$v_{d}>c$, the impurity is beyond the deterministic lightcone of the medium
and backflow is impossible. Here the Markovian results are reproduced exactly
and the same two phases appear: a percolating phase where the steady state
value of $\overline{N_{I}(t)}$ is greater than zero and an absorbing phase
where it vanishes. These phases survive when $v^{*}<v_{d}<c$ but the critical
swap rate, $p_{c}$, is shifted. The long-range bonds formed in this region are
suppressed exponentially in time and their only affect is to renormalize the
probability of the short-range bond formation. This renormalization diverges
as we approach the critical velocity, $v_{d}=v^{*}$, and the probability of
long-range bond formation becomes constant. This occurs because, at this
velocity, the impurity is perfectly following the centre of the diffuse
lightcone boundary within the medium such that from its reference frame the
local operator weight is constant in time. Unless we combat this enhancement
by breaking bonds at a comparable, unphysical, rate then the impurity will
always percolate.

##  5 Information transition

We have shown that an impurity moving in a 1D medium can exhibit a phase
transition in operator Markovianity by studying the OTOC dynamics of the
system with the impurity reset operation shown in Fig. 4. Operator dynamics
has been linked to the spread of information [1, 14], and we will now explore
how the phase transition in operator non-Markovianity manifests in the
dynamics of quantum information. Specifically we study the coherent
information transmitted through two complementary channels. These channels are
those highlighted in Figure 4 as Yellow$\to$ Blue for the echo protocol and
Yellow$\to$ Green for the Teleportation protocol. The results in this section
were numerically simulated using the QuantumClifford.jl software package [81].

###  5.1 Echo protocol

In the thought experiment, Alice prepares the impurity qubit, $I$, in a
particular state, $\rho_{I}$, and then allows it to move through the chaotic
medium, $M$, evolving under the circuit described in Section 2. At an
intermediate time she discards her qubit and replaces it with a maximally
mixed qubit, $R$. At the end of the experiment, a second party, Bob, obtains
access to both the discarded qubit, $D$, and the medium $M$. For clarity, we
emphasise that Bob cannot access the impurity and has no knowledge of the
input states.

We now show that the transition in operator backflow can also be observed in
the information Bob has about the state, $\rho_{I}$, that Alice prepared in
the impurity qubit at the beginning of the experiment. We quantify this
information using the coherent information [82]. The coherent information is
most easily calculated by introducing an ancilla qubit, $A$, that is maximally
entangled with the initial state of Alice’s qubit (the impurity).
Conceptually, this ancilla acts a memory of the initial state of the impurity,
and allows us to track how much of that information remains on the impurity as
it interacts with the medium. Note that this protocol is complementary to the
one in which Alice prepares a specific state $\rho_{I}$ on the impurity. The
coherent information accessible by Bob is then given [82] as

$I(A\to\text{Bob})=S(\rho_{\text{Bob}})-S(\rho_{\text{Bob}\cup A}),$ (9)

where $S(\rho_{C})=-\mathrm{Tr}[\rho_{C}\log_{2}\rho_{C}]$ is the von-Neumann
entanglement entropy of the reduced density matrix of subspace $C$. The
subspace labelled “Bob” contains the qubits that Bob can access, namely the
qubit discarded by Alice and the final state of the medium: $\text{Bob}\equiv
D\cup M$. For a circuit which allows Bob to recover Alice’s state $\rho_{I}$
perfectly, his qubits will end up maximally entangled with the ancilla $A$
such that $I(A\to\text{Bob})=1$. While if Bob ends up with no information
about Alice’s state, his qubits and the ancilla will decouple, and
$I(A\to\text{Bob})\leq 0$.

$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$$-0.5$$0$$0.5$$1$$v_{d}$$I(A\to\text{Bob})$
Figure 8: Manifestation of backflow phase transition in coherent information.
Here we set $t_{1}=100$, $t_{2}=1000$ and $p=0.1$. This is the result of an
average over 400 disorder realizations.

During the first stage of the protocol, the initial state is entirely encoded
in the impurity and the medium subspaces. Immediately after this impurity is
discarded, prior to the occurrence of any swaps, this is still true and Bob
has perfect access to Alice’s initial qubit such that $I(A\to\text{Bob})=1$.
As long as this remains true, Bob will always have complete information on the
initial state. This is certainly true if the dynamics are Markovian. Here
there is no operator back flow, the information remains in the medium and
again Bob has perfect access to the information in Alice’s initial qubit. Once
the operator dynamics become non-Markovian, however, some information flows
back into Alice’s impurity and Bob loses the ability to decode the initial
state of Alice’s qubits. This is shown in Figure 8 where we plot
$I(A\to\text{Bob})$ as a function of $v_{d}$. There we see a transition
occurring at the same critical velocity as the backflow transition observed by
the OTOC. We again find the diffusive scaling exponents as shown in Figure 9,
with the relevant scaling quantity being $1-I(A\to\text{Bob})$. This quantity
is zero only in the region of zero backflow and increases as the velocity is
decreased into the phase of non-zero backflow i.e. it acts as a witness of
operator non-Markovianity.

####  5.1.1 Echo decoder

The information accessible to Bob is maximal in the Markovian limit as all of
the information about the Alice’s qubit has flowed to Bob. We will now
describe a decoder which yields a perfect fidelity in this limit and can be
used to observe the transition. The decoder is depicted in Figure 10 as the
green unitary operations. The construction is similar to the decoder in Ref.
[14]; here Bob inserts a new impurity qubit, $I^{\prime}$, prepared in an
infinite temperature state and carries out the time-reversal of the evolution
up to that point. The additional reset step in reverse consists of tracing out
the impurity and replacing it with Alice’s discarded impurity, $D$. The
fidelity of this decoder is then given by the probability of a Bell pair
forming between the final state of $D$ and $A$. Perfect fidelity is achieved
when $\mathrm{Tr}_{R}[U_{2}\rho_{R}]$, and by extension
$\mathrm{Tr}_{I}[U_{2}^{\dagger}\rho_{I}]$, is unitary, which implies no
information is transferred from $M$ to $R$. When this occurs, such as in the
Markovian limit, $U_{1}^{\dagger}$ is trivially able to cancel the effect of
$U_{1}$ and the initial state is recovered.

$1$$1.1$$1.2$$1.3$$1.4$$10^{-2}$$10^{-1}$$10^{0}$$v_{d}$$1-I(A\rightarrow\text{Bob})$$-6$$-4$$-2$$0$$2$$4$$10^{-2}$$10^{-1}$$10^{0}$$(v_{d}-v^{*})\sqrt{T}$$1-I(A\rightarrow\text{Bob})$$t_{2}$1252505001000
Figure 9: Left: Information phase transition around critical point. Right:
Data collapse under a finite-time rescaling. Here we set $t_{1}=100$ and
$p=0.1$. This is the result of an average over 400 disorder realizations.
Figure 10:  Diagram of the decoding protocol. When Alice reliquishes control
of the impurity Bob performs a reversal of the dynamics up to that point
except he only has access to the medium, $M$, and the impurity discarded by
Alice in the refresh step, $D$.

There is a connection between the coherent information and the decoder
fidelity for a given circuit realization given by

$I(A\to\text{Bob})=1+\log_{2}\mathcal{F},$ (10)

we derive this in Appendix B. In the supersonic phase the coherent information
approaches unity and the decoder achieves perfect fidelity, while in the
subsonic phase some information of the initial state flows back onto $R$
meaning the decoder can no longer perfectly recover the initial state. We will
now outline how a different protocol witnesses the same transition.

###  5.2 Teleportation protocol

Here we make use of a fundamental property of bipartite entropies to derive a
distinct protocol which exhibits an equivalent transition in coherent
information but for a channel complementary to the previous protocol. In this
new protocol Alice carries out an equivalent role i.e. she prepares the
impurity qubit, $I$, in a specific state, allows it to interact with the
medium, $M$, and then replaces it with a new qubit, $R$, in an infinite
temperature state. Throughout this the total evolution is unitary, as such any
information which Bob cannot access from the discarded qubit, $D$, or the
final state of $M$ must be accessible elsewhere in the system. With this in
mind we now introduce a third party, Charlie, who is also attempting to deduce
information about the initial state of the impurity. Charlie has qubits that
are maximally entangled with the initial states of $M$ and $R$, $A_{M}$ and
$A_{R}$ respectively, as well as having access to the final state of $R$.
These new ancilla, $A_{M}$ and $A_{R}$, play a similar role to $A$ in that
they perfectly remember the initial state of the medium and the new qubit.
Charlie having access to these implies he can make use of this information to
help decode Alice’s information. However, unlike Bob, Charlie has no knowledge
of the discarded qubit, $D$, or the final state of $M$. The degrees of freedom
Charlie can access are complementary to those accessible by Bob. This allows
us to greatly simplify the calculation of the coherent information available
to Charlie.

For pure states, like the one describing the quantum correlations between
Alice, Charlie and Bob, all bipartite entanglement entropies satisfy

$S(\rho_{B})=S(\rho_{B^{\perp}})$ (11)

where $B^{\perp}$ is the complement to the subspace $B$. Take a generic
tripartite space consisting of $X\cup Y\cup Z$ and consider the coherent
informations $I(X\to Y)$ and $I(X\to Z)$. Making use of Eq. (11) allows us to
relate these in the following way:

$\displaystyle I(X\to Y)$ $\displaystyle=S(\rho_{Y})-S(\rho_{X\cup Y})$
$\displaystyle=S(\rho_{Y^{\perp}})-S(\rho_{(X\cup Y)^{\perp}})$
$\displaystyle=S(\rho_{X\cup Z})-S(\rho_{Z})$ $\displaystyle=-I(X\to Z).$ (12)

Applying this relation to the relevant channels of Bob and Charlie then yields

$I(A\to\text{Charlie})=-I(A\to\text{Bob}).$ (13)

This equivalence implies that $I(A\to\text{Charlie})$ must also witness the
backflow transition when $v_{d}$ is varied. In contrast to Bob, Charlie has no
information about Alice’s qubits in the Markovian regime (when Bob can decode
perfectly). While, in the regime of operator Non-Markovianity, Charlie has
partial information about Alice’s qubits.

Figure 11:  Diagrams depicting the complementary decoding protocol performed
by Charlie given he only has access to the final state of the impurity along
with the initial states of the medium and the impurity inserted at the reset
state. The blue arrow indicates the goal of the protocol: to teleport
information from the initial state of Alice’s qubits to Charlie’s ancillas.

####  5.2.1 Teleportation decoder

As before, we can construct a decoding protocol that reconstructs this partial
information as shown in Figure 11. In that figure, the decoder is shown in
green, and we show the diagram for the corresponding fidelity,
$\tilde{\mathcal{F}}$, in Appendix C. There is a clear similarity with the
Hayden-Preskill protocol decoder proposed by Yoshida and Kitaev [10]; in both
cases a Bell measurement is performed to facilitate the teleportation of
Alice’s information to Charlie’s qubits. The teleportation that Charlie hopes
to achieve in this case is indicated by the blue arrow in Figure 11. The
success of the teleportation improves when the bell measurement probability is
lower (see Appendix C). Given that $R$ and $A_{R}$ are initially in a Bell
state, the decoder has the worst fidelity when
$\mathrm{Tr}_{M}[U_{2}\rho_{M}(t_{1})]$ is unitary, i.e. $U_{2}$ and
$U_{2}^{*}$ do nothing to disentangle $R$ and $A_{R}$.

As in the previous protocol, we can relate Charlie’s decoder fidelity,
$\tilde{\mathcal{F}}$ to the coherent information (details in Appendix C):

$I(A\to\text{Charlie})=1+\log_{2}\tilde{\mathcal{F}}.$ (14)

The fidelity of this decoder, therefore also observes the backflow transition.
Note, that while Bob’s decoder works perfectly in the Markovian phase,
Charlie’s decoder only partially succeeds in the non-Markovian phase. This is
the main difference with the Hayden-Preskill protocol, and highlights the fact
that the scrambling of operators back on to the impurity is not maximal, as is
assumed for the black hole in the Hayden-Preskill protocol.

The phase transition in the OTOC dynamics explored in Section 4.1 is related
to a revival in operator support on the impurity due to non-Markovian feedback
from the medium. However, the non-Markovian effects only arise because of the
echo step carried out in the OTOC calculation. This new protocol we have
identified undergoes a phase transition in non-Markovianity as it is more
traditionally defined: in terms of the backflow of information from
environment to system. While the decoding is facilitated by Charlie initially
having access to the medium, this access is lost during the dynamics and the
enhancement of the decoder fidelity is driven by feedback of information onto
the impurity.

##  6 Conclusion

Within the simple framework of random unitary circuits we have studied an
impurity moving through a chaotic medium and the flow of information between
them as characterized by an OTOC. Information deposited into the medium is
rapidly scrambled away from the deposition site at the speed of sound. We show
that the supersonic and subsonic impurity velocities correspond to regimes of
zero and non-zero operator backflow respectively. This was determined by
defining a protocol in which we swapped the impurity qubit with a fresh one
uncorrelated with the environment and computing the OTOC on this new qubit.
The scaling exponents associated with this transition was shown to be related
to the diffusive evolution of the OTOC in the medium.

We considered the implications of the backflow transition on the scrambling
transition previously observed in a 1D impurity interacting with a Markovian
environment [14]. As expected from that work, this transition is only possible
at supersonic velocities where there isn’t operator backflow. There can be no
absorbing phase if there is a non-vanishing probability for operator weight to
flow back into the impurity.

We went on to show how this transition is also manifest in a quantum channel
capacity, namely that between the initial impurity state and the final state
of the environment plus the qubit discarded in the backflow protocol. We
associated with this channel a decoder whose fidelity was closely related to
the channel capacity and became ideal in the Markovian limit. By considering
the complement to this quantum channel we then derived an alternative protocol
which resembles the proposed in [10] for the Hayden-Preskill protocol. In this
case the performance of the decoder was tied to the degree of backflow from
the environment becoming optimal in the non-Markovian limit.

We have presented a relatively simple model that undergoes a phase transition
in information backflow. Similar to the scrambling transition [14] and the
measurement induced phase transition (MIPT) [83, 34, 35, 84], the transition
occurs in the dynamics of information. Unlike the MIPT, but similar to the
scrambling transition, this transition does not require the post selection of
measurement outcomes. Similar to the scrambling transition, the dynamics of
information in the backflow transition can be observed using one of the two
simple decoder shown above. The main gain in comparison the the scrambling
transition (which requires storing $L^{2}$ qubits of the medium) is that the
backflow transition occurs in 1D medium, and only requires storing $L$ qubits
during the time evolution and decoding. This difference may be of importance
when simulating the transitions on near term quantum computers.

Beyond realizing this transition on modern near-term devices other promising
directions would be to extend this to a more realistic open quantum model such
as a spin-boson type model [85, 86] or a Kondo model [87]. In this regard,
recent techniques developed to treat non-Markovian dynamics will be of great
assistance [88, 89, 90, 91, 92].

## Acknowledgements

The authors gratefully acknowledge the computing time granted on the
supercomputer MOGON 2 at Johannes Gutenberg-University Mainz (hpc.uni-
mainz.de).

##### Funding information

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - TRR 288 - 422213477 (project B09), TRR 306 QuCoLiMa
(“Quantum Cooperativity of Light and Matter”), Project-ID 429529648 (project
D04) and in part by the QuantERA II Programme that has received funding from
the European Union’s Horizon 2020 research and innovation programme under
Grant Agreement No 101017733 (“QuSiED”) and by the DFG (project number
499037529), and by the Dynamics and Topology Centre funded by the State of
Rhineland Palatinate.

##  Appendix A Alternate 1D circuit

Here we show that the backflow transition, with modified exponents, also
occurs when the 1D model of the main text is slightly modified. We now allow
for scrambling to only occur up to the position of the impurity as depicted in
Figure 12. In this case the backflow order parameter on variation of $v_{d}$
is shown in Figure 13.

Figure 12: Cartoon of the modified boundary conidition where scrambling in the
environment, as indicated by the green region, can only occur up to the
position of the impurity, the dashed line.

The change to the circuit modifies the spread of the operator within the
environment and in particular the weight of the OTOC at the impurity position
which is instead given by

$P_{1}(v_{d},t)=\frac{\exp\left(\frac{-v^{2}t}{4D}\right)}{\sqrt{\pi
Dt}\left[1+\mathrm{erf}\left(\frac{v\sqrt{t}}{2\sqrt{D}}\right)\right]}.$ (15)

This in turn satisfies a distinct scaling relation:

$P_{1}(v,t)=\sqrt{t}F_{1}(v\sqrt{t}),$ (16)

and we see this manifest as well in the scaling of the backflow order
parameter in Figure 14

$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$$0$$0.2$$0.4$$0.6$$0.8$$v_{d}$$B$
Figure 13: Backflow order parameter for the alternate circuit on variation of
impurity drift velocity. Here we set $t_{1}=100$, $t_{2}=1000$ and $p=0.1$.
This is the result of an average over $10^{4}$ disorder realizations.
$1$$1.1$$1.2$$1.3$$1.4$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$v_{d}$$B$$-6$$-4$$-2$$0$$2$$4$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$(v_{d}-v^{*})\sqrt{T}$$B\sqrt{T}$$t_{2}$1252505001000
Figure 14: Left: Expanded view of the backflow phase transition for the
alternate circuit near the critical point. Right: Data collapse under a
finite-time rescaling. Here we set $t_{1}=100$ and $p=0.1$. This is the result
of an average over $10^{4}$ disorder realizations.

##  Appendix B Derivation of Eq. (10)

The fidelity of the decoder for Bob’s protocol is related to the purity of the
reduced density matrix $\rho^{A\cup\text{Bob}}$ prior to decoding via

$\mathcal{F}=d_{M}^{-1}\mathrm{Tr}(\rho_{A\cup\text{Bob}}^{2}),$ (17)

where $d_{M}$ is the Hilbert space dimension of the medium; we show this
diagrammatically in Figure 15. By manipulating the fidelity diagram the
decoder becomes a replica in the purity diagram. These two diagrams correspond
when $\rho_{M}$, $\rho_{I^{\prime}}$ and $\rho_{R}$ are infinite temperature
states. Infinite temperature states in the fidelity are related to trace
operations in the purity (and _vice versa_) via

$\rho_{B}^{\infty}\leftrightarrow d_{B}^{-1}\mathrm{Tr}_{B}[\cdot],$ (18)

where $d_{B}$ is the Hilbert space dimension of $B$. These dimensional factors
mostly cancel in our case except for $d_{M}$.

Figure 15: Left: The fidelity of Bob’s decoding protocol for a given unitary
circuit, $U$, and initial states: $\rho_{M}$, $\rho_{R}$ and
$\rho_{I^{\prime}}$. Right: The purity of the reduced density matrix
$\rho_{A\cup\text{Bob}}$. If $\rho_{M}$, $\rho_{I^{\prime}}$ and $\rho_{R}$
are all infinite temperature states then these two diagrams are equivalent
down to a numerical factor.

Now, the entanglement spectrum for Clifford circuits is flat[]; this results
in an equivalence between Renyi entropies, $S_{n}(\rho_{C})$, defined by

$S_{n}(\rho_{C})=\frac{1}{n-1}\log_{2}\mathrm{Tr}[\rho_{C}^{n}].$ (19)

This combined with the relation $S(\rho_{C})=\lim_{n\to 1}S_{n}(\rho_{C})$
allows us to make the following substitutions:

$\displaystyle I(A\to\text{Bob})$
$\displaystyle=S_{2}(\rho_{\text{Bob}})-S_{2}(\rho_{A\cup\text{Bob}})$
$\displaystyle=-\log_{2}\mathrm{Tr}[\rho_{\text{Bob}}^{2}]+\log_{2}\mathrm{Tr}[\rho_{A\cup\text{Bob}}^{2}]$
$\displaystyle=1+\log_{2}\mathcal{F},$ (20)

where to arrive at the last line we used
$S_{2}(\rho_{\text{Bob}})=-\log_{2}(d_{M}d_{I})=1-\log_{2}(d_{M})$.

##  Appendix C Details of HP fidelity and derivation of Eq. (14)

The fidelity for Charlie’s decoding protocol is shown diagrammatically in
Figure 16 where the only non-trivial factor is given by the probability of
measuring a Bell pair, $P_{\text{EPR}}$. By comparing the diagram for
$P_{\text{EPR}}$ and that for the original decoder’s fidelity, $\mathcal{F}$
in Figure 15, it is apparent that

$P_{\text{EPR}}\equiv\mathcal{F}.$ (21)

The numerator in $\tilde{\mathcal{F}}$ trivally contracts to yield a factor of
$d_{I}^{-2}$ where $d_{I}$ is the Hilbert space dimension of the impurity.
Altogether we have the following relation between the two fidelities:

$\mathcal{F}=\frac{1}{\tilde{\mathcal{F}}d_{I}^{2}}.$ (22)

Substituting this into Eq. (10) and making use of Eq. (13) (along with
$d_{I}=2$) then yields Eq. (14).

Figure 16: The fidelity of the second Hayden-Preskill-like protocol of the
main text. The numerator trivially contracts to yield a constant factor and
the non-trivial dependence on the unitary appears in the probability of
measuring a bell pair, $P_{\text{EPR}}$

.

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# One-dimensional topological theories with defects and linear generating
functions

Mee Seong Im  Department of Mathematics, United States Naval Academy,
Annapolis, MD 21402, USA<EMAIL_ADDRESS>and  Paul Zimmer  Department of
Mathematics, United States Naval Academy, Annapolis, MD 21402, USA
<EMAIL_ADDRESS>
###### Abstract.

We study the Gram determinant and construct bases of hom spaces for the one-
dimensional topological theory of decorated unoriented one-dimensional
cobordisms, as recently defined by Khovanov, when the pair of generating
functions is linear.

##  1\. Introduction

Atiyah’s celebrated topological quantum field theories framework [Ati88] can
be augmented by non-monoidal (in general) topological theories coming from the
universal constructions as in [BHMV95, FKN+05, Kho20b, Kho20a, KS20]. In these
constructions one starts with an evaluation of closed $n$-manifolds to
elements of the ground field and associates state spaces to closed
$(n-1)$-manifolds and maps between spaces to cobordisms between them. Khovanov
[Kho20a] have shown that the resulting theories are interesting even in
dimension one, if manifolds are enriched by defects, such as submanifolds or
other decorations. In dimension one, defects are zero-dimensional submanifolds
that may carry additional labels. Universal theories in this case relate to
noncommutative power series [Kho20a].

Without the labels and in the one-dimensional case, one considers one-
manifolds equipped with zero-dimensional submanifolds. They can be visualised
by dots inside one-manifolds. A closed connected 1-manifold with such defects
is diffeomorphic to a circle with some number $k$ of dots. Multiplicative
evaluation function $\alpha$ as in [Kho20a] is then determined by its values
on such circles, one for each $k\geq 0$, and may be encoded by power series

(1) $Z_{\alpha}(T)=\sum_{k\geq 0}\gamma_{k}T^{k},\ \ \
\gamma_{k}=\alpha(\mathrm{circle\ with}\ k\ \mathrm{dots})\in\mathbf{k},$

where $\mathbf{k}$ is the ground field. As explained in [Kho20a], one can
furthermore consider cobordisms that may have two types of boundary points, so
that components may ”end” in the middle of the cobordism, see Figure 2.1.1
below. This allows to view intervals as closed cobordisms, so that evaluation
$\alpha$ acquires additional parameters, its values on an interval with $k$
dots. Then $\alpha$ is encoded by two generating functions, $Z_{\alpha}(T)$ in
(1) and

(2) $Z_{1,\alpha}(T)=\sum_{k\geq 0}\beta_{k}T^{k},\ \ \
\beta_{k}=\alpha(\mathrm{interval\ with}\ k\ \mathrm{dots})\in\mathbf{k}.$

Below, we relabel $Z_{\alpha}(T)$ to $Z_{0,\alpha}(T)$.

To a generating function, there are several categories of linearized
cobordisms associated [KS20, Kho20a]. They have not been understood, and in
this paper we treat a particular case when each of the two generating
functions is linear. In this case we’re able to compute the Gram determinant
for the spanning set of cobordisms from the empty 0-manifold to $n$ points and
show that it’s nonzero. This gives a basis of hom spaces between any two
objects $n,m$ of monoidal category $\mathsf{Cob}_{\alpha}$, the topological
theory associated to $\alpha$, with easily computable composition of basis
elements.

##  2\. Background

###  2.1. Setup and the main result

We follow the framework of [Kho20a] and consider the category of one-
dimensional unoriented dotted cobordisms $\mathsf{Cob}_{1}$ with boundary. We
work with the skeletal version of this category where objects are nonnegative
integers $n\in\mathbb{Z}_{+}$. Morphisms from $n$ to $m$ are unoriented dotted
one-dimensional manifolds $M$, with the boundary subdivided into the _inner_
and _outer_ parts,

$\partial M=(\partial M)_{\mathsf{inn}}\sqcup(\partial M)_{\mathsf{out}},$

and a fixed bijection between the outer part and the union of $n+m$ points,

$(\partial M)_{\mathsf{out}}\cong\\{1,2,\dots,n+m\\}.$

One thinks of $M$ as a cobordism from $n$ to $m$ ordered points, see Figure
2.1.1.

$1$$2$$n$$n+1$$n+m$ Figure 2.1.1. An unoriented dotted cobordism from $n$ to
$m$ points. Defects are represented by large dots and boundary points of
cobordisms are represented by small dots.

These cobordisms are composed in the usual way. Cobordisms are considered up
to rel outer boundary diffeomorphisms and can be described combinatorially.
Each cobordism may have _closed_ and _boundary_ components. Closed components
(with empty intersection with the outer boundary) are classified by their
topological type (an interval or a circle) together with the number of dots on
them (possibly none), see Figure 2.1.2. Boundary components may have one or
two outer boundary points and will be called _$1$ -arcs_, respectively, _$2$
-arcs_. Each boundary component may carry dots as well.

$1$$2$$3$$n$$1$$2$$3$$n$

Figure 2.1.2. On the left: possible floating connected components of a
cobordism include an arc with $n\geq 0$ dots and a circle with $n\geq 0$ dots.
On the right: five types of components which contain one or two outer boundary
points, including two types of 1-arcs and three types of 2-arcs.

An evaluation function or series $\alpha$ is a map from the set of
diffeomorphism classes of closed connected diagrams to the ground field
$\mathbf{k}$. It can be encoded by two power series

(3) $Z_{0,\alpha}(T)=\sum_{n\geq 0}\beta_{n}T^{n},\qquad
Z_{1,\alpha}(T)=\sum_{n\geq 0}\gamma_{n}T^{n},$

where $\beta_{n}$, respectively $\gamma_{n}$, is the value of $\alpha$ on an
interval with $n$ dots, respectively on a circle with $n$ dots, $n\geq 0$, see
Figure 2.1.3.

$\stackrel{{\scriptstyle\mbox{\Large$\alpha$}}}{{\mbox{\Large$\longrightarrow$}}}$$\beta_{0}$

$1$$2$$n$${\stackrel{{\scriptstyle\mbox{\Large$\alpha$}}}{{\mbox{\Large$\longrightarrow$}}}}$$\beta_{n}$

$\stackrel{{\scriptstyle\mbox{\Large$\alpha$}}}{{\mbox{\Large$\longrightarrow$}}}$$\gamma_{1}$

$1$$2$$3$$n$$\stackrel{{\scriptstyle\mbox{\Large$\alpha$}}}{{\mbox{\Large$\longrightarrow$}}}$$\gamma_{n}$

Figure 2.1.3.  Evaluation function $\alpha$.

To an evaluation $\alpha$ one can associate several categories and functors
between them as in [KS20, Kho20a]. To $n$ points there’s also associated a
vector space $A(n)$. One considers a vector space $V(n)$ with a basis of all
cobordisms $x$ as above from $0$ to $n$. Denote by $\overline{x}$ the
reflection of $x$, a cobordism from $n$ to $0$, see Figure 2.1.4 top.

$x$

$\overline{x}$

$\overline{y}$

$\overline{y}x$$\simeq$

Figure 2.1.4.  An example of a cobordism $x$ from $0$ to $4$, reflections
$\overline{x},\overline{y}$ and coupling $\overline{y}x$.

For two cobordisms $x,y$ from $0$ to $n$, the composition $\overline{y}x$ is a
cobordism from $0$ to $0$ (a closed cobordism) and can be evaluated via
$\alpha$, see an example on Figure 2.1.4.

This induces a bilinear form on $V(n)$,

(4) $(\>\>,\>\>)\ :\ V(n)\times V(n)\longrightarrow\mathbf{k},\hskip
11.38109pt(x,y)=\alpha(\overline{y}x).$

Define $A(n)$ as the quotient of $V(n)$ by the kernel of the bilinear form,

$A(n)\ :=\ V(n)/\mathrm{ker}((\>\cdot,\cdot\>)).$

From vector spaces $A(n)$ one can build monoidal category
$\mathsf{Cob}_{\alpha}$ as in [Kho20a] by identifying homs from $n$ to $m$ in
the latter category with $A(n+m)$ and forming suitable compositions of
morphisms.

In this manuscript, we consider a very special evaluation $\alpha$; it is
nonzero only on four diffeomorphism classes of decorated connected 1-manifolds
and

  * •

takes values $\beta_{0},\beta_{1}$ on an interval with no dots and one dot,
respectively;

  * •

takes values $\gamma_{0},\gamma_{1}$ on a circle with no dots and one dot,
respectively;

  * •

evaluates any connected 1-manifold with at least two dots to $0$;

see Figure 2.1.5.

$\mapsto$$\beta_{0}$

$\mapsto$$\beta_{1}$

$\mapsto$$\gamma_{0}$

$\mapsto$$\gamma_{1}$

$\mapsto$$0$$\vdots$$\vdots$

Figure 2.1.5. Closed diagrams under the evaluation map $\alpha$.

We study vector spaces $A(n)$ and associated categories for this special
evaluation $\alpha$, for which both $Z_{0,\alpha},Z_{1,\alpha}$ are linear
functions,

(5) $Z_{0}(\alpha)=\beta_{0}+\beta_{1}T,\qquad
Z_{1}(\alpha)=\gamma_{0}+\gamma_{1}T,\qquad\beta_{0},\beta_{1},\gamma_{0},\gamma_{1}\in\mathbf{k}.$

The evaluation $\alpha$ is zero once a closed cobordism has a component with
at least two dots, see Figure 2.1.5 right, since quadratic and all higher
order terms of $Z_{0,\alpha},Z_{1,\alpha}$ are $0$. This means there’s a
relation that two dots on the line evaluate to $0$, both in the skein category
and in $\mathsf{Cob}_{\alpha}$. In particular, in $\mathsf{Cob}_{\alpha}$ a
morphism can be reduced to a linear combination of cobordisms with no closed
components and with at most one dot on each component. Likewise, vector space
$A(n)$ has a spanning set of such cobordisms (from $0$ to $n$). We call these
cobordisms _$n$ -diagrams_ and denote their set by $\mathcal{D}_{n}$. A
connected component of an $n$-diagram can be one of the four types, see Figure
2.1.6:

  1. (1)

a dotless arc,

  2. (2)

arc with one dot,

  3. (3)

a dotless cup,

  4. (4)

a cup with one dot.

Types (1) and (2) may also be called 1-arcs, types (3) and (4) may be called
2-arcs.

$(1)$$(2)$$(3)$$(4)$ Figure 2.1.6. Four types of components of an $n$-diagram.
Types (1) and (2) are 1-arcs, types (3) and (4) are 2-arcs, see earlier.
Figure 2.1.7. Example of a diagram in the orbit $\mathcal{O}(2,1,2,1)$.

The symmetric group $S_{n}$ acts on the set of $n$-diagrams by permuting the
endpoints. Denote the set of orbits of the action by $\mathsf{Orb}_{n}$. For
an $n$-diagram $X\in\mathcal{D}_{n}$ let

  * •

$a$ be the number of dotless strands,

  * •

$b$ be the number of strands with a single dot,

  * •

$c$ be the number of dotless cups, and

  * •

$d$ be the number of cups with a single dot in a diagram.

We have

$a+b+2c+2d=n,$

and the quadruple $(a,b,c,d)$ is a complete invariant of an orbit of $S_{n}$
on $\mathcal{D}_{n}$. Figure 2.1.7 shows an example of a 9-diagram with
parameters $(2,1,2,1)$.

We define $\mathcal{O}(a,b,c,d)=S_{n}X\in\mathsf{Orb}_{n}$ as the orbit which
consist of all $n$-diagrams with the parameters $(a,b,c,d)$. Then

$\mathcal{D}_{n}=\coprod_{a,b,c,d}\mathcal{O}(a,b,c,d),\hskip
11.38109pta+b+2c+2d=n,\hskip 11.38109pta,b,c,d\geq 0.$

For a diagram $X\in\mathcal{O}(a,b,c,d)$, define

  * •

$\mathsf{def}(X)=b+d$, the number of defect points in the diagram $X$,

  * •

$\mathsf{comp}(X)=a+b+c+d$, the total number of connected components of $X$.

Diagrams in $\mathcal{D}_{n}$ span $A(n)$. To see whether they constitute a
basis of $A(n)$ we need to form the Gram matrix by pairing these diagrams and
see whether it’s nondenegenerate.

Recall that the reflection $\overline{X}$ of an $n$-diagram $X$ is given by
reflecting $X$ in a horizontal line, so it becomes a cobordism from $n$ to
$0$. Given two $n$-diagrams $X,Y$, we can compose $\overline{Y}$ and $X$ to
get a closed diagram $\overline{Y}X$. It can then be evaluated via $\alpha$.
Thus, we define

(6) $\alpha(X,Y)\ :=\ \alpha(\overline{Y}X)\in\mathbf{k}.$

An example can be found in Figure 2.1.4, for $n=4$ and diagrams $x$ and $y$
with parameters $(1,1,0,1)$ and $(0,0,2,0)$, respectively. Evaluation
$\alpha(\overline{y}x)=\beta_{1}\gamma_{1}$.

Picking an order on the set $\mathcal{D}_{n}$ of $n$-diagrams gives us the
Gram matrix $G$. We now explain how to compute the determinant of $G$. It
turns out that, after suitable permutations of rows and columns, $G$ becomes
lower-triangular. To see this, we first define the notion of the dual of an
$n$-diagram.

######  Definition 2.1.

The dual $\iota(X)$ of a diagram $X\in\mathcal{D}_{n}$ is the diagram with the
same arcs and cups as $X$, with the dots removed from single-dotted components
and a single dot added to each dotless component.

$x$

$\iota(x)$

Figure 2.1.8. Example of a diagram $x$ and its dual $\iota(x)$ when $n=5$.

An example can be found in Figure 2.1.8. Map
$\iota:\mathcal{D}_{n}\longrightarrow\mathcal{D}_{n}$ is an involution on the
set of $n$-diagrams. It descends to an involution on the set of
$S_{n}$-orbits, also denoted $\iota$.

It is convenient to twist the pairing $\alpha(\>\>,\>\>)$ by $\iota$ and
define $\alpha_{\iota}(X,Y):=\alpha(X,\iota(Y))=\alpha(\overline{\iota(Y)}X)$.
An example of twisted pairing is computed in Figure 2.2.1.

Both pairings are maps
$\alpha,\alpha_{\iota}:\mathcal{D}_{n}\times\mathcal{D}_{n}\longrightarrow\mathbf{k}$.

###  2.2. Understanding the Gram matrix

Up to a possible sign, instead of computing determinant of the Gram matrix $G$
we can compute the determinant of the twisted matrix $G_{\iota}$ with the
entry on the intersection of $X$-row and $Y$-column
$\alpha_{\iota}(X,Y)=\alpha(\overline{\iota(Y)}X).$

If one of the components of closed diagram $\overline{\iota(Y)}X$ contains at
least two dots, the evaluation is zero. Otherwise it is
$\beta_{0}^{k_{0}}\beta_{1}^{k_{1}}\gamma_{0}^{k_{2}}\gamma_{1}^{k_{3}}$,
where $k_{0},k_{1},k_{2},k_{3}$ are the number of free (dotless) arcs, arcs
with a dot, free circles, and circles with a dot, respectively, see Figure
2.2.1 for an example.

$\alpha_{\iota}$,$=$$\alpha$$=$$\alpha$$=$$0$. Figure 2.2.1. An example of
evaluation $\alpha_{\iota}(X,Y)=\alpha(\overline{\iota(Y)}X)$, with
$X\in\mathcal{O}(1,1,1,0)$ and $Y\in\mathcal{O}(2,0,0,1)$.

######  Definition 2.2.

Let $X\in\mathcal{O}(a,b,c,d)$ and
$Y\in\mathcal{O}(a^{\prime},b^{\prime},c^{\prime},d^{\prime})$. We define a
strict total ordering on the diagrams, and write $X<Y$, if

  1. (1)

$\mathsf{def}(X)>\mathsf{def}(Y)$, or

  2. (2)

$\mathsf{def}(X)=\mathsf{def}(Y)$ and $\mathsf{comp}(X)<\mathsf{comp}(Y)$, or

  3. (3)

$\mathsf{def}(X)=\mathsf{def}(Y)$ and $\mathsf{comp}(X)=\mathsf{comp}(Y)$, and
$d>d^{\prime}$.

Note that, if $\mathsf{def}(X)=\mathsf{def}(Y)$,
$\mathsf{comp}(X)=\mathsf{comp}(Y)$, and $d=d^{\prime}$, then $X=Y$. Orbits
with more defects (dots) appear earlier in the order than orbits with fewer
defects, according to our definition.

For each orbit $\mathcal{O}(a,b,c,d)\in\mathsf{Orb}_{n}$ we choose an
arbitrary total ordering of its elements. Next, order the diagrams in
$\mathcal{D}_{n}$ using the strict total ordering on orbits in Definition 2.2
and the arbitrary choice of ordering we’ve made within each orbit. Diagrams
from each orbit $\mathcal{O}(a,b,c,d)$ are grouped together in the strict
total ordering.

We use this order for the rows in the Gram matrix, going from the smallest to
the greatest $n$-diagram relative to the order. To label columns, we use
involution $\iota$. Thus, if the $i$-row is labelled by diagram $X$, then
$i$-th column is labelled by diagram $\iota(X)$. We denote this permuted Gram
matrix by $G_{\iota}$. It can also be thought of as a block matrix, with
blocks enumerated by pairs of orbits of $S_{n}$ on $\mathcal{D}_{n}$.

######  Proposition 2.3.

Suppose $\mathcal{O}>\mathcal{O}^{\prime}$ for orbits
$\mathcal{O},\mathcal{O}^{\prime}\in\mathsf{Orb}_{n}$. Then
$\alpha(\overline{\iota(Y)}X)=0$ for any
$X\in\mathcal{O},Y\in\mathcal{O}^{\prime}$.

###### Proof.

Let $\mathcal{O}=\mathcal{O}(a,b,c,d)$ and
$\mathcal{O}^{\prime}=\mathcal{O}(a^{\prime},b^{\prime},c^{\prime},d^{\prime})$.
Then $\iota(Y)\in\mathcal{O}(b^{\prime},a^{\prime},d^{\prime},c^{\prime})$.
Condition $\mathcal{O}>\mathcal{O}^{\prime}$ says that one of the following
three conditions hold:

  * •

$\mathsf{def}(X)>\mathsf{def}(Y)$, or $b+d>b^{\prime}+d^{\prime}$. Diagram
$\iota(Y)$ has $a^{\prime}+c^{\prime}$ dots, and the coupling diagram
$\overline{(}\iota(Y))X$ has
$b+d+a^{\prime}+c^{\prime}>b^{\prime}+d^{\prime}+c^{\prime}+a^{\prime}=\mathsf{comp}(Y)$
dots, making it evaluate to zero, since one of the components has at least two
dots.

  * •

$\mathsf{def}(X)=\mathsf{def}(Y)$ and $\mathsf{comp}(X)<\mathsf{comp}(Y)$.
This means $b+d=b^{\prime}+d^{\prime}$ and
$a+b+c+d<a^{\prime}+b^{\prime}+c^{\prime}+d^{\prime}$. We can rewrite the
inequality as $a+c<a^{\prime}+c^{\prime}$ (or that $X$ has fewer arcs than
$Y$). Diagram $\iota(Y)$ has $a^{\prime}+c^{\prime}$ dots and the coupling
$\overline{\iota(Y)}X$ has exactly
$b+d+a^{\prime}+c^{\prime}=a^{\prime}+b^{\prime}+c^{\prime}+d^{\prime}=\mathsf{comp}(Y)$
dots. In the coupling diagram $\overline{\iota(Y)}X$ we can slide the dots
away from components of $\iota(Y)$ onto adjacent components of $X$. Since
$\mathsf{comp}(X)<\mathsf{comp}(Y)$ and there are $\mathsf{comp}(Y)$ dots in
the diagram, one of the components of $X$ will carry more than one dot.
Consequently, the diagram $\overline{\iota(Y)}X$ will evaluate to $0$ via
$\alpha$.

  * •

$\mathsf{def}(X)=\mathsf{def}(Y),$ $\mathsf{comp}(X)=\mathsf{comp}(Y)$ and
$d>d^{\prime}$. Repeating the arguments from the previous case, the diagram
$X\cdot\iota(Y)$ has exactly $\mathsf{comp}(X)=\mathsf{comp}(Y)$ dots and they
are distributed so that there’s one dot on each component. In the coupling
arcs of $X$ must pair to (reflected) arcs of $Y$ and cups of $X$ to
(reflected) cups of $Y$, for otherwise $X\cdot\iota(Y)$ has less than
$\mathsf{comp}(X)=\mathsf{comp}(Y)$ connected components and evaluates to $0$.
Diagram $\overline{\iota(Y)}X$ consists of $c+d=c^{\prime}+d^{\prime}$ circles
and $a+b=a^{\prime}+b^{\prime}$ arcs. The number of dots on cups of $X$ and
$Y$ is $c^{\prime}+d>c^{\prime}+d^{\prime}$, so it’s greater than the number
of circles of $\overline{\iota(Y)}X$, and in the latter coupling one of the
circles carries more than one dot, evaluating to $0$.

∎

######  Proposition 2.4.

Suppose that $X$ and $Y$ belong to the same orbit $\mathcal{O}(a,b,c,d)$. Then

(7)
$\alpha(\overline{\iota(Y)}X)=\begin{cases}\beta_{1}^{a+b}\gamma_{1}^{c+d}&\textrm{
if }X=Y,\\\ 0&\textrm{ otherwise.}\end{cases}$

###### Proof.

The number of dots on $\overline{\iota(Y)}X$ is
$a+b+c+d=\mathsf{comp}(X)=\mathsf{comp}(Y)$. If arcs and cups of $X$ don’t
perfectly match those of $Y$ (which happens if and only if $X=Y$), the coupled
diagram $\overline{\iota(Y)}X$ will have fewer than $\mathsf{comp}(X)$
components and evaluate to zero, due to the presence of $\mathsf{comp}(X)$
dots on it. The diagonal case $X=Y$ is clear, since $\overline{\iota(X)}X$
contains $a+b$ single-dotted intervals and $c+d$ single-dotted circles. ∎

######  Theorem 2.5.

Twisted Gram matrix $G_{\iota}$ is lower-triangular.

###### Proof.

Consider $n$-diagrams $X,Y$ with $X<Y$. They belong to orbits
$\mathcal{O},\mathcal{O}^{\prime}$, respectively, with
$\mathcal{O}\leq\mathcal{O}^{\prime}$. Let $i$ and $j$ be the rows
corresponding to $X$ and $Y$, respectively. The $(i,j)$ entry of $G_{\iota}$
is $\alpha(\overline{\iota(Y)}X)$.

If $\mathcal{O}<\mathcal{O}^{\prime}$ then $\alpha(\overline{\iota(Y)}X)=0$ by
Proposition 2.3. If $\mathcal{O}=\mathcal{O}^{\prime}$ then $X\not=Y$ belong
to same orbit and $\alpha(\overline{\iota(Y)}X)=0$ by Proposition 2.4. ∎

######  Corollary 2.6.

For a diagram $X\in\mathcal{O}(a,b,c,d)$ the diagonal entry is

(8) $\alpha(\iota(X)X)=\gamma_{1}^{a+b}\beta_{1}^{c+d}.$

###### Proof.

This is part of Proposition 2.4. ∎

$\Big{\\{}$,$\Big{\\}}$$\Big{\\{}$,$\Big{\\}}$$\gamma_{1}^{2}$$\beta_{1}$$\gamma_{1}^{2}$$\gamma_{1}^{2}$$\beta_{1}$$\gamma_{1}^{2}$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$\gamma_{1}^{2}$$\gamma_{0}\gamma_{1}$$\gamma_{0}\gamma_{1}$$\gamma_{0}$$\gamma_{0}^{2}$$\gamma_{1}$$\gamma_{1}$$\beta_{0}$$\gamma_{0}$$0$$\gamma_{1}$$\gamma_{0}\gamma_{1}$$\gamma_{1}$$\gamma_{0}\gamma_{1}$$\gamma_{1}$
Figure 2.2.2. Twisted Gram matrix for $n=2$. The determinant is
$\beta_{1}^{2}\gamma_{1}^{8}$.

An example of the twisted Gram matrix is shown in Figure 2.2.2 for $n=2$. In
this case only one orbit has more than one element, and there are two choices
for the total order. Each of them produces an upper-triangular matrix.
Diagrams $X$ and $\iota(X)$ are displayed in the corresponding rows and
columns. Table 1 shows how the rows and columns are ordered.

Rows | Columns
---|---
$(a,b,c,d)$ | $\mathsf{def}$ | $\mathsf{comp}$ | $(b,a,d,c)$ | $\mathsf{def}$ | $\mathsf{comp}$
$(0,2,0,0)$ | $2$ | $2$ | $(2,0,0,0)$ | $0$ | $2$
$(0,0,0,1)$ | $1$ | $1$ | $(0,0,1,0)$ | $0$ | $1$
$(1,1,0,0)$ | $1$ | $2$ | $(1,1,0,0)$ | $1$ | $2$
$(0,0,1,0)$ | $0$ | $1$ | $(0,0,0,1)$ | $1$ | $1$
$(2,0,0,0)$ | $0$ | $2$ | $(0,2,0,0)$ | $2$ | $2$
Table 1. Ordering of the orbits for the twisted Gram matrix for $n=2$.

######  Proposition 2.7.

Let $|\mathcal{O}(a,b,c,d)|$ denote the cardinality of the orbit
$\mathcal{O}(a,b,c,d)$. Then

${\left|\mathcal{O}(a,b,c,d)\right|=\frac{n!}{a!\>b!\>c!\>d!\>2^{c+d}}}.$

###### Proof.

The cardinality of the orbit is the order of $S_{n}$ divided by the order of
the stabilizer of any representative of the orbit. So
$\left|\mathcal{O}(a,b,c,d)\right|=\displaystyle{\frac{|S_{n}|}{|S_{nx}|}}$.
Given a representative of $\mathcal{O}(a,b,c,d)$, we can can permute any of
the un-dotted strands, which contributed a factor of $a!$, dotted strands,
which contributes a factor of $b!$, un-dotted cups, which contributes a factor
of $c!$, or dotted cups, which contributes a factor of $d!$. Moreover we can
swap the two endpoints of any cup, which contributes a factor of $2^{c+d}$. ∎

The determinant of the diagonal block for the orbit $\mathcal{O}(a,b,c,d)$ is

$\left(\gamma_{1}^{a+b}\beta_{1}^{c+d}\right)^{\left|\mathcal{O}(a,b,c,d)\right|}$

Taking the product over all orbits gives the determinant of the twisted Gram
matrix:

$\det
G_{\iota}=\prod_{A+2B=n}\left(\gamma_{1}^{A}\beta_{1}^{B}\right)^{\displaystyle{\sum_{\stackrel{{\scriptstyle
a+b=A,}}{{c+d=B}}}}|\mathcal{O}(a,b,c,d)|}$

This determinant may differ by a sign from the determinant of the original
Gram matrix, $\det G_{\iota}=\pm\det G$.

######  Proposition 2.8.

For $n\geq 2$ the set $\mathcal{D}_{n}$ of $n$-diagrams constitutes a basis of
the state space $A(n)$ if and only if $\beta_{1}\gamma_{1}\not=0$.

###### Proof.

This is immediate since the determinant is nonzero exactly when
$\beta_{1}\gamma_{1}\not=0$. For $n=1$, the determinant is $-\beta_{1}^{2}$,
and the corresponding condition is $\beta_{1}\not=0$. ∎

This proposition implies that, when $\beta_{1}\gamma_{1}\not=0$, there is no
difference between the skein category for this evaluation $\alpha$, as defined
in [Kho20a], and the quotient $\mathsf{Cob}_{\alpha}$ of that category. In
both categories, the defining relations are the evaluations of closed diagrams
with one or no dots via $\alpha$ and the relation that two dots on a strand
equal $0$.

Furthermore, in these categories, spaces of homs from $n$ to $m$ can be
naturally identified with $A(n+m)$ by moving the $n$ points from the source to
the target of the cobordism via $n$ arcs. Consequently, the above proposition
gives bases of homs between all pairs of objects $(n,m)$ in
$\mathsf{Cob}_{\alpha}$. Composition of basic morphisms is computed by
concatenating them, evaluating closed components via $\alpha$, and further
evaluating to $0$ if one of the remaining components contains more than one
dot.

When $\beta_{1}=\gamma_{1}=0$, a dot evaluates to $0$, the resulting
categories have smaller hom spaces and relate to rook Brauer algebras [Hd14].
It may also be interesting to investigate the case when only one of
$\beta_{1},\gamma_{1}$ equals zero.

$x$$\iota(x)$orbitsizeorbitsize$1$$3$$3$$3$$3$$3$$1$$1$$1$$3$$3$$3$$3$$3$$3$$1$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$\beta_{1}^{3}$ $\mathsf{I}_{1}$$*$$*$$*$$*$$*$$*$$*$$\beta_{1}^{3}$ $\mathsf{I}_{3}$$0$$*$$*$$*$$*$$*$$\beta_{1}\gamma_{1}$ $\mathsf{I}_{3}$$*$$*$$*$$*$$*$$\beta_{1}\gamma_{1}$ $\mathsf{I}_{3}$$0$$*$$*$$*$$\beta_{1}\gamma_{1}$ $\mathsf{I}_{3}$$*$$*$$*$$\beta_{1}^{3}$ $\mathsf{I}_{3}$$*$$*$$\beta_{1}\gamma_{1}$ $\mathsf{I}_{3}$$*$$\beta_{1}^{3}$ $\mathsf{I}_{1}$ Figure 2.2.3. Twisted Gram block matrix when $n=3$, where rows represent orbits, and the columns dual orbits. One element of the orbit is shown in each row and column. The cardinality of the orbit is shown to the left or above each orbit representative. $\mathsf{I}_{k}$ denotes the $k\times k$ identity matrix. The determinant is $\beta_{1}^{36}\gamma_{1}^{12}$. Along the row | Along the column
---|---
$(a,b,c,d)$ | $\mathsf{def}$ | $\mathsf{comp}$ | $(b,a,d,c)$ | $\mathsf{def}$ | $\mathsf{comp}$
$(0,3,0,0)$ | $3$ | $3$ | $(3,0,0,0)$ | $0$ | $3$
$(0,1,0,1)$ | $2$ | $2$ | $(1,0,1,0)$ | $0$ | $2$
$(1,2,0,0)$ | $2$ | $3$ | $(2,1,0,0)$ | $1$ | $3$
$(1,0,0,1)$ | $1$ | $2$ | $(0,1,1,0)$ | $1$ | $2$
$(0,1,1,0)$ | $1$ | $2$ | $(1,0,0,1)$ | $1$ | $2$
$(2,1,0,0)$ | $1$ | $3$ | $(1,2,0,0)$ | $2$ | $3$
$(1,0,1,0)$ | $0$ | $2$ | $(0,1,0,1)$ | $2$ | $2$
$(3,0,0,0)$ | $0$ | $3$ | $(0,3,0,0)$ | $3$ | $3$
Table 2. Ordering of the orbits for the Gram matrix for $n=3$. $x$$\iota(x)$orbitsizeorbitsize$1$$6$$4$$3$$12$$6$$6$$6$$6$$12$$4$$3$$6$$1$$1$$6$$4$$3$$12$$6$$6$$6$$6$$12$$4$$3$$6$$1$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$0$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$0$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$0$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$0$$*$$*$$*$$*$$*$$*$$*$$*$$0$$*$$*$$*$$*$$*$$\beta_{1}^{4}\mathsf{I}_{1}$$\beta_{1}^{2}\gamma_{1}\mathsf{I}_{6}$$\beta_{1}^{4}\mathsf{I}_{4}$$\gamma_{1}^{2}\mathsf{I}_{3}$$\beta_{1}^{2}\gamma_{1}\mathsf{I}_{12}$$\beta_{1}^{2}\gamma_{1}\mathsf{I}_{6}$$\beta_{1}^{4}\mathsf{I}_{6}$$\gamma_{1}^{2}\mathsf{I}_{6}$$\beta_{1}^{2}\gamma_{1}\mathsf{I}_{6}$$\beta_{1}^{2}\gamma_{1}\mathsf{I}_{12}$$\beta_{1}^{4}\mathsf{I}_{4}$$\gamma_{1}^{2}\mathsf{I}_{3}$$\beta_{1}^{2}\gamma_{1}\mathsf{I}_{6}$$\beta_{1}^{4}\mathsf{I}_{1}$ Figure 2.2.4. Twisted Gram block matrix when $n=4$. The determinant is $\beta_{1}^{160}\gamma_{1}^{72}$. Along the row | Along the column
---|---
$(a,b,c,d)$ | $\mathsf{def}$ | $\mathsf{comp}$ | $(b,a,d,c)$ | $\mathsf{def}$ | $\mathsf{comp}$
$(0,4,0,0)$ | $4$ | $4$ | $(4,0,0,0)$ | $0$ | $4$
$(0,2,0,1)$ | $3$ | $3$ | $(2,0,1,0)$ | $0$ | $3$
$(1,3,0,0)$ | $3$ | $4$ | $(3,1,0,0)$ | $1$ | $4$
$(0,0,0,2)$ | $2$ | $2$ | $(0,0,2,0)$ | $0$ | $2$
$(0,2,1,0)$ | $2$ | $3$ | $(2,0,0,1)$ | $1$ | $3$
$(1,1,0,1)$ | $2$ | $3$ | $(1,1,1,0)$ | $1$ | $3$
$(2,2,0,0)$ | $2$ | $4$ | $(2,2,0,0)$ | $2$ | $4$
$(0,0,1,1)$ | $1$ | $2$ | $(0,0,1,1)$ | $1$ | $2$
$(1,1,1,0)$ | $1$ | $3$ | $(1,1,0,1)$ | $2$ | $3$
$(2,0,0,1)$ | $1$ | $3$ | $(0,2,1,0)$ | $2$ | $3$
$(3,1,0,0)$ | $1$ | $4$ | $(1,3,0,0)$ | $3$ | $4$
$(0,0,2,0)$ | $0$ | $2$ | $(0,0,0,2)$ | $2$ | $2$
$(2,0,1,0)$ | $0$ | $3$ | $(0,2,0,1)$ | $3$ | $3$
$(4,0,0,0)$ | $0$ | $4$ | $(0,4,0,0)$ | $4$ | $4$
Table 3. Ordering of the orbits for the Gram matrix for $n=4$.

## Acknowledgments

The authors would like to thank Mikhail Khovanov for introducing the authors
to decorated low-dimensional cobordisms, and for guidance and feedback on this
manuscript.

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