# Persistent nonlocality in an ultracold-atom environment

Bradley Longstaff  Jonatan Bohr Brask  Department of Physics, Technical
University of Denmark, Fysikvej, 2800 Kgs. Lyngby, Denmark

###### Abstract

We investigate nonlocal quantum correlations arising between multiple two-
level impurity atoms coupled to an ultracold bosonic gas. We find that the
environment-induced dynamics of the impurity subsystem can generate nonlocal
states that are robust against noise and violate a multipartite Bell
inequality when projective spin measurements are made. Genuine multipartite
nonlocality is also observed in a system of three impurities. We show that
non-Markovian effects, and the persistence of coherences in the impurity
subsystem, are crucial for preventing complete loss of nonlocality and allow
for nonlocal correlations to be generated and maintained for extended periods
of time.

Quantum physics allows for correlations that have no classical counterpart. In
modern quantum information science, entanglement plays a key role as a
resource [1], enabling communication protocols such as quantum teleportation,
quantum-over-classical speed-ups in computation [2], and measurement precision
beyond the classical shot-noise limit [3]. The strongest quantum correlations
defy explanation in terms of a local theory [4, 5]. As shown by Bell [5], the
assumption of local causality places constraints on the possible observations
for parties that do not communicate, known as Bell inequalities. These
constraints can be violated in quantum mechanics when the parties share an
entangled state on which they each make local measurements, thus allowing the
observation of Bell nonlocality [6]. Nonlocality has been decisively
demonstrated in experiment [7, 8, 9] and is recognised to enable information
processing at an unprecedented level of security, such as device-independent
random number generation [10, 11] and quantum key distribution [12].

Understanding how and what quantum correlations arise in interacting systems
thus presents an intriguing question, and generating and stabilising them has
many potential applications. In particular, the study of correlations is key
to the characterisation of many-body systems and many-body systems can support
a wide range of correlations [13]. The problem of generating and detecting
nonlocality is related but distinct from that of entanglement, as entanglement
is necessary but not sufficient for nonlocality [14, 15]. Nonlocality in many-
body systems is more challenging to observe [16, 17, 18, 19] but does arise,
as shown in a number of works, e.g. [20, 21, 22, 23, 24, 25, 26]. At the same
time, engineered environments have enabled dissipative entangled state
preparation, where the effects of noise and loss are used to an advantage
rather than being a hindrance [27, 28]. Steady-state entanglement generation
is also possible with these techniques [29, 30, 31, 32] and can even be
achieved via incoherent interactions with thermal environments [33, 34]. In
the latter case, it has recently been suggested that the generated
entanglement be quantified based on its usefulness for particular tasks,
including Bell inequality violation [35]. However, similar techniques for
nonlocality generation remain largely unexplored [36].

In this work we consider the generation of nonlocality between two-level
impurity atoms coupled to an ultracold bosonic gas, which is trapped in a one-
dimensional optical lattice and described by the Bose-Hubbard model [37]. The
subsystem of impurity atoms can be viewed as an open system in a controllable
environment [38]. Previous analyses of a single impurity atom embedded in a
Bose-Hubbard lattice have found that the pure-dephasing dynamics of the
impurity can be non-Markovian [38, 39, 40]. This observation partially
motivates our choice of this model for nonlocality generation. Suppose that
the system-environment interactions can generate nonlocality in the impurity
subsystem. By viewing non-Markovianity in terms of information backflow from
the environment to the system [41], one might suspect that memory effects
could suppress the destruction of nonlocal correlations due to decoherence.
Indeed, previous work examining entanglement generation between two impurities
reported on the possibility of entanglement trapping [42], where non-Markovian
effects completely halt the dephasing and thus the loss of entanglement.

Here we demonstrate that memory effects can lead to persistent nonlocality, in
which the reduced state of the impurity atoms evolves from a separable state
into a nonlocal state and remains nonlocal for an extended period of time. Not
only is the resulting nonlocality found to be robust against noise but it also
survives in large systems. Note that the focus is on nonlocality generation
without the precise dynamical control usually involved in realising gates [43,
44]. In this respect, our work is similar in spirit to autonomous steady-state
entanglement generation [29, 30, 31, 32, 33, 34].

A system of $N$ ultracold atoms in a one-dimensional optical lattice is well
described by the Bose-Hubbard Hamiltonian

$\hat{H}_{BH}=-J\sum_{j=1}^{M}\left(\hat{a}_{j+1}^{\dagger}\hat{a}_{j}+\hat{a}_{j}^{\dagger}\hat{a}_{j+1}\right)+\frac{U}{2}\sum_{j=1}^{M}{\hat{a}_{j}^{\dagger}}{}^{2}\hat{a}_{j}^{2},$
(1)

where $j$ labels the lattice sites and the bosonic operators
$\hat{a}_{j}^{\dagger}$ ($\hat{a}_{j}$) create (annihilate) an atom localised
on site $j$. The hopping between neighbouring lattice sites is quantified by
$J>0$, and $U>0$ is the pairwise on-site interaction strength. We consider a
ring-shaped lattice of size $M$, such that $\hat{a}_{M+1}=\hat{a}_{1}$.
Impurity atoms of a different species are then embedded in the Bose-Hubbard
lattice [45], with at most one impurity atom per lattice site. Setups
introducing multiple atomic impurities into a large Bose gas have been
demonstrated experimentally [46, 47, 48]. Only the two lowest internal states
of each impurity are assumed to contribute to the dynamics, so that each
impurity can be treated as a qubit. The excited and ground states are labelled
$|1\rangle$ and $|0\rangle$ respectively. If the qubits are coupled to the
Bose gas via a contact density-density interaction then the total Hamiltonian
has the form [49, 50]

$\hat{H}=\hat{H}_{BH}+\frac{\omega_{0}}{2}\sum_{j=1}^{d}\hat{\sigma}^{z}_{j}+\eta\sum_{j=1}^{d}|1\rangle\langle
1|_{j}\otimes\hat{n}_{l_{j}},$ (2)

where $d$ is the number of impurity atoms.

The first and second terms are the free Hamiltonians of the Bose gas (1) and
the qubits respectively. The third term describes qubit-gas interactions, with
the excited state of the qubit on site $l_{j}$ coupling to the bosonic number
operator $\hat{n}_{l_{j}}=\hat{a}^{\dagger}_{l_{j}}\hat{a}_{l_{j}}$ with
strength $\eta\geq 0$. For simplicity, we assume that the qubits are
degenerate and couple with equal strength to the gas, such that $\omega_{0}$
and $\eta$ are the same on all lattice sites.

The dynamics generated by the Bose-Hubbard Hamiltonian are generally
complicated and approximations are needed in order to obtain analytic results.
To this end, we assume that the gas is prepared in its ground state and work
in the superfluid regime $U/J\ll 1$. In this case the Bogoliubov approximation
[51] can be applied to bring the Hamiltonian (2) into the form

$\displaystyle\hat{H}=\sum_{k}\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}+\frac{\overline{\omega}_{0}}{2}\sum_{j=1}^{d}\hat{\sigma}_{j}^{z}$
(3)
$\displaystyle+\eta\sum_{j=1}^{d}\sum_{k}\sqrt{\frac{\epsilon_{k}n_{0}}{\omega_{k}M}}|1\rangle\langle
1|_{j}\otimes\left(\hat{b}_{k}^{\dagger}\mathrm{e}^{ikal_{j}}+\hat{b}_{k}\mathrm{e}^{-ikal_{j}}\right),$

where we have neglected an overall constant term and introduced the
renormalised transition frequency $\overline{\omega}_{0}=\omega_{0}+\eta
n_{0}$ in terms of the quasicondensate density $n_{0}$. The operators
$\hat{b}_{k}^{\dagger}$ ($\hat{b}_{k}$) create (annihilate) Bogoliubov
quasiparticles with quasimomentum $k$ and satisfy the canonical commutation
relations. The sums are taken over $k\neq 0$ and the quasiparticle dispersion
relation $\omega_{k}=\sqrt{\epsilon_{k}^{2}+2Un_{0}\epsilon_{k}}$ depends on
the single-particle energies $\epsilon_{k}=4J\sin^{2}\left(ka/2\right)$, where
$a$ is the lattice constant.

From here on we shall often refer to the ultracold gas as the ‘environment’
and the subsystem of qubits as the ‘system’. We assume that the environment
and the system are initially uncorrelated
$\hat{\rho}(0)=\hat{\rho}_{S}(0)\otimes\hat{\rho}_{E}$, where
$\hat{\rho}_{S}(0)$ is an arbitrary $d$-qubit state and $\hat{\rho}_{E}$ is
the ground state of the Bose-Hubbard Hamiltonian (1). With the Bogoliubov
approximation it is then possible to derive an analytic expression for system
state
$\hat{\rho}_{S}(t)=\textnormal{tr}_{E}[\mathrm{e}^{-it\hat{H}}\hat{\rho}(0)\mathrm{e}^{it\hat{H}}]$,
where $\hat{H}$ is the Hamiltonian (3) and the partial trace is taken over the
environment. The matrix elements of the system state in the eigenbasis of the
free qubit Hamiltonian are found to be of the form

$[\hat{\rho}_{S}(t)]_{ij}=\mathrm{e}^{-\gamma_{ij}(t)}\mathrm{e}^{i\varphi_{ij}(t)}[\hat{\rho}_{S}(0)]_{ij},$
(4)

where $i=(i_{1},\ldots,i_{d})$ labels a basis vector with $i_{j}\in\\{0,1\\}$,
and the time-dependent parameters $\gamma_{ij}(t)$ and $\varphi_{ij}(t)$ are
real valued. A derivation of this result is provided in Appendix A but there
are several details worth noting here.

First, the time evolution of $\hat{\rho}_{S}$ consists of a unitary and a non-
unitary part. In the unitary part, we find that the coupling of the qubits to
the ultracold gas induces a time-dependent qubit-qubit interaction with terms
of the form $c_{ij}(t)\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z}$. That is, each
qubit couples to all the others through a $ZZ$ interaction with a time-
dependent strength that varies between different pairs of qubits. We shall see
that this interaction can generate a nonlocal state from a separable initial
state. To give some insight into how this can happen, consider an interaction
of the form $g\hat{\sigma}_{1}^{z}\hat{\sigma}_{2}^{z}$ with $g\in\mathbb{R}$.
Two qubits initialised in the spin-$x$ up state $|+,+\rangle$ are mapped to
the state $\cos(gt)|+,+\rangle-i\sin(gt)|-,-\rangle$. When $gt=\pi/4$ the
separable initial state has evolved into a maximally entangled state.

Figure 1: Generation of tripartite nonlocality (A) and genuine tripartite
nonlocality (B) between three qubits coupled to a three-site Bose-Hubbard
ring. Bell inequality violation vs. time and coupling strength is shown on the
left of each panel. Positive values indicate violation. Noise robustness is
shown on the right as $p^{\ast}$ vs. time. Smaller $p^{\ast}$ indicates
greater robustness. The results are for an interacting ultracold gas of
$N=100$ particles and interaction strength $UN=2J$. The qubit energy is
$\omega_{0}=1$.

On the other hand, the non-unitary part gives rise to dephasing described by
the parameters

$\gamma_{ij}(t)=\sum_{k}\nu_{k}\sin^{2}\left(\frac{\omega_{k}t}{2}\right)\left|\sum_{m=1}^{d}\mathrm{e}^{ikal_{m}}(i_{m}-j_{m})\right|^{2},$
(5)

with $\nu_{k}=2\eta^{2}n_{0}\epsilon_{k}/\omega_{k}^{3}M$. This only affects
the off-diagonal elements of the state and generally degrades any nonlocal
correlations that may have built up. However, due to the sinusoidal factor,
these parameters need not be monotonically increasing. There can be times when
the time rate of change $\dot{\gamma}_{ij}(t)<0$, corresponding to a backflow
of information from the environment to the system.

To understand this, recall that the trace distance can be viewed as a measure
of distinguishability between two quantum states. If the trace distance, and
hence distinguishablity, between two states is increasing at some instant of
time, then there must be a flow of information from the environment to the
system [41]. In Appendix B we show that if $\dot{\gamma}_{ij}(t)<0$ at some
time $t$ for some pair of indices $(i,j)$, then one can always find a pair of
initial states such that the trace distance distance between them is
increasing at time $t$.

The parameters $\gamma_{ij}(t)$ are proportional to $\eta^{2}$ and, as
expected, the strength of the dephasing is amplified when the interaction
between the system and environment is increased. Furthermore, the coefficients
$c_{ij}(t)$ in the $ZZ$ interaction are also proportional to $\eta^{2}$. We
therefore expect there to be a trade-off between generating nonlocality slowly
over an extended period of time, and quickly generating nonlocality that only
lasts for a short period of time, due to a large dephasing rate.

Finally, from the solution (4) we see that the eigenstates of
$\hat{\sigma}_{1}^{z}\dots\hat{\sigma}_{d}^{z}$ are stationary states of the
approximate dynamics. Thus, to observe nonlocality generation, and non-trivial
time evolution, the initial state cannot be a product of spin-$z$ eigenstates.
Guided by the remarks above equation (5) we shall assume that all the qubits
are initially prepared in the spin-$x$ up state $|+,\dots,+\rangle$.

In general, to determine whether a state is nonlocal is a challenging problem
in the multipartite setting [6]. All tight correlation Bell inequalities for
$d$ parties, each with a choice of two dichotomic measurements, were derived
by Werner and Wolf [52] and Żukowski and Brukner [53]. These $2^{2^{d}}$
inequalities can be expressed as a single nonlinear inequality

$\sum_{r}|\tilde{\xi}(r)|\leq 1$ (6)

which we refer to as the WWZB inequality. Here
$\tilde{\xi}(r)=2^{-d}\sum_{s}(-1)^{r\cdot s}\xi(s)$, $r,s$ are vectors in
$\\{0,1\\}^{d}$ and
$\xi(s)=\langle\hat{A}_{s_{1}}^{(1)}\dots\hat{A}_{s_{d}}^{(d)}\rangle$, where
the components $s_{j}$ label the choice of the $\pm 1$-valued observable
$\hat{A}_{s_{j}}^{(j)}$ made on subsystem $j$.

Various forms of nonlocality can be distinguished in the multipartite setting.
For instance, it is possible for nonlocal correlations to be local with
respect to some bipartition. On the other hand, when all parties are
nonlocally correlated the correlations are said to be genuine multipartite
nonlocal (GMNL). This is the strongest form of multipartite nonlocality.
However, it is generally difficult to test if a state is GMNL [54, 6]. Here we
make use of the Bell inequality introduced in [55], a violation of which
detects genuine tripartite nonlocality (GTNL), and find that GTNL can be
established between three impurity atoms. For brevity we refer to this
inequality as the GTNL inequality and provide the definition in Appendix C.

Let us first examine a three-site Bose-Hubbard ring with a single impurity
atom coupled to each site. In this case, we can numerically solve the
Schrödinger equation with the many-body Hamiltonian (2) exactly to obtain the
reduced state of the three qubits. We test the WWZB inequality (6) using
projective spin measurements
$\hat{A}_{s_{j}}^{(j)}=\mathbf{u}_{s_{j}}\cdot\hat{\boldsymbol{\sigma}}_{j}$
on each of the $j\in\\{1,2,3\\}$ qubits. Here $\hat{\boldsymbol{\sigma}}_{j}$
is a vector of Pauli operators acting on qubit $j$ and the unit vector
$\mathbf{u}_{s_{j}}$ is the Bloch vector of the measurement. We also test the
GTNL inequality with projective spin measurements (see Appendix C). At each
instant of time a maximisation of each inequality was performed over the
twelve angles parameterising the measurements.

In panels A and B of Fig. 1 we plot respectively the degree of violation of
the WWZB and GTNL inequalities against coupling strength and time. We see that
nonlocal states can be generated from separable states across a wide range of
parameters. Furthermore, across the entire parameter space we see oscillations
in the degree of Bell inequality violation, which is a non-Markovian effect
due to information backflow from the environment. Using (5) we find that
$\dot{\gamma}_{ij}(t)<0$ if and only if $\sin(\omega_{k}t)<0$, where
$\omega_{k}$ is evaluated at $k=2\pi/3a$. Information thus flows back and
forth between the system and environment, and information flow from the
environment to the system vanishes at the times $t=2n\pi/\omega_{k}$ with
$n\in\mathbb{N}$. These are indicated by dashed-black vertical lines and
coincide with the revivals of nonlocality.

For coupling strength $\eta\approx 0.2J$, we observe that nonlocality is
generated and persists for a long period of time. This illustrates the trade-
off between dephasing and unitary dynamics. Due to the non-Markovianity of the
dynamics, the information lost to the environment quickly returns and complete
loss of nonlocality is avoided.

The noise robustness of nonlocality can be quantified by subjecting
$\hat{\rho}_{S}(t)$ to completely depolarising noise [56]

$\tilde{\rho}(p)=p\hat{\rho}_{S}(t)+(1-p)\frac{\hat{I}}{2^{d}}.$ (7)

The smallest value $p^{\ast}$ such that $\tilde{\rho}(p^{\ast})$ is nonlocal
provides a measure of robustness. Panel A in Fig. 1 illustrates that robust
nonlocality ($p^{\ast}\approx 0.5$) can be generated quickly for larger $\eta$
but is short lived. For weaker coupling strength, nonlocality builds up more
slowly but remains robust over a longer time. In frame B of Fig. 1 we see that
the GTNL is less robust against noise but the behaviour with respect to $\eta$
is similar to the tripartite nonlocal case.

Using the approximate solution (4), we are also able to investigate
nonlocality generation beyond the regime where exact numerical solutions are
possible. The approximate solution is expected to be valid provided that the
ultracold gas remains close to the ground state. We compared the approximate
and exact numerical solutions for small, numerically tractable systems and
found excellent agreement between the two. In Fig. 2 we consider a five-site
Bose-Hubbard ring with one qubit coupled to each site. We again observe that
separable states can evolve into robust nonlocal states that violate the WWZB
inequality for projective spin measurements.

To illustrate the importance of non-Markovianity on the nonlocality generation
we also plot the Breuer-Laine-Piilo (BLP) measure of non-Markovianity
$\mathcal{N}$ [41] in the right panel of Fig. 2. This quantifies the total
amount of information flowing back into the system from the environment and
can be written as

$\mathcal{N}=\max_{\hat{\rho}_{1,2}(0)}\sum_{i}\left[D(\hat{\rho}_{1}(b_{i}),\hat{\rho}_{2}(b_{i}))-D(\hat{\rho}_{1}(a_{i}),\hat{\rho}_{2}(a_{i}))\right].$
(8)

The sum is taken over all time intervals $(a_{i},b_{i})$ where the trace
distance $D(\hat{\rho}_{1}(t),\hat{\rho}_{2}(t))$ between the two initial
states states $\hat{\rho}_{1,2}(0)$ is increasing. Thus, whenever
$\mathcal{N}$ increases there is information flowing from the environment to
the system. In Fig. 2 we observe that the oscillations in the robustness
parameter $p^{\ast}$ coincide with the information flow into and out of the
system. This confirms the intuition that memory effects are crucial in
preventing the complete loss of nonlocality.

Figure 2: Generation of nonlocality among five qubits coupled to a five-site
Bose-Hubbard ring with strength $\eta=0.05J$. (Left) The noise robustness of
nonlocality vs. time. Smaller $p^{\ast}$ indicates greater robustness. (Right)
Non-Markovianity vs. time. Results are for a gas of $N=1000$ particles with
interaction strength $UN=2J$ and qubit energy $\omega_{0}=1$. Figure 3: Two
impurity atoms embedded in a lattice of size $M$ at sites $1$ and $M$
respectively. (Left) The noise robustness of nonlocality vs. time when the
non-unitary part of the dynamics is neglected. Smaller $p^{\ast}$ indicates
greater robustness. (Right) Non-Markovianity vs. time. Results are for a gas
of $N=1000$ particles with interaction strength $UN=2J$, coupling strength
$\eta=0.04J$ and qubit energy $\omega_{0}=1$.

The examples above focused on the case $d=M$, with one impurity embedded in
each lattice site. When the number of lattice sites is increased, with the
number of impurities and other parameters fixed, generating nonlocal
correlations in a short time can become difficult. There are two main reasons
for this. The first concerns the unitary part of the dynamics, generated by
the effective Hamiltonian (see Appendix A)

$\hat{h}(t)=\sum_{j=1}^{d}\omega_{j}(t)\hat{\sigma}_{j}^{z}+\sum_{j>m}^{d}\sum_{m=1}^{d-1}c_{jm}(t)\hat{\sigma}_{j}^{z}\hat{\sigma}_{m}^{z}.$
(9)

This has a complicated dependence on the system parameters, as illustrated in
the left panel of Fig. 3 for two impurities embedded on sites $1$ and $M$
respectively. Within the time scale considered, we observe that the unitary
part generates increasingly weaker nonlocality as the lattice size increases
from $M=2$ to $M=6$, at which point essentially zero nonlocality is generated.
Nonlocal correlations appear again when $M$ is increased beyond this point.
Furthermore, the dynamics of the impurities become ‘more Markovian’ (with a
smaller BLP measure of non-Markovianity) when the lattice size increases [57].
This is illustrated in the right panel of Fig. 3. A larger $M$ results in
longer time intervals of information loss to the environment and less
information backflow. Fig. 3 shows how these two effects can combine together
to suppress nonlocality generation at short times. However, at longer times
strong persistent nonlocal correlations can be generated, even when the
separation between the impurities is large. This raises the question of
whether persistent nonlocality is possible in large lattices.

There is no condensate in an infinite one-dimensional lattice. Taking the
continuum limit with the substitutions $\sum_{k}\to M\int_{-1/2}^{1/2}dq$ and
$k\to 2\pi q/a$ leads to a divergent momentum integral in the expression for
the condensate density [51]. In a large but finite system, this divergence can
be avoided by introducing a low-momentum cut off $q_{0}=1/M$, where the number
of lattice sites $M$ is large. This yields the dephasing parameters

$\gamma_{ij}(t)=4\eta^{2}n_{0}\int_{q_{0}}^{1/2}\frac{\epsilon_{q}}{\omega_{q}^{3}}\sin^{2}\left(\frac{\omega_{q}t}{2}\right)S_{ij}(q)\,dq.$
(10)

Here $\epsilon_{q}=4J\sin^{2}(\pi q)$,
$\omega_{q}=\sqrt{\epsilon_{q}^{2}+2Un_{0}\epsilon_{q}}$ and

$S_{ij}(q)=\left|\sum_{m=1}^{d}\mathrm{e}^{2i\pi
ql_{m}}(i_{m}-j_{m})\right|^{2}.$ (11)

The same substitutions can also be used to obtain a large lattice expression
for the term $\varphi_{ij}(t)$ in the approximate solution (4).

For simplicity we focus on the dynamics of two impurity atoms embedded in a
large lattice. In this case we can apply the Horodecki criterion to quantify
the degree of Bell inequality violation [58, 59] (see Appendix D), avoiding
the need to optimise over measurements. For a pair of qubits there are three
different decoherence parameters that appear in the density matrix. The first,
$\Gamma_{0}(t)\equiv\gamma_{(1,1),(1,0)}(t)$, is given by

$\Gamma_{0}(t)=4\eta^{2}n_{0}\int_{q_{0}}^{1/2}\frac{\epsilon_{q}}{\omega_{q}^{3}}\sin^{2}\left(\frac{\omega_{q}t}{2}\right)dq$
(12)

and is the dephasing rate that a single impurity atom experiences. The other
two, $\Gamma_{\pm}(t)$, which are defined as $\gamma_{(1,1),(0,0)}(t)$ and
$\gamma_{(1,0),(0,1)}(t)$ respectively, have the form
$\Gamma_{\pm}(t)=2\Gamma_{0}(t)\pm\Gamma(t)$ with

$\Gamma(t)=8\eta^{2}n_{0}\int_{q_{0}}^{1/2}\frac{\epsilon_{q}}{\omega_{q}^{3}}\sin^{2}\left(\frac{\omega_{q}t}{2}\right)\cos\left(2\pi
q(l_{1}-l_{2})\right)\,dq.$ (13)

The decoherence rates $\Gamma_{\pm}(t)$ are not simply twice the single
impurity dephasing rate. Rather, one is larger and one is smaller than
$\Gamma_{0}(t)$. The system is said to exhibit super and subdecoherence. This
feature of collective decoherence is known to occur when two qubits interact
with a shared bosonic environment via a spin-boson type interaction [60, 61].

The time evolution of $\Gamma_{\pm}(t)$ is plotted in Fig. 4 for a pair of
qubits embedded in neighbouring lattice sites. The rate $\Gamma_{-}(t)$
exhibits small oscillations, indicating (weak) non-Markovian behaviour. More
importantly, the rates level out around a finite value, resulting in the
persistence of coherences in the impurity subsystem at long times. A similar
effect has previously been observed in a set up consisting of impurity atoms,
each trapped in a double-well potential and interacting with a homogeneous
Bose gas [61]. Because the coherences are preserved, long-time generation and
persistence of nonlocality is possible. This is illustrated in Fig. 5 for two
impurities coupled to a large lattice with $10^{6}$ sites.

Figure 4: Decoherence rates $\Gamma_{+}$ (left) and $\Gamma_{-}$ (right) vs.
time. Results show the long-time evolution for a gas with $n=1$ atoms per
lattice site, interaction strength $U=0.04J$, coupling strength $\eta=0.03J$
and qubit energy $\omega_{0}=1$. The low-momentum cut off $q_{0}=10^{-6}$,
corresponding to $10^{6}$ lattice sites. Figure 5: Nonlocality generation in a
large system. Measure of Bell inequality violation $\mathcal{B}$ vs. time.
Values greater than zero indicate Bell inequality violation, and
$\mathcal{B}=1$ corresponds to a maximal violation of the CHSH inequality for
some set of measurements. The parameters are the same as Fig. 4

In conclusion, we have demonstrated that the system-environment coupling of
impurity qubits in a cold Bose gas can generate multipartite, nonlocal
correlations that persist over long times. The setup can be realised with
current technology and the procedure is simple in the sense that no dynamic
driving or control is needed, except for initialisation and read out. We have
shown that robust 3- and 5-partite nonlocality can be observed (tolerating up
to $\sim 50\%$ depolarising noise) as well as genuinely tripartite
nonlocality. We note that the observation of (genuine) multipartite
nonlocality also witnesses the presence of (genuine) multipartite
entanglement. The number of parties studied here is limited by numerics, but
we expect that nonlocality between an arbitrary number of parties is possible.

To gain insight into the many-body dynamics we made the Bogoliubov
approximation and derived an approximate solution for the reduced state of the
impurities. The time evolution was found to consist of two parts: an effective
unitary dynamics, generating the nonlocality; and a non-unitary part,
resulting in dephasing that tends to destroy the nonlocal correlations. We
observed that non-Markovian effects in the reduced-state dynamics are crucial
for maintaining nonlocality over long periods of time. Here, memory effects
suppress the complete loss of information to the environment. This feature
could be particularly useful in experimental setups that require flexibility
in the manipulation of the impurities. Furthermore, we demonstrated that
persistent nonlocality can still occur in large systems. This is possible due
to the saturation of decoherence rates, resulting in sustained coherences in
the impurity subsystem.

The analytic solution obtained using the Bogoliubov approximation is only
expected to be valid provided that the ultracold gas remains close to its
ground state. Surprisingly, in our numerical simulations we observed that the
approximation can still give a reliable indication of nonlocality generation
in situations where the trace distance between the exact numerical and
approximate solutions is large. This suggests that the properties of the total
state that the Bogoliubov approximation fails to account for may not play a
crucial role in determining whether the reduced state is nonlocal. Exploring
this connection could be beneficial for further understanding nonlocality
generation in many-body systems.

Finally, there has been growing interest in the use of impurity atoms embedded
in many-body environments, from nondestructive measurements of Bose-Einstein
condensate phase fluctuations [62] and correlations in ultracold-atom systems
[49, 50], to temperature measurements of Bose-Einstein condensates [63]. In
each of these examples information about a many-body system is extracted by
making measurements on a probe system that typically has a simple structure,
e.g., a two-level impurity atom. An interesting future research direction
would be to examine whether nonlocal probe states can provide an advantage in
applications of this kind.

We gratefully acknowledge support from the Carlsberg Foundation CF19-0313 and
the Independent Research Fund Denmark 7027-00044B.

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##  Appendix A Derivation of the approximate solution

Here we provide a derivation of the approximate solution (4) in the main text.
We refer to the subsystem of impurities as the ‘system’ and the ultracold gas
as the ‘environment’. We work in the zero-temperature limit and assume that
the system and environment are initially uncorrelated
$\hat{\rho}(0)=\hat{\rho}_{S}(0)\otimes\hat{\rho}_{E}$. Here
$\hat{\rho}_{S}(0)$ is an arbitrary $d$-qubit state and
$\hat{\rho}_{E}=|v\rangle\langle v|$ is the Bogoliubov vacuum state with
$|v\rangle=|0,\ldots,0\rangle$.

The dynamics of the total system-environment state are generated by the
Hamiltonian (3) in the main text, where the Bogoliubov approximation has been
made. Setting

$\hat{H}_{0}=\sum_{k}\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}+\frac{\overline{\omega}_{0}}{2}\sum_{j=1}^{d}\hat{\sigma}_{j}^{z}$
(14)

yields the interaction-picture Hamiltonian

$\hat{H}_{I}(t)=\sum_{j=1}^{d}\sum_{k}\xi_{k}|1\rangle\langle
1|_{j}\otimes\left(\hat{b}_{k}^{\dagger}\mathrm{e}^{i(\omega_{k}t+kal_{j})}+h.c.\right),$
(15)

where we have defined $\xi_{k}=\eta\sqrt{n_{0}\epsilon_{k}/\omega_{k}M}$. In
order to solve the Schrödinger equation we use the Magnus expansion [64] to
obtain the time-evolution operator $\hat{U}_{I}(t)$. Fortunately, in this case
the Magnus series truncates at the second term and we find
$\hat{U}_{I}(t)=\exp(\hat{\Omega}_{1}(t)+\hat{\Omega}_{2}(t))$, where

$\hat{\Omega}_{1}(t)=\sum_{j=1}^{d}\sum_{k}|1\rangle\langle
1|_{j}\otimes\left(\beta_{jk}(t)\hat{b}_{k}^{\dagger}-h.c.\right),$ (16)

with
$\beta_{jk}(t)={\xi_{k}}\mathrm{e}^{ikal_{j}}\left(1-\mathrm{e}^{i\omega_{k}t}\right)/{\omega_{k}}$,
and

$\displaystyle\hat{\Omega}_{2}(t)=-i\sum_{j=1}^{d}$
$\displaystyle\left(\sum_{m=1}^{d}c_{jm}(t)\right)\hat{\sigma}_{j}^{z}$
$\displaystyle-i\sum_{j>m}^{d}\sum_{m=1}^{d-1}c_{jm}(t)\hat{\sigma}_{j}^{z}\hat{\sigma}_{m}^{z},$
(17)

where we have neglected a term proportional to the identity and defined the
time-dependent coefficients

$c_{jm}(t)=\frac{1}{2}\sum_{k}\left(\frac{\xi_{k}}{\omega_{k}}\right)^{2}\left(\sin(\omega_{k}t)-\omega_{k}t\right)\cos(ka\left(l_{j}-l_{m}\right)).$
(18)

Note that $\hat{\Omega}_{1}$ and $\hat{\Omega}_{2}$ commute with each other.
Furthermore, $\hat{\Omega}_{2}$ only acts on the system and also commutes with
$\hat{H}_{0}$. Going back to the Schrödinger picture we therefore find that
the coupling to the environment induces a unitary evolution
$\hat{U}(t)=\exp(-i\hat{h}(t))$ on the subsystem of qubits with

$\hat{h}(t)=\sum_{j=1}^{d}\omega_{j}(t)\hat{\sigma}_{j}^{z}+\sum_{j>m}^{d}\sum_{m=1}^{d-1}c_{jm}(t)\hat{\sigma}_{j}^{z}\hat{\sigma}_{m}^{z},$
(19)

where $\omega_{j}(t)=\overline{\omega}_{0}t/2+\sum_{m}c_{jm}(t)$.

To simplify the notation we define
$\hat{\sigma}(t)=\hat{U}(t)\hat{\rho}_{S}(0)\hat{U}^{\dagger}(t)$, which is
the system state following the unitary evolution above. The total system-
environment state at time $t$ can then be written as

$\hat{\rho}(t)=\hat{K}(t)\left(\hat{\sigma}(t)\otimes|v\rangle\langle
v|\right)\hat{K}^{\dagger}(t)$ (20)

where
$\hat{K}(t)=\exp(-it\sum_{k}\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k})\exp(\hat{\Omega}_{1}(t))$.
The system state is obtained by taking the partial trace over the environment

$\hat{\rho}_{S}(t)=\sum_{n=0}^{\infty}\langle
n|\mathrm{e}^{\hat{\Omega}_{1}(t)}|v\rangle\hat{\sigma}(t)\langle
v|\mathrm{e}^{\hat{\Omega}_{1}^{\dagger}(t)}|n\rangle,$ (21)

where the sum is taken over all $n=(n_{1},\ldots,n_{M-1})$. Because each of
the different $k$ modes in $\hat{\Omega}_{1}(t)$ commute we can write

$\mathrm{e}^{\hat{\Omega}_{1}(t)}=\prod_{k}\exp\left(\hat{X}_{k}\otimes\hat{b}_{k}^{\dagger}-\hat{X}_{k}^{\dagger}\otimes\hat{b}_{k}\right)$
(22)

in terms of the operators
$\hat{X}_{k}=\sum_{j=1}^{d}\beta_{jk}(t)|1\rangle\langle 1|_{j}$, where the
$\beta_{jk}(t)$ are defined in (16). The operators $\langle
n|\mathrm{e}^{\hat{\Omega}_{1}(t)}|v\rangle$ appearing in (21) therefore
factorise into products of operators of the form $\langle
n_{k}|\exp\left(\hat{X}_{k}\otimes\hat{b}_{k}^{\dagger}-\hat{X}_{k}^{\dagger}\otimes\hat{b}_{k}\right)|0\rangle$,
which, after applying the Baker-Campbell-Hausdorff formula, can be written as

$\frac{1}{\sqrt{n_{k}!}}\mathrm{e}^{-\frac{1}{2}\hat{X}_{k}^{\dagger}\hat{X}_{k}}\hat{X}_{k}^{n_{k}}.$
(23)

The $\hat{X}_{k}$ are diagonal in the standard basis, with time-dependent
matrix elements

$x_{i}^{k}(t)=\frac{\xi_{k}}{\omega_{k}}\left(1-\mathrm{e}^{i\omega_{k}t}\right)\sum_{m=1}^{d}i_{m}\mathrm{e}^{ikal_{m}}.$
(24)

Here $i=(i_{1},\ldots,i_{d})$ with $i_{j}\in\\{0,1\\}$ is the matrix element
index and $k$ labels the quasimomentum value.

After a little algebra we obtain the matrix elements of the system state in
the standard basis

$[\hat{\rho}_{S}(t)]_{ij}=\prod_{k}\langle
x_{j}^{k}(t)|x_{i}^{k}(t)\rangle[\hat{\sigma}(t)]_{ij},$ (25)

where $|x_{j}^{k}(t)\rangle$ is a Glauber coherent state. It is
straightforward to work out the action of the unitary part (19) on the system
and obtain an expression for $[\hat{\sigma}(t)]_{ij}$. Inserting this into
(25) finally gives

$[\hat{\rho}_{S}(t)]_{ij}=\mathrm{e}^{-\gamma_{ij}(t)}\mathrm{e}^{i\varphi_{ij}(t)}[\hat{\rho}_{S}(0)]_{ij},$
(26)

where the $\gamma_{ij}(t)$ are defined in equation (5) of the main text and

$\displaystyle\varphi_{ij}(t)=\sum_{s=1}^{d}\omega_{s}(t)\left((-1)^{i_{s}}-(-1)^{j_{s}}\right)$
$\displaystyle+\sum_{r>s}^{d}\sum_{s=1}^{d-1}c_{rs}(t)\left((-1)^{j_{r}+j_{s}}-(-1)^{i_{r}+i_{s}}\right)$
$\displaystyle+\sum_{k}\sum_{r,s=1}^{d}f_{k}^{2}(t)\sin(ka(l_{r}-l_{s}))i_{r}j_{s},$
(27)

with $f_{k}(t)=2\xi_{k}\sin\left(\omega_{k}t/2\right)/\omega_{k}$.

##  Appendix B Negative rates and non-Markovian dynamics

We show that if $\dot{\gamma}_{ij}(t)<0$ at some time $t$ for some pair of
indices $(i,j)$, then the dynamics are non-Markovian. We shall do this by
finding a pair of initial states such that their trace distance is increasing
at time $t$.

First recall that the indices $i=(i_{1},\ldots,i_{d})$ and
$j=(j_{1},\ldots,j_{d})$ label standard basis vectors, which we denote
$|i\rangle$ and $|j\rangle$ respectively. Consider the pair of pure initial
states

$|\psi_{1}(0)\rangle=\frac{1}{\sqrt{2}}\left(|i\rangle+|j\rangle\right),\quad|\psi_{2}(0)\rangle=\frac{1}{\sqrt{2}}\left(|i\rangle-|j\rangle\right),$
(28)

where the difference between the corresponding density operators is given by

$\hat{\Delta}(0)=\hat{\rho}_{1}(0)-\hat{\rho}_{2}(0)=|j\rangle\langle
i|+|i\rangle\langle j|.$ (29)

We see that in the standard basis $\hat{\Delta}(0)$ is a matrix with $1$ in
elements $(i,j)$ and $(j,i)$, and zeros elsewhere.

The time evolution of the difference is given by (26) and is of the form

$\Delta(t)=\mathrm{e}^{-\gamma_{ij}(t)}A(t).$ (30)

Here $A(t)$ is a Hermitian matrix with $\mathrm{e}^{i\varphi_{ij}(t)}$ in
element $(i,j)$, $\mathrm{e}^{-i\varphi_{ij}(t)}$ in element $(j,i)$, and
zeros elsewhere. The trace norm $\lVert
A(t)\rVert_{1}=\tr\left[\sqrt{A^{\dagger}(t)A(t)}\right]=2$. It then follows
that the trace distance between the two states $\hat{\rho}_{1}(t)$ and
$\hat{\rho}_{2}(t)$ is given by

$D(\hat{\rho}_{1}(t),\hat{\rho}_{2}(t))=\frac{1}{2}\lVert\hat{\Delta}(t)\rVert_{1}=\mathrm{e}^{-\gamma_{ij}(t)}.$
(31)

Finally, we see that

$\frac{d}{dt}D(\hat{\rho}_{1}(t),\hat{\rho}_{2}(t))=-\dot{\gamma}_{ij}(t)\mathrm{e}^{-\gamma_{ij}(t)}>0$
(32)

at time $t$ by hypothesis. The dynamics are therefore non-Markovian according
to the BLP measure.

##  Appendix C Genuine tripartite nonlocality

Here we provide the Bell-type inequality derived in [55] to test for genuine
tripartite nonlocality. Consider three parties, Alice, Bob and Charlie, each
with a choice of two measurements labelled $x,y,z\in\\{0,1\\}$ with outputs
$a,b,c\in\\{0,1\\}$. A violation of the inequality

$\displaystyle I$
$\displaystyle=-2P(A_{1}B_{1})-2P(B_{1}C_{1})-2P(A_{1}C_{1})$
$\displaystyle-P(A_{0}B_{0}C_{1})-P(A_{0}B_{1}C_{0})-P(A_{1}B_{0}C_{0})$
$\displaystyle+2P(A_{1}B_{1}C_{0})+2P(A_{1}B_{0}C_{1})+2P(A_{0}B_{1}C_{1})$
$\displaystyle+2P(A_{1}B_{1}C_{1})\leq 0,$ (33)

detects genuine tripartite nonlocality. Here,
$P(A_{i}B_{j})=P(a=0,b=0|x=i,y=j)$ and
$P(A_{i}B_{j}C_{k})=P(a=0,b=0,c=0|x=i,y=j,z=k)$.

The measurement operators defined in the main text have outcomes $\pm 1$, as
required for testing the WWZB inequality. Measurement operators for projective
spin measurements corresponding to output labels $a\in\\{0,1\\}$ take the form

$\hat{\Pi}_{a|x}=\frac{1}{2}\left[\hat{I}+(-1)^{a}\mathbf{u}_{x}\cdot\hat{\boldsymbol{\sigma}}\right],$
(34)

where the input $x\in\\{0,1\\}$ selects the unit vector $\mathbf{u}_{x}$ (the
Bloch vector of the measurement).

##  Appendix D The Horodecki criterion

In this appendix we provide the Horodecki criterion and a quantity to measure
the degree of Bell inequality violation in a two-qubit system. The Horodecki
criterion [58] provides a necessary and sufficient condition for a state
$\hat{\rho}$ to violate the CHSH inequality [65] for some set of measurements.
Given a two-qubit state $\hat{\rho}$, construct the $3\times 3$ matrix $T$
with elements $T_{ij}=\tr[\hat{\sigma}_{i}\otimes\hat{\sigma}_{j}\hat{\rho}]$,
where $\hat{\sigma}_{i}$ are the Pauli spin operators. Define
$\mathcal{M}(\hat{\rho})$ as the sum of the two largest eigenvalues of
$T^{\dagger}T$. The Horodecki criterion states that $\hat{\rho}$ violates the
CHSH inequality for some set of measurements iff $\mathcal{M}(\hat{\rho})>1$.
To quantify the degree of Bell inequality violation we use the measure
proposed in [59]

$\mathcal{B}(\hat{\rho})=\sqrt{\max\\{0,\mathcal{M}(\hat{\rho})-1\\}}.$ (35)

When $\mathcal{B}(\hat{\rho})=0$ there exists a local model and
$\mathcal{B}(\hat{\rho})=1$ corresponds to a maximal violation of the CHSH
inequality. The larger the value of $\mathcal{B}(\hat{\rho})>0$ the greater
the violation of the CHSH inequality.


Dagger Linear Logic and Categorical Quantum Mechanics

Priyaa Varshinee Srinivasan
Graduate Program in Computer Science
Doctor of Philosophy

This thesis develops the categorical proof theory for the non-compact multiplicative dagger ($\dagger$) linear logic,
and investigates its applications to Categorical Quantum Mechanics (CQM).

The existing frameworks of CQM are categorical proof theories of compact $\dagger$-linear logic, and
are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces.
This thesis describes a new non-compact framework called Mixed Unitary Categories which can
accommodate infinite dimensional systems, and develops models for the framework. To this end, it builds
on linearly distributive categories, and $*$-autonomous categories which are categorical proof theories
of (non-compact) multiplicative linear logic. The proof theory of non-compact $\dagger$-linear logic is
obtained from the basic setting of an LDC by adding a dagger functor satisfying appropriate coherences
to give a $\dagger$-LDC.

From every (isomix) $\dagger$-LDC one can extract a canonical “unitary core” which up to equivalence is the traditional
CQM framework of $\dagger$-monoidal categories. This leads to the framework of Mixed Unitary Categories (MUCs):
every MUC contains a (compact) unitary core which is extended by a (non-compact) isomix $\dagger$-LDC.
Various models of MUCs based on Finiteness Spaces, Chu spaces, Hopf modules, etc., are developed in this thesis.
This thesis also generalizes the key algebraic structures of CQM, such as observables, measurement, and
complementarity, to MUC framework. Furthermore, using the MUC framework, this thesis establishes a
connection between the complementary observables of quantum mechanics and the exponential modalities of linear logic.

CHAPTER: ACKNOWLEDGEMENTS

I thank my supervisor Dr. Robin Cockett for the trust he placed in me even though
I was not a mathematician and had no clue about Category Theory when I joined his group.
I am eternally grateful for his unwavering support and patience in helping me refine my art of learning,
and, dually, my art of teaching. I am also grateful to him for providing an inclusive, safe and a comfortable
environment for me to learn Mathematics and to do research. Thank you, Robin,
for all the proofs we worked out over a cup of tea and the group hikes.

I am thankful to Jonathan, my friend, for the countless hours he spent in teaching
me Category Theory and helping me write proofs  in the initial days of my Ph.D., without
which this journey would not have been possible. Thank you, Jonathan!

I thank my co-supervisor Dr. Gilad Gour for his patience and kind support
throughout my degree. I thank Dr. Kristine Bauer for her support as my interim supervisor
during Dr. Cockett's sabbatical and for all the fun conversations
we shared during the weekly seminars and symposiums.
I thank Dr. Renate Scheidler for her excellent mentorship during
the practicum of my SAGES teaching certification. I thank Dr. Jalal Kawash
for the kind guidance I received as his teaching assistant.  I thank Dr. Barry Sanders
for the financial support he provided me during the first three years of my degree.
I thank Dr. Bob Coecke, Dr. Jamie Vicary,  and Dr. Chris Heunen for hosting me at  the
University of Oxford and at the University of Edinburgh during my Mitacs Globallink internship.
I thank my examiners Dr. Peter Selinger, Dr. Philipp Woelfel, Dr. Carlo Maria Scandolo, and
Dr. Kristine Bauer immensely for their time and effort in reading my thesis, and their feedback.

I thank my friend and colleague Jean Simon Lemay (JS) for many examples
in this thesis, and for making my research visit to Oxford quite enjoyable. Thank you, JS,
for lending me your bicycle during the visit! Thank you, Jonathan, Amolak, JS, and Ben for proof-reading
my thesis. Thank you, Cole, for the fun and successful collaborations on the CNOT and the dagger linear logic papers.
A big thanks to all my colleagues, in the past and in the present, from Dr. Cockett's and Dr. Gour's
group for all the enlightening conversations and support. It was an honour for me to work with you all.

A ton of thanks to my friends, for making me feel home in Calgary and for all the beautiful memories -
Sunday afternoons at coffee shops, night walks at the Confederation park, dumpling parties,
road trip to Okanagan, hiking in the Rockies, hot tea, home-made delicacies, shared lunch,
weeding bees, and many more. I thank all my overseas friends for being there for me over Skype
and other mediums whenever I was in need of you.

Many thanks to Maryam, Jackie, and Katie at the front office of the Computer Science Department
for providing excellent administrative support throughout my degree, and for their gentle smile
whenever I turned up at their desk with a request.

Last, but most of all, I thank my family for their extra-ordinary support,
and encouragement in all my endeavors, and for making possible this
first ever doctorate degree in my family tree.

To my parents Shanthi and Srinivasan,

and to all the front-line workers in this world and in heaven (COVID-19).

CHAPTER: SYMBOLS, ABBREVIATIONS AND NOMENCLATURE

Symbol     Definition

$\cup$, $\bigcup$     Set union

$\sqcup$     Disjoint union of sets

$\cap$, $\bigcap$     Set intersection

$\in$      A member of

$\llbracket x_1, x_2, \cdots, x_n \rrbracket$     finite multiset

$\emptyset$     Empty set

$\{ * \}$     Singleton set

$\{ x \mid $ Condition on $x \}$     defining a set by a condition

$\X, \Y$     Categories except for $\C$ and $\R$ which sometimes mean

complex numbers, and real numbers respectively

$A, B, C, \cdots$     Objects in a category

$A \simeq B$     Object $A$ is isomorphic to object $B$

$f, g, h$     Maps in a category

$F, G $     Functors between categories

$\ox$     Tensor product

$\oa$     Par product (in linear logic)

Symbol     Definition

$\prod$     Product

$\coprod$     Coproduct

$a_\ox$     Associator natural isomorphism  for the tensor product

$a_\oa$     Associator natural isomorphism for the par product

$u_\ox^l $     Left unitor natural isomorphism for the tensor product

$\top$     Unit of the tensor product

$\bot$     Unit of the par product

$\partial^L$     Left linear distributor

$\partial^R$     Right linear distributor

$m$, $\nabla$     Multiplication for a monoid

$u$, $\triunit{0.65}$     Unit for a monoid

$d$, $\Delta$     Comultiplication for a comonoid

$e$, $\tricounit{0.65}$     Counit for a comonoid

$\dagger$, $\ddagger$     Dagger functor

$(-)^*$     Dual functor

$\overline{(-)}$     Conjugation functor

$:=$     The left-hand side is defined to the right-hand side of the equation

$\implies$     Implication

$\Leftrightarrow$     If and only if

$\exists$     there exists

$\forall $     For all

$\N$     Natural numbers

$\Z$     Integers

CHAPTER: EPIGRAPH

Dagger Linear Logic

Application of dagger linear logic to categorical quantum mechanics

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Cambridge University Press

titleHopf Algebras

titleBig toy models

journaltitleJournal of philosophical logic
titleCoalgebras, Chu spaces, and representations of physical systems

Washington, DC, USA

IEEE Computer Society

booktitleProceedings of the 19th Annual IEEE Symposium on Logic in Computer Science
seriesLICS '04
titleA Categorical Semantics of Quantum Protocols

booktitleIn the Proceedings of the 2nd International Workshop on Quantum Programming Languages (QPL '04)
titleA categorical quantum logic

journaltitleClifford Lectures
titleH*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics

IOP Publishing

journaltitleJournal of Physics A: Mathematical and Theoretical
titleCartoon computation: quantum-like computing without quantum mechanics

journaltitleTransactions of the American Mathematical Society
titleStructure theorems for a special class of Banach algebras

Oxford University Press, Inc.

titleCategory Theory

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titleMaking the stabilizer ZX-calculus complete for scalars

journaltitleLecture Notes in Physics
titlePhysics, Topology, Logic and Computation: A Rosetta Stone

Springer Berlin Heidelberg

journaltitleLecture Notes in Physics
titlePhysics, Topology, Logic and Computation: A Rosetta Stone

journaltitleElectronic Notes in Theoretical Computer Science
titleInformation processing in convex operational theories

Cambridge University Press

journaltitleMathematical Structures in Computer Science
title*-Autonomous categories and linear logic

Cambridge University Press

journaltitleMathematical Structures in Computer Science
title*-Autonomous categories and linear logic

journaltitleJournal of Pure and Applied Algebra
titleAccessible categories and models of linear logic

Prentice-Hall, Inc.

titleCategory Theory for Computing Science

journaltitleTheory and Applications of Categories
titleThe Chu construction: history of an idea

titleElectrodynamics: A Modern Geometric Approach

Berlin, Heidelberg

booktitleProceedings of the Second International Conference on Typed Lambda Calculi and Applications
seriesTLCA '95
titleWhat is a categorical model of intuitionistic linear logic?

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleHopf algebras and linear logic

journaltitleMathematical Structures in Computer Science
title! and?-Storage  as tensorial strength

journaltitleTheory and Applications of Categories
titleCartesian differential categories with storage

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleDifferential categories

journaltitleJournal of Pure and Applied Algebra
titleFeedback for linearly distributive categories: traces and fixpoints

booktitleProceedings of Ninth Conf. on Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science
titleFock space: a model of linear exponential types

journaltitleApplied Categorical Structures
titleDeep inference and probabilistic coherence spaces

journaltitleLinear logic in Computer Science
titleCategory theory for linear logicians

journaltitleApplied Categorical Structures
titleDifferential categories revisited

journaltitleJournal of Pure and Applied Algebra
titleNatural deduction and coherence for weakly distributive categories

titleNo-cloning implies unalterability of the past

Cambridge Univ. Press

titleHandbook of Categorical Algebra 1 – Basic Category Theory

journaltitleLinear Algebra and its Applications
titleCompletely positive linear maps on complex matrices

titleDagger linear logic for categorical quantum mechanics

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleIntroduction to linear bicategories

journaltitleJournal of Pure and Applied Algebra
titleLinearly distributive functors

journaltitleTheory and Applications of categories
titleProof theory for full intuitionistic linear logic, bilinear logic, and mix categories

journaltitleJournal of Pure and Applied Algebra
titleWeakly distributive categories

note“To appear in the proceedings of 4th International Conference on Applied Category Theory, 2021"
titleExponential modalities and complementarity

journaltitlearXiv preprint arXiv:1905.08877
titleComplete Positivity for Mixed Unitary Categories

journaltitleElectronic Notes in Theoretical Computer Science
noteProceedings of the 4th International Workshop on Quantum Programming Languages (QPL 2006)
titleAxiomatic Description of Mixed States From Selinger's CPM-construction

titleIntroducing categories to the practicing physicist

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titleTerminality implies non-signalling

IOP Publishing

journaltitleNew Journal of Physics
titleInteracting quantum observables: categorical algebra and diagrammatics

Elsevier BV

journaltitleElectronic Notes in Theoretical Computer Science
titlePhase Groups and the Origin of Non-locality for Qubits

journaltitleInformation and Computation
titlePictures of complete positivity in arbitrary dimension

Cambridge University Press

titlePicturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning

booktitleMathematics of Quantum Computation and Quantum Technology
titleQuantum measurements without sums

Cambridge University Press

journaltitleMathematical structures in computer science
titleA new description of orthogonal bases

booktitleInternational Workshop on Computer Science Logic
titleEnvironment and classical channels in categorical quantum mechanics

journaltitle2012 27th Annual IEEE Symposium on Logic in Computer Science
titleStrong Complementarity and Non-locality in Categorical Quantum Mechanics

booktitleIn Esparza J., Murawski A. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2017. Lecture Notes in Computer Science
titleThe Free Exponential Modality of Probabilistic Coherence Spaces.

IOP Publishing

journaltitleJournal of Physics A: Mathematical and Theoretical
titleElementary gates for cartoon computation

journaltitleArchive for Mathematical logic
titleThe structure of multiplicatives

journaltitleNew York J. Math
titleNote on Frobenius monoidal functors

North Holland

titleVon Neumann Algebras

Cambridge University Press

titleGeometric Algebra for Physicists

Cambridge University Press

titleGeometric Algebra for Physicists

Oxford University

titleTypes for Quantum Computing

New York, NY, USA

Association for Computing Machinery

booktitleProceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science
seriesLICS '16
titleInteracting Frobenius Algebras Are Hopf

journaltitleTheory and Applications of Categories
titleOn involutive monoidal categories

Mount Allison University, Department of Mathematics
Computer Science

journaltitleTheory and Applications of Categories
titleThe Frobenius relations meet linear distributivity.

journaltitleJournal of Pure and Applied Algebra
titleOn cyclic star-autonomous categories

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleFiniteness spaces

booktitleInternational Conference on Typed Lambda Calculi and Applications
titleDifferential structure in models of multiplicative biadditive intuitionistic linear logic

Elsevier BV

journaltitleAnnals of Physics
titleGeometric algebra techniques for general relativity

North Holland

titleCategories, Allegories

journaltitleTheoretical computer science
titleLinear logic

titleLinear Logic: Its Syntax And Semantics

Open Publishing Association

booktitleProceedings 13th International Conference on Quantum Physics and Logic, Glasgow, Scotland, 6-10 June 2016
seriesElectronic Proceedings in Theoretical Computer Science
titleInfinite-dimensional Categorical Quantum Mechanics

Cambridge University Press

titleIntroduction to quantum mechanics

booktitleProceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158)
titleCorrectness of multiplicative proof nets is linear

Springer Science & Business Media

titleClifford algebra to geometric calculus: a unified language for mathematics and physics

journaltitleLogical Methods in Computer Science
titleCompactly accessible categories and quantum key distribution

journaltitleLogical Methods in Computer Science
titleCompactly accessible categories and quantum key distribution

Oxford University Press

titleCategories for Quantum Theory: An Introduction

journaltitleCommunications in Mathematical Physics
titleFrobenius structures over Hilbert C*-modules

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titleCategories of relations as models of quantum theory

journaltitleJournal of Pure and Applied Algebra
titleNuclearity in the category of complete semilattices

titleThe Clifford Algebra approach to Quantum Mechanics A: The Schrödinger and Pauli Particles

Elsevier Science Publishers Ltd.

journaltitleTheoretical Computer Science
titleGlueing and orthogonality for models of linear logic

The Academy of Science of the Royal Society of Canada

journaltitleMathematical Reports
titleFree bicomplete categories

The Academy of Science of the Royal Society of Canada

journaltitleMathematical Reports
titleFree bicomplete categories

journaltitleJournal of Algebra
titleOn MacLane's conditions for coherence of natural associativities, commutativities, etc.

journaltitleJournal of Pure and Applied Algebra
titleCoherence for compact closed categories

booktitleCategory Seminar
titleDoctrinal adjunction

journaltitleLogical Methods Computer Science
titleA categorical semantics for causal structure

journaltitleTheory and Applications of Categories
titleAn extended view of the Chu-construction

Université Paris Diderot - Paris

[S.l.] : [s.n.], 1988

titleLogiques, catégories & machines: Implantation de langages de programmation guidé par la logique catégorique

journaltitleLogical Methods in Computer Science
titleLifting coalgebra modalities and MELL Model Structure to Eilenberg-Moore categories

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titleWhy FHilb is not an interesting (co)differential category

titleA Graphical Calculus for Multi-Qudit Computations with Generalized Clifford Algebras

titleAn Algebraic Framework for Multi-Qudit Computations with Generalized Clifford Algebras

titleClifford algebra, geometric algebra, and applications

Berlin/Heidelberg, Germany

Springer Science & Business Media

titleCategories for the Working Mathematician

journaltitleArchive for Mathematical Logic
titleModularity of proof-nets

Cambridge University Press

titleFoundations of Quantum Group Theory

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleAn explicit formula for the free exponential modality of linear logic

Physics Today

journaltitleMathematical Structures in Computer Science
titleWhat's wrong with this pillow?

journaltitleQuantum Inf. Comput.
titleQuantum Latin squares and unitary error bases

Princeton University Press

editionNED - New edition
titleMathematical Foundations of Quantum Mechanics: New Edition

Cambridge University Press

titleQuantum Computation and Quantum Information: 10th Anniversary Edition

Mathematical Sciences Publishers

journaltitleAlgebra & Number Theory
titleWeak Hopf monoids in braided monoidal categories

journaltitleElectronic Notes in Theoretical Computer Science
titleRelating toy models of quantum computation: comprehension, complementarity and dagger mix autonomous categories

journaltitleQuaestiones Mathematicae
titleOn categories of monoids, comonoids, and bimonoids

booktitleIn Proc. Workshop on Physics and Computation (PhysComp'92
titleLinear Logic for Generalized Quantum Mechanics

Princeton University Press

titleNon-standard analysis

noteManuscript downloadable from <www.cs.man.ac.uk/ schalk/notes/llmodel.pdf>
titleWhat is a categorical model for Linear Logic?

booktitleNew Structures for Physics
titleA survey of graphical languages for monoidal categories

journaltitleElectronic Notes in Theoretical Computer Science
titleDagger compact closed categories and completely positive maps

journaltitleElectron. Notes Theor. Comput. Sci.
titleIdempotents in Dagger Categories: (Extended Abstract)

journaltitleMathematical Structures in Computer Science
titleTowards a quantum programming language

journaltitleMathematical Structures in Computer Science
titleOn Banach spaces of sequences and free linear logic exponential modality

journaltitlePhysics Letters A
titleExpressing the operations of quantum computing in multiparticle geometric algebra

Université Paris-Diderot - Paris VII

titleSémantiques et Syntaxes Vectorielles de la Logique Linéaire
Linear Logic ; Finiteness spaces ; Linearised topology ; Quantitative semantics ; Syntactic Taylor formula ; Algebraic lambda-calculus ; Differential nets. ; Logique linéaire ; Espaces de finitude ; Topologie linéarisée ; Sémantique quantitative ; Formule de Taylor syntaxique ; lambda-calcul algébrique ; Réseaux différentiels

Oxford University Press

titleAn Introduction to Clifford Algebras and Spinors

journaltitleInternational Journal of Theoretical Physics
titleA categorical framework for the quantum harmonic oscillator

Springer Science
Business Media LLC

journaltitleCommunications in Mathematical Physics
titleCategorical formulation of finite-dimensional quantum algebras

American Physical Society (APS)

journaltitlePhysical Review A
titleClifford algebras and universal sets of quantum gates

titleQuantum gates and Clifford algebras

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Cambridge University Press

titleHopf Algebras

titleBig toy models

journaltitleJournal of philosophical logic
titleCoalgebras, Chu spaces, and representations of physical systems

Washington, DC, USA

IEEE Computer Society

booktitleProceedings of the 19th Annual IEEE Symposium on Logic in Computer Science
seriesLICS '04
titleA Categorical Semantics of Quantum Protocols

booktitleIn the Proceedings of the 2nd International Workshop on Quantum Programming Languages (QPL '04)
titleA categorical quantum logic

journaltitleClifford Lectures
titleH*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics

IOP Publishing

journaltitleJournal of Physics A: Mathematical and Theoretical
titleCartoon computation: quantum-like computing without quantum mechanics

journaltitleTransactions of the American Mathematical Society
titleStructure theorems for a special class of Banach algebras

Oxford University Press, Inc.

titleCategory Theory

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titleMaking the stabilizer ZX-calculus complete for scalars

journaltitleLecture Notes in Physics
titlePhysics, Topology, Logic and Computation: A Rosetta Stone

Springer Berlin Heidelberg

journaltitleLecture Notes in Physics
titlePhysics, Topology, Logic and Computation: A Rosetta Stone

journaltitleElectronic Notes in Theoretical Computer Science
titleInformation processing in convex operational theories

Cambridge University Press

journaltitleMathematical Structures in Computer Science
title*-Autonomous categories and linear logic

Cambridge University Press

journaltitleMathematical Structures in Computer Science
title*-Autonomous categories and linear logic

journaltitleJournal of Pure and Applied Algebra
titleAccessible categories and models of linear logic

Prentice-Hall, Inc.

titleCategory Theory for Computing Science

journaltitleTheory and Applications of Categories
titleThe Chu construction: history of an idea

titleElectrodynamics: A Modern Geometric Approach

Berlin, Heidelberg

booktitleProceedings of the Second International Conference on Typed Lambda Calculi and Applications
seriesTLCA '95
titleWhat is a categorical model of intuitionistic linear logic?

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleHopf algebras and linear logic

journaltitleMathematical Structures in Computer Science
title! and?-Storage  as tensorial strength

journaltitleTheory and Applications of Categories
titleCartesian differential categories with storage

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleDifferential categories

journaltitleJournal of Pure and Applied Algebra
titleFeedback for linearly distributive categories: traces and fixpoints

booktitleProceedings of Ninth Conf. on Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science
titleFock space: a model of linear exponential types

journaltitleApplied Categorical Structures
titleDeep inference and probabilistic coherence spaces

journaltitleLinear logic in Computer Science
titleCategory theory for linear logicians

journaltitleApplied Categorical Structures
titleDifferential categories revisited

journaltitleJournal of Pure and Applied Algebra
titleNatural deduction and coherence for weakly distributive categories

titleNo-cloning implies unalterability of the past

Cambridge Univ. Press

titleHandbook of Categorical Algebra 1 – Basic Category Theory

journaltitleLinear Algebra and its Applications
titleCompletely positive linear maps on complex matrices

titleDagger linear logic for categorical quantum mechanics

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleIntroduction to linear bicategories

journaltitleJournal of Pure and Applied Algebra
titleLinearly distributive functors

journaltitleTheory and Applications of categories
titleProof theory for full intuitionistic linear logic, bilinear logic, and mix categories

journaltitleJournal of Pure and Applied Algebra
titleWeakly distributive categories

note“To appear in the proceedings of 4th International Conference on Applied Category Theory, 2021"
titleExponential modalities and complementarity

journaltitlearXiv preprint arXiv:1905.08877
titleComplete Positivity for Mixed Unitary Categories

journaltitleElectronic Notes in Theoretical Computer Science
noteProceedings of the 4th International Workshop on Quantum Programming Languages (QPL 2006)
titleAxiomatic Description of Mixed States From Selinger's CPM-construction

titleIntroducing categories to the practicing physicist

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titleTerminality implies non-signalling

IOP Publishing

journaltitleNew Journal of Physics
titleInteracting quantum observables: categorical algebra and diagrammatics

Elsevier BV

journaltitleElectronic Notes in Theoretical Computer Science
titlePhase Groups and the Origin of Non-locality for Qubits

journaltitleInformation and Computation
titlePictures of complete positivity in arbitrary dimension

Cambridge University Press

titlePicturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning

booktitleMathematics of Quantum Computation and Quantum Technology
titleQuantum measurements without sums

Cambridge University Press

journaltitleMathematical structures in computer science
titleA new description of orthogonal bases

booktitleInternational Workshop on Computer Science Logic
titleEnvironment and classical channels in categorical quantum mechanics

journaltitle2012 27th Annual IEEE Symposium on Logic in Computer Science
titleStrong Complementarity and Non-locality in Categorical Quantum Mechanics

booktitleIn Esparza J., Murawski A. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2017. Lecture Notes in Computer Science
titleThe Free Exponential Modality of Probabilistic Coherence Spaces.

IOP Publishing

journaltitleJournal of Physics A: Mathematical and Theoretical
titleElementary gates for cartoon computation

journaltitleArchive for Mathematical logic
titleThe structure of multiplicatives

journaltitleNew York J. Math
titleNote on Frobenius monoidal functors

North Holland

titleVon Neumann Algebras

Cambridge University Press

titleGeometric Algebra for Physicists

Cambridge University Press

titleGeometric Algebra for Physicists

Oxford University

titleTypes for Quantum Computing

New York, NY, USA

Association for Computing Machinery

booktitleProceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science
seriesLICS '16
titleInteracting Frobenius Algebras Are Hopf

journaltitleTheory and Applications of Categories
titleOn involutive monoidal categories

Mount Allison University, Department of Mathematics
Computer Science

journaltitleTheory and Applications of Categories
titleThe Frobenius relations meet linear distributivity.

journaltitleJournal of Pure and Applied Algebra
titleOn cyclic star-autonomous categories

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleFiniteness spaces

booktitleInternational Conference on Typed Lambda Calculi and Applications
titleDifferential structure in models of multiplicative biadditive intuitionistic linear logic

Elsevier BV

journaltitleAnnals of Physics
titleGeometric algebra techniques for general relativity

North Holland

titleCategories, Allegories

journaltitleTheoretical computer science
titleLinear logic

titleLinear Logic: Its Syntax And Semantics

Open Publishing Association

booktitleProceedings 13th International Conference on Quantum Physics and Logic, Glasgow, Scotland, 6-10 June 2016
seriesElectronic Proceedings in Theoretical Computer Science
titleInfinite-dimensional Categorical Quantum Mechanics

Cambridge University Press

titleIntroduction to quantum mechanics

booktitleProceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158)
titleCorrectness of multiplicative proof nets is linear

Springer Science & Business Media

titleClifford algebra to geometric calculus: a unified language for mathematics and physics

journaltitleLogical Methods in Computer Science
titleCompactly accessible categories and quantum key distribution

journaltitleLogical Methods in Computer Science
titleCompactly accessible categories and quantum key distribution

Oxford University Press

titleCategories for Quantum Theory: An Introduction

journaltitleCommunications in Mathematical Physics
titleFrobenius structures over Hilbert C*-modules

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titleCategories of relations as models of quantum theory

journaltitleJournal of Pure and Applied Algebra
titleNuclearity in the category of complete semilattices

titleThe Clifford Algebra approach to Quantum Mechanics A: The Schrödinger and Pauli Particles

Elsevier Science Publishers Ltd.

journaltitleTheoretical Computer Science
titleGlueing and orthogonality for models of linear logic

The Academy of Science of the Royal Society of Canada

journaltitleMathematical Reports
titleFree bicomplete categories

The Academy of Science of the Royal Society of Canada

journaltitleMathematical Reports
titleFree bicomplete categories

journaltitleJournal of Algebra
titleOn MacLane's conditions for coherence of natural associativities, commutativities, etc.

journaltitleJournal of Pure and Applied Algebra
titleCoherence for compact closed categories

booktitleCategory Seminar
titleDoctrinal adjunction

journaltitleLogical Methods Computer Science
titleA categorical semantics for causal structure

journaltitleTheory and Applications of Categories
titleAn extended view of the Chu-construction

Université Paris Diderot - Paris

[S.l.] : [s.n.], 1988

titleLogiques, catégories & machines: Implantation de langages de programmation guidé par la logique catégorique

journaltitleLogical Methods in Computer Science
titleLifting coalgebra modalities and MELL Model Structure to Eilenberg-Moore categories

Open Publishing Association

journaltitleElectronic Proceedings in Theoretical Computer Science
titleWhy FHilb is not an interesting (co)differential category

titleA Graphical Calculus for Multi-Qudit Computations with Generalized Clifford Algebras

titleAn Algebraic Framework for Multi-Qudit Computations with Generalized Clifford Algebras

titleClifford algebra, geometric algebra, and applications

Berlin/Heidelberg, Germany

Springer Science & Business Media

titleCategories for the Working Mathematician

journaltitleArchive for Mathematical Logic
titleModularity of proof-nets

Cambridge University Press

titleFoundations of Quantum Group Theory

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleAn explicit formula for the free exponential modality of linear logic

Physics Today

journaltitleMathematical Structures in Computer Science
titleWhat's wrong with this pillow?

journaltitleQuantum Inf. Comput.
titleQuantum Latin squares and unitary error bases

Princeton University Press

editionNED - New edition
titleMathematical Foundations of Quantum Mechanics: New Edition

Cambridge University Press

titleQuantum Computation and Quantum Information: 10th Anniversary Edition

Mathematical Sciences Publishers

journaltitleAlgebra & Number Theory
titleWeak Hopf monoids in braided monoidal categories

journaltitleElectronic Notes in Theoretical Computer Science
titleRelating toy models of quantum computation: comprehension, complementarity and dagger mix autonomous categories

journaltitleQuaestiones Mathematicae
titleOn categories of monoids, comonoids, and bimonoids

booktitleIn Proc. Workshop on Physics and Computation (PhysComp'92
titleLinear Logic for Generalized Quantum Mechanics

Princeton University Press

titleNon-standard analysis

noteManuscript downloadable from <www.cs.man.ac.uk/ schalk/notes/llmodel.pdf>
titleWhat is a categorical model for Linear Logic?

booktitleNew Structures for Physics
titleA survey of graphical languages for monoidal categories

journaltitleElectronic Notes in Theoretical Computer Science
titleDagger compact closed categories and completely positive maps

journaltitleElectron. Notes Theor. Comput. Sci.
titleIdempotents in Dagger Categories: (Extended Abstract)

journaltitleMathematical Structures in Computer Science
titleTowards a quantum programming language

journaltitleMathematical Structures in Computer Science
titleOn Banach spaces of sequences and free linear logic exponential modality

journaltitlePhysics Letters A
titleExpressing the operations of quantum computing in multiparticle geometric algebra

Université Paris-Diderot - Paris VII

titleSémantiques et Syntaxes Vectorielles de la Logique Linéaire
Linear Logic ; Finiteness spaces ; Linearised topology ; Quantitative semantics ; Syntactic Taylor formula ; Algebraic lambda-calculus ; Differential nets. ; Logique linéaire ; Espaces de finitude ; Topologie linéarisée ; Sémantique quantitative ; Formule de Taylor syntaxique ; lambda-calcul algébrique ; Réseaux différentiels

Oxford University Press

titleAn Introduction to Clifford Algebras and Spinors

journaltitleInternational Journal of Theoretical Physics
titleA categorical framework for the quantum harmonic oscillator

Springer Science
Business Media LLC

journaltitleCommunications in Mathematical Physics
titleCategorical formulation of finite-dimensional quantum algebras

American Physical Society (APS)

journaltitlePhysical Review A
titleClifford algebras and universal sets of quantum gates

titleQuantum gates and Clifford algebras