# Single-shot energetic-based estimator for entanglement in a half-parity
measurement setup

Cyril Elouard  Department of Physics and Astronomy, University of Rochester,
Rochester, NY 14627, USA  CNRS and Université Grenoble Alpes, Institut Néel,
F-38042 Grenoble, France  Alexia Auffèves  CNRS and Université Grenoble Alpes,
Institut Néel, F-38042 Grenoble, France  Géraldine Haack
<EMAIL_ADDRESS>Université de Genève, Department of Applied Physics,
Chemin de Pinchat 22, CH-1211 Genève 4, Switzerland

###### Abstract

Producing and certifying entanglement between distant qubits is a highly
desirable skill for quantum information technologies. Here we propose a new
strategy to monitor and characterize entanglement genesis in a half parity
measurement setup, that relies on the continuous readout of an energetic
observable which is the half-parity observable itself. Based on a quantum-
trajectory approach, we theoretically analyze the statistics of energetic
fluctuations for a pair of continuously monitored qubits. We quantitatively
relate these energetic fluctuations to the rate of entanglement produced
between the qubits, and build an energetic-based estimator to assess the
presence of entanglement in the circuit. Remarkably, this estimator is valid
at the single-trajectory level and shows to be robust against finite detection
efficiency. Our work paves the road towards a fundamental understanding of the
stochastic energetic processes associated with entanglement genesis, and opens
new perspectives for witnessing quantum correlations thanks to quantum
thermodynamic quantities.

##  1 Introduction

Entanglement is a cornerstone of quantum information technologies: Entangled
pairs of qubits have for long been a resource for secure quantum key
distribution [1], quantum teleportation [2], quantum repeaters for long
distance quantum communication [3], while large scale entanglement can be
exploited through cluster states to perform measurement-based quantum
computing [4].

From a practical point of view, entangled pairs of qubits can be produced,
e.g. by using quantum gates based on non-linear interactions [5], or
performing measurement-based protocols of the parity observable. The latter is
known to induce entanglement between two qubits initially in a separable state
and has been extensively investigated in the last decade within various
physical systems. Proposals have been made considering superconducting qubits
jointly measured by a cavity mode [6, 7, 8, 9] and semiconductor quantum dots
jointly measured by a quantum point contact [10, 11] or by an electronic Mach-
Zehnder interferometer in quantum transport experiments [12, 13]. All these
works have derived the specific conditions under which these setups can be
operated as parity meters. Those conditions include having a fine tuning of
the coupling parameters between the qubits and the detector such that only the
parity degree of freedom of the qubits is measured. In addition, having each
of the two qubits initially in a maximal coherent superposition state is
required to generate maximally entangled states. In particular, Ref. [11]
investigated the stochastic generation of entanglement from a weak continuous
measurement of the parity operator, putting forward the measurement-induced
entanglement genesis. Since then, measurement-induced entanglement has
eventually been implemented within circuit QED experiments [14, 15, 8].
Interestingly, these platforms also provide the technological know-how to
access the quantum trajectories of individual quantum systems both subject to
local measurements [16, 17, 18] or to joint-measurements [15, 8]. Hence, Refs.
[15, 8] not only implemented a parity-measurement based protocol onto two
qubits, but could also access the stochastic trajectories followed by the
joint state of the qubits along the entanglement generation process.

Recently, it was shown that the measurements allowing to reconstruct the pure
state trajectories of the monitored systems are associated with energetic
fluctuations of genuinely quantum origin called quantum heat. These energetic
fluctuations can be turned into work in various protocols [19, 20, 21, 22] and
have been related to entropy production of quantum origin [23, 24] and provide
new merit criteria to assess the performances of a feedback loop [25, 18]. So
far, a thermodynamic analysis of entanglement genesis based on stochastic
trajectories has remained elusive.

In this work, we theoretically analyze the statistics of energetic
fluctuations for a pair of continuously monitored half spins subject to a
half-parity measurement. This is achieved within the framework of stochastic
quantum thermodynamics presented in [23, 26] and it allows us to derive and
highlight for the first time energetic signatures associated with measurement-
induced entanglement genesis. We quantitatively relate these energetic quantum
fluctuations to the rate of entanglement produced between the qubits. We then
exploit our results to propose a new practical application of these energetic
fluctuations by building an energetic-based estimator to attest the presence
of entanglement in the circuit. Remarkably, this latter quantity holds at the
single trajectory level and does not rely on the measurement record itself.

The paper is organized as follows. In Sec. 2, we present our system and recall
the basics of a half-parity measurement protocol. In Sec. 3, we relate parity
measurement and energy measurement, and we introduce the quantum energetic
fluctuations associated with the quantum heat. We define and compute the
stochastic energetic quantities involved in the continuous measurement case.
In Sec. 4, we relate the fluctuations of quantum heat to the rate of
entanglement induced between the qubits. This one is investigated through the
time-derivative of the concurrence, a monotone measure for two-qubit
entanglement. We derive upper and lower bounds for the rate of entanglement
genesis, which we exploit in Sec. 5 to build an estimator assessing the
presence of entanglement. The latter does not depend explicitly on the
measurement record, it is solely based on energetic quantities and is valid at
the single-trajectory level. We also investigate its robustness in presence of
finite detection efficiency.

Figure 1: Model and motivation. a) Weak continuous half-parity measurement.
Two qubits are weakly coupled to a meter able to measure continuously the
joint observable $\hat{\Phi}$, for instance in circuit QED setups, see Refs.
[14, 15]. The corresponding measurement record $I_{\gamma}(t)$ is stochastic.
Two realizations are shown: trajectory leading the qubits into a maximally
entangled state (red) or into a product state (blue). b) After post-selection,
time-dependent single-trajectory concurrence and single-trajectory quantum
heat fluctuations. Vertical dashed lines highlight similar times for the
concurrence and the quantum heat fluctuations to reach their long-time limit
value. In this work we derive formal relations between those quantities and
exploit them to derive an energetic-based estimator, an alternative to quantum
state tomography.

##  2 Model

System– The system is made of two two-level systems (qubits) with identical
energy splitting $\epsilon$ described by their Hamiltonian $\hat{H}_{{\cal
S}}$

$\displaystyle\hat{H}_{{\cal
S}}=\epsilon\left(\hat{\sigma}_{z}^{(1)}\otimes\mathds{1}+\mathds{1}\otimes\hat{\sigma}_{z}^{(2)}\right)\,,$
(1)

where $\sigma_{z}^{(i)}$ denotes the z-Pauli matrix for qubit $i$. We assume
the two qubits to be initially in a separable state, and more specifically in
a maximal superposition state:

$\displaystyle|\psi(0)\rangle=\frac{1}{2}\left(|\\!\uparrow\rangle+|\\!\downarrow\rangle\right)\otimes\left(|\\!\uparrow\rangle+|\\!\downarrow\rangle\right)\,.$
(2)

This choice is motivated by previous works showing that this state belongs to
the set of optimal states that lead to maximally entangled final states (i.e.
Bell states) when the qubits are subject to a (half-) parity measurement [12].
Hence, this choice of initial state will allow us to investigate the energetic
signatures associated to the generation of entanglement in optimal conditions.
In addition, the qubits are subject to a weak continuous measurement of the
joint observable $\hat{\Phi}$, that implements a half-parity measurement, see
Fig. 1. Within the two-qubit computational basis
$\\{{|\\!\\!\uparrow\uparrow\rangle},{|\\!\\!\uparrow\downarrow\rangle},{|\\!\\!\downarrow\uparrow\rangle},{|\\!\\!\downarrow\downarrow\rangle}\\}$,
the collective operator $\hat{\Phi}=\sum_{i=1,2}\sigma_{z}^{(i)}$ is defined
by
${|\\!\\!\uparrow\uparrow\rangle}{\langle\uparrow\uparrow\\!\\!|}-{|\\!\\!\downarrow\downarrow\rangle}{\langle\downarrow\downarrow\\!\\!|}$
and has three eigenvalues $\pm 1$ and $0$ with eigenstates
${|\\!\\!\uparrow\uparrow\rangle}$, ${|\\!\\!\downarrow\downarrow\rangle}$ and
$({|\\!\\!\uparrow\downarrow\rangle}\pm{|\\!\\!\downarrow\uparrow\rangle})/\sqrt{2}$
respectively. The eigenvalue 0 is degenerate, implying that this outcome does
not allow one to distinguish between the two odd states
$\\{{|\\!\\!\uparrow\downarrow\rangle},{|\\!\\!\downarrow\uparrow\rangle}\\}$.
Consequently, when outcome $0$ is obtained from the measurement, the two
qubits are driven into a coherent superposition of those states, which leads
to entanglement. On the contrary, the measurement of the eigenvalues $\pm 1$
allows to distinguish the two even states, leaving the qubits in a product
state. Hence, the half-parity measurement presents the specificity of
producing in a probabilistic way entangled and product states. It is clear
from the expression of $\hat{\Phi}$ that its measurement does not put the
qubits into an entangled state (spanned by the odd states) if these ones are
not in a coherent superposition initially. Although the forms of $\hat{\Phi}$
and of the parity operator $\hat{P}=\sigma_{z}^{(1)}\otimes\sigma_{z}^{(2)}$
seem to be specific, these joint operators are the only ones up to local
unitary operations that can generate correlations between two qubits. This
explains the high interest in the past years for this parity measurement
protocol within solid-state setups, also known within the quantum information
community as a Bell measurement. This one was for instance proposed and
realized for quantum teleportation [2, 27].

Quantum trajectories– Assuming a weak continuous measurement of $\hat{\Phi}$,
each realization of the measurement procedure is associated to a quantum
stochastic trajectory followed by the qubits and labelled $\gamma$ in the rest
of the manuscript [28, 29]. We assume the initial state of the qubits is a
known pure state, see Eq.(2), such that the trajectory $\gamma$ is made of a
sequence of pure states $\\{|\psi_{\gamma}(t)\rangle\\}$. To each trajectory
$\gamma$ corresponds a stochastic measurement record $I_{\gamma}(t)$:

$\displaystyle
I_{\gamma}(t)=\langle\hat{\Phi}(t)\rangle_{\gamma}+\frac{dW_{\gamma}(t)}{2\sqrt{\Gamma}dt}\,.$
(3)

Here,
$\langle\hat{\Phi}(t)\rangle_{\gamma}=\langle\psi_{\gamma}(t)|\hat{\Phi}|\psi_{\gamma}(t)\rangle$
is the expectation value of the half-parity operator w.r.t. the state
$|\psi_{\gamma}(t)\rangle$, and $\Gamma$ corresponds to the detector
measurement rate, i.e. the rate at which one is able to distinguish the
measurement outcomes from the detector’s shot noise [30, 31, 12, 13]. The
infinitesimal Wiener increment $dW_{\gamma}(t)$ is a stochastic variable
characterized by a zero average and variance $dt$, i.e. $\langle\\!\langle
dW_{\gamma}\rangle\\!\rangle_{t}=0$ and $\langle\\!\langle
dW_{\gamma}^{2}\rangle\\!\rangle_{t}=dt$, where
$\langle\\!\langle\cdot\rangle\\!\rangle_{t}$ denotes the average over all
realizations of the measurement during the time interval $[0,t]$. Note that
throughout this work, we use Ito’s convention for stochastic differential
calculus [32, 33]. This infinitesimal Wiener increment encodes Gaussian
fluctuations of the measurement record $I_{\gamma}(t)$ around its expectation
value $\langle\hat{\Phi}(t)\rangle_{\gamma}$ and therefore captures the
detector’s shot noise in the weak coupling limit [31, 29]. Based upon the
knowledge of $I_{\gamma}$, the conditional dynamics of the two qubits subject
to the weak measurement of the half-parity observable $\hat{\Phi}$ is captured
by the stochastic Schrödinger equation

$\displaystyle d|\psi_{\gamma}(t)\rangle$ $\displaystyle=$
$\displaystyle\left[-i\hat{H}_{{\cal
S}}dt-\frac{\Gamma}{2}dt(\hat{\Phi}-\langle\hat{\Phi}(t)\rangle_{\gamma})^{2}\right.$
(4)
$\displaystyle\left.\;+\sqrt{\Gamma}\,dW_{\gamma}(t)(\hat{\Phi}-\langle\hat{\Phi}(t)\rangle_{\gamma})\right]|\psi_{\gamma}(t)\rangle.$

Figure 2: Post-selected average concurrence $\mathcal{C}_{\gamma}$ and quantum
heat $Q_{\gamma}$ (as a function of time i.u. of $[1/\Gamma]$). Red dashed
curves are obtained by averaging over all odd trajectories, i.e. that lead the
qubits into the maximally entangled state
$(|{\uparrow\downarrow}\rangle+|{\downarrow\uparrow}\rangle)/\sqrt{2}$. Blue
(full, dashed) curves are obtained by averaging over all even trajectories,
i.e. that lead the qubits into a product state
$(|{\uparrow\uparrow}\rangle,|{\downarrow\downarrow}\rangle)$. The half-parity
measurement induces entanglement, but also serves as energy filter. When
averaged over all trajectories (even and odd), $\langle\\!\langle
Q_{\gamma}\rangle\\!\rangle=0$ as expected for a QND-measurement
$[\hat{H}_{S},\hat{\Phi}]=0$. Total number of simulated quantum trajectories:
800

Using $\hat{H}_{{\cal S}}=\epsilon\hat{\Phi}$, see Eq. (1), one can solve
analytically Eq.(4) at any time $t$ as a function of the stochastic
measurement record:

$\displaystyle|\psi(J_{\gamma},t)\rangle=\frac{1}{N_{\gamma}(t)}\left(e^{(-i\epsilon+2\Gamma
J_{\gamma}\,)\hat{\Phi}t-\Gamma\hat{\Phi}^{2}t}\right)|\psi(0)\rangle\,,$ (5)

with $N_{\gamma}(t)$ the time-dependent normalization factor
$N_{\gamma}(t)=\sqrt{(1+e^{-2\Gamma t}\cosh(4\Gamma J_{\gamma}t))/2}$.
Following Ref. [29], this solution is expressed in terms of the measurement
outcome $J_{\gamma}=1/t\,\int_{0}^{t}I_{\gamma}(\tau)\,d\tau$:

$\displaystyle
J_{\gamma}(t)=\frac{1}{t}\int_{0}^{t}\langle\hat{\Phi}(\tau)\rangle_{\gamma}\,d\tau\,+\frac{W_{\gamma}(t)}{2\sqrt{\Gamma}t}\,,$
(6)

with $W_{\gamma}(t)=\int_{0}^{t}dW_{\gamma}(t^{\prime})$ a Gaussian random
variable with mean zero and variance $t$. The time integral corresponds to a
finite resolution time, typically set by experimental constraints. The
probability distribution of $J_{\gamma}(t)$ is a sum of three Gaussian
functions of variance $1/4\Gamma t$, each peaked around one of the eigenvalues
$\\{0,\pm 1\\}$ of the measured observable $\hat{\Phi}$:

$\displaystyle P(J_{\gamma},t)=\frac{1}{3}\sum_{j=-1}^{1}\sqrt{\frac{2\Gamma
t}{\pi}}e^{-(J_{\gamma}-j)^{2}/(2\tilde{\sigma}_{0}^{2})}\,,$ (7)

with the variance $\tilde{\sigma}_{0}\equiv\sqrt{\Gamma t}$ setting the
measurement strength. In the long time limit $t\gg(4\Gamma)^{-1}$,
$J_{\gamma}(t)$ only takes one of three possible outcomes which are the
eigenvalues of the half-parity measurement operator: $J_{0}\equiv 0$, $J_{\pm
1}\equiv\pm 1$. At those long times, the measurement becomes projective. In
the following of the work, we choose to label the different trajectories with
their long-time value of $J_{\gamma}$; this defines three subsets of
trajectories, leading the qubits either in an entangled state ($J_{\gamma}=0$)
or in a product state ($J_{\gamma}=\pm 1$).

Measure of entanglement– We quantify the presence of entanglement at any time
$t$ using the concurrence, a monotone measure for two-qubit entangled states
[34]. The qubits being in a pure state at each instant of time along their
trajectory $\gamma$, we can make use of a simpler definition
$\mathcal{C}_{\gamma}(t)=\text{max}\left\\{0,2|ad-bc|\right\\}$ where
$a,b,c,d$ are the amplitudes of $|\psi_{\gamma}(t)\rangle$ with respect to the
computational states
${|\\!\\!\uparrow\uparrow\rangle},{|\\!\\!\uparrow\downarrow\rangle},{|\\!\\!\downarrow\uparrow\rangle},{|\\!\\!\downarrow\downarrow\rangle}$
respectively. Inserting Eq.(5), we get a stochastic concurrence which depends
on the measurement outcome $J_{\gamma}$:

$\displaystyle\mathcal{C}_{\gamma}(t)$ $\displaystyle=$
$\displaystyle\frac{1-e^{-2\Gamma t}}{1+e^{-2\Gamma t}\cosh(4\Gamma
J_{\gamma}t)}.$ (8)

Figure 2 a) shows the time evolution of the average post-selected concurrence
$\mathcal{C}_{\gamma}$ according to the three subsets of trajectories, labeled
by the final outcomes $J_{\gamma}\to\\{J_{0},J_{\pm 1}\\}$. The average is
made over all realizations that belong to a given subset. At long times,
$t>6/\Gamma$, the concurrence can be directly obtained by replacing in Eq.(8)
$J_{\gamma}$ by the $\hat{\Phi}$-eigenvalues $0,\pm 1$ as the measurement
becomes projective. Under optimal conditions as considered here (QND-
measurement, no additional dephasing processes, ideal detection scheme), the
concurrence reaches the value 1 when qubits are driven into the odd subspace,
meaning that the qubits end up in a maximally entangled state. Non-ideal
conditions encountered in experiments lead to a lower maximal value of
concurrence, and eventually to a long-time state that is separable if
dephasing processes are too important. This was for instance the case in Ref.
[15], and can be accounted for theoretically.

In this work, the goal is to establish fundamental links between energetic
signatures and the generation of entanglement. This is why we focus below on
the realization of weak continuous half-parity measurement under optimal
conditions. Non-ideal conditions could also be accounted for, but would
prevent us to draw clear conclusions on the origin of these energetic
signatures.

##  3 Half-parity measurement seen as an energy measurement (filter)

It is remarkable that the half-parity observable also serves as energy filter
within our model: $\hat{\Phi}\propto\hat{H}_{S}$, both being related by the
energy gap of the qubits $\epsilon$ (see Eq. (1)). Hence, the half-parity
measurement provides a direct access to the energetics of the qubits along the
quantum trajectories generated by the weak measurement of $\hat{\Phi}$. The
internal energy $U_{\gamma}(t)$ of the two qubits along a given trajectory
$\gamma$ is given by:

$\displaystyle U_{\gamma}(t)=\langle\psi_{\gamma}(t)|\hat{H}_{{\cal
S}}|\psi_{\gamma}(t)\rangle=\epsilon\langle\psi_{\gamma}(t)|\hat{\Phi}|\psi_{\gamma}(t)\rangle\,.$
(9)

Considering the initial state $|\psi(0)\rangle$ (see Eq.(2)), the initial
internal energy $U(0)$ is zero and constitutes the energy reference. In the
absence of driving, $\hat{H}_{{\cal S}}$ is time-independent. Hence no work is
performed onto the qubits. In addition, there is no thermal reservoir involved
in the problem so that a change in the internal energy of the qubits can only
arise from the measurement process itself. This form of energy exchange has no
classical counter-part [35, 25, 23, 36, 37, 38] and will be, in the line of
[23], referred to as quantum heat and denoted $Q$ in this article.

Long times: projective energetic measurement– At long times,
$t>(6\Gamma)^{-1}$, the change of internal energy (associated here to a net
quantum heat $Q$) takes three different values depending on the measurement
outcomes:

$\displaystyle\Delta U\equiv Q=\left\\{\begin{array}[]{c}0\quad\text{for}\quad
J_{\gamma}=0\\\ \epsilon\quad\text{for}\quad J_{\gamma}=1\\\
-\epsilon\quad\text{for}\quad J_{\gamma}=-1.\end{array}\right.$ (13)

When averaged over all trajectories, $\left\langle\\!\left\langle\Delta
U\right\rangle\\!\right\rangle_{\gamma}=0$, which equals the reference
internal energy at initial time $U(0)=0$. This equality follows from
$[\hat{H}_{S},\hat{\Phi}]=0$ that characterizes a QND-measurement. Indeed, as
both observables commute, there must be no change of internal energy on
average and this must hold at all times. At long times, the weak continuous
measurement of $\hat{\Phi}$ is equivalent to a projective energy measurement,
and this explains the long-time values of $\Delta U$ for the different subsets
of trajectories, equal to the energy of the odd states
$\\{|{\uparrow\downarrow}\rangle,|{\downarrow\uparrow}\rangle\\}$ and even
states $|{\uparrow\uparrow}\rangle$ and $|{\downarrow\downarrow}\rangle$
respectively.

Intermediate times– At arbitrary time $t$, the quantum heat exchange depends
on the exact outcome $J_{\gamma}(t)$:

$\displaystyle Q_{\gamma}(t)$ $\displaystyle=$ $\displaystyle
U_{\gamma}(t)-U(0)$ (14) $\displaystyle=$
$\displaystyle\epsilon\,\frac{e^{-2\Gamma t}\sinh(4\Gamma
tJ_{\gamma})}{1+e^{-2\Gamma t}\cosh(4\Gamma tJ_{\gamma})\,.}$

This quantum heat exchange $Q_{\gamma}(t)$, after post-selection, is plotted
in Fig. 2 b) as a function of time. As for the concurrence, an average is made
within each subset of trajectories defined by the long-time limit value of the
measurement record $J_{\gamma}=0,\pm 1$. Because the half-parity measurement
amounts to an energetic measurement, an access to the internal energy is
sufficient to determine whether the qubits are entangled or not. A record
equal to 0 implies that the qubits ended up in the odd subspace, in a coherent
superposition of the odd states
$(|{\uparrow\downarrow}\rangle+|{\downarrow\uparrow}\rangle)\sqrt{2}$. This
first part demonstrates that, on top of being a way to generate entangled
state in a heralded way (the outcome is enough to know whether the final state
is entangled) as demonstrated in previous works, the half-parity measurement
also gives access to the joint internal energy of the qubits. Let us note that
the rate at which the quantum heat exchange converges to its final value
corresponds to the measurement-induced dephasing rates derived in previous
works [12, 13, 14, 15].

Stochastic fluctuations at arbitrary times– To derive the fluctuations of the
quantum heat, let us first introduce the increment $\delta Q_{\gamma}(t)$ that
corresponds to the stochastic infinitesimal variation of the internal energy
$U_{\gamma}(t)$ between times $t$ and $t+dt$ along a trajectory $\gamma$. It
is related to the total quantum heat exchange $Q_{\gamma}(t)$ up to time $t$
via:

$\displaystyle Q_{\gamma}(t)=\int_{0}^{t}\delta Q_{\gamma}(t^{\prime})\,,$
(15)

and is defined as

$\displaystyle\delta Q_{\gamma}(t)$ $\displaystyle\equiv$ $\displaystyle
d\left(\langle\psi_{\gamma}(t)|\hat{H}_{{\cal
S}}|\psi_{\gamma}(t)\rangle\right)$ (17) $\displaystyle=$
$\displaystyle\Big{(}d\langle\psi_{\gamma}(t)|\Big{)}\hat{H}_{{\cal
S}}|\psi_{\gamma}(t)\rangle+\langle\psi_{\gamma}(t)|\hat{H}_{{\cal
S}}\Big{(}d|\psi_{\gamma}(t)\rangle\Big{)}$
$\displaystyle+\Big{(}d\langle\psi_{\gamma}(t)|\Big{)}\hat{H}_{{\cal
S}}\Big{(}d|\psi_{\gamma}(t)\rangle\Big{)}$

Inserting Eqs. (1) and (5) into Eq. (17) and expanding the last term up to the
first order in $dt$, we obtain

$\displaystyle\delta Q_{\gamma}(t)$ $\displaystyle=$ $\displaystyle
2\epsilon\sqrt{\Gamma}dW_{\gamma}(t)\left(4P_{\uparrow\uparrow}P_{\downarrow\downarrow}+P_{o}P_{e}\right)$
(18) $\displaystyle\equiv$ $\displaystyle\delta Q^{(e)}_{\gamma}(t)+\delta
Q^{(eo)}_{\gamma}(t)\,,$

with

$\displaystyle\delta Q^{(e)}_{\gamma}(t)$ $\displaystyle=$ $\displaystyle
8\epsilon\sqrt{\Gamma}dW_{\gamma}(t)P_{\uparrow\uparrow}P_{\downarrow\downarrow}$
(19) $\displaystyle\delta Q^{(eo)}_{\gamma}(t)$ $\displaystyle=$
$\displaystyle 2\epsilon\sqrt{\Gamma}dW_{\gamma}(t)P_{e}P_{o}\,.$ (20)

Here, we have introduced the populations $P_{ij}$ for the 4 two-qubit states
defined as $P_{ij}\equiv P_{ij,\gamma}(t)=|\langle
ij|\psi_{\gamma}(t)\rangle|^{2}\quad i,j=\uparrow,\downarrow$ and we denote
$P_{e}=P_{\uparrow\uparrow}+P_{\downarrow\downarrow}$ and
$P_{o}=P_{\uparrow\downarrow}+P_{\downarrow\uparrow}$ the populations within
the even and odd parity subspaces respectively. From those definitions, the
products $P_{\uparrow\uparrow}P_{\downarrow\downarrow}$ and $P_{e}P_{o}$
correspond respectively to the squared coherences (off-diagonal elements in
the two-qubit density matrix) within the even subspace and between the even
and odd subspaces. Let us recall that a (half-) parity measurement generates
entanglement by distinguishing the even states form the odd ones. It is
therefore the loss of coherence between the two parity subspaces that brings
the two qubits in a coherent superposition of odd states when outcome
$J_{\gamma}=0$ is obtained. Hence, we claim that the product $P_{o}P_{e}$
(equivalently $\delta Q^{(eo)}_{\gamma}(t)$) in Eq.(20) reflects entanglement
genesis, and so does the total heat increment $\delta Q_{\gamma}(t)$. We
demonstrate this claim in the following sections.

##  4 Energetic bounds for the entanglement genesis rate

To validate our claim, we define the standard deviation of the quantum heat
increment between times $t$ and $t+dt$ as

$\displaystyle\sigma_{\gamma}(t)$ $\displaystyle=$
$\displaystyle\sqrt{\langle\\!\langle\delta
Q_{\gamma}^{2}(t)\rangle\\!\rangle_{dt}}$ (21) $\displaystyle=$ $\displaystyle
2\epsilon\sqrt{\Gamma dt}\,\frac{e^{-4\Gamma t}+e^{-2\Gamma t}\cosh(4\Gamma
tJ_{\gamma})}{\left(1+e^{-2\Gamma t}\cosh(4\Gamma tJ_{\gamma})\right)^{2}}\,,$

where we made use of $\langle\\!\langle dW(t)^{2}\rangle\\!\rangle_{dt}=dt$
and $\langle\\!\langle dW(t)\rangle\\!\rangle_{dt}=0$. For simplicity, we will
work in the following with dimensionless quantities:

$\displaystyle\tilde{\sigma}_{\gamma}(t)=\frac{\sigma_{\gamma}(t)}{2\epsilon\sqrt{\Gamma
dt}}\quad\text{and}\quad\tilde{Q}_{\gamma}(t)$ $\displaystyle=$
$\displaystyle\frac{Q_{\gamma}(t)}{\epsilon}\,.$ (22)

$\sqrt{\Gamma dt}$ refers to the variance of the Gaussian distribution of the
measurement record $I_{\gamma}(t)$ during the discrete time interval $dt$, see
the variance $\tilde{\sigma}_{0}$ defined in Eq. (7). Similarly to Eq. (18),
we define two contributions to the standard deviation of the stochastic
quantum heat increment:

$\displaystyle\tilde{\sigma}_{\gamma}(t)=\tilde{\sigma}_{\gamma}^{(eo)}(t)+\tilde{\sigma}_{\gamma}^{(e)}(t)$
(23)

Figure 3: Infinitesimal quantum heat fluctuations
$\tilde{\sigma}_{\gamma}^{(eo)}$ as a function of the outcome $J_{\gamma}$ and
the duration of the measurement $t$. The contour plot (solid black lines)
corresponds to the lines of constant concurrence (from 0.2 to 0.8). Regions
for $\mathcal{C}=0,1$ correspond to the blue ones and are explicit on the
figure. There exists a one to one correspondance between a finite derivative
of the concurrence ($\mathcal{C}$ is changing) and finite fluctuations of the
increment of quantum heat exchange. At long times, measurement outcome
$J_{\gamma}$ converges to $0$ or $\pm 1$, shown with dashed lines (not anymore
continuous along the x-axis). Fluctuations are not present
($\tilde{\sigma}_{0,\pm 1}^{(eo)}=0$)and $\mathcal{C}=\\{1,0\\}$ for
$J_{\gamma}=\\{0,\pm 1\\}$.

Figure 3 illustrates our claim. The contour plot corresponding to constant
values of the concurrence is superimposed onto the density plot of
$\tilde{\sigma}^{(eo)}(t)$. It highlights that an increase in the concurrence
is associated with non-zero fluctuations of $\delta Q^{(eo)}_{\gamma}(t)$. In
contrast, the concurrence plateaus correspond to the areas where
$\tilde{\sigma}^{(eo)}$ vanishes. This observation is supported by a strict
equality between the concurrence variation (the concurrence derivative) and
$\tilde{\sigma}_{\gamma}^{(eo)}$ for the trajectories leading the qubits into
an entangled state $J_{\gamma}=0$ (see App. A for detailed derivation):

$\displaystyle\left.\frac{{d\mathcal{C}}_{\gamma}}{dt}\right|_{J_{\gamma}=0}=4\Gamma\left.\tilde{\sigma}_{\gamma}^{(eo)}\right|_{J_{\gamma}=0}\,.$
(24)

Because it constitutes at the same time the source of entanglement and a
direct access to the energy and its fluctuations, it follows that the energy
fluctuations must be related to the creation of entanglement. However, the
practical interest of Eq. (24) is limited as it would be impossible in a heat-
sensitive measurement to distinguish the two contributions $\delta
Q^{(eo)}_{\gamma}$ and $\delta Q^{(e)}_{\gamma}$.

Nevertheless, at intermediate times, we demonstrate that the concurrence
derivative, that characterizes the rate at which entanglement is induced by
the measurement, can be upper and lower bounded by energetic quantities
solely. Starting from

$\displaystyle\frac{d\mathcal{C}_{\gamma}(t)}{dt}$ $\displaystyle=$
$\displaystyle 2\Gamma\Bigg{[}\frac{e^{-2\Gamma t}\left(1+\cosh(4\Gamma
tJ_{\gamma})\right)}{(1+e^{-2\Gamma t}\cosh(4\Gamma tJ_{\gamma})^{2}}$
$\displaystyle-\frac{2\mathcal{C}_{\gamma}(t)Q_{\gamma}(t)}{\epsilon}\left(\langle\hat{\Phi}(t)\rangle_{\gamma}+\frac{1}{2\sqrt{\Gamma}}\frac{dW_{\gamma}(t)}{dt}\right)\Bigg{]}\,,$

we then perform an an ensemble average over all trajectories (or equivalently
over the measurement record $I_{\gamma}(t)$) during the time interval
$[t,t+dt]$, keeping the past records
$\\{I_{\gamma}(t^{\prime})\\}_{t^{\prime}<t}$ fixed. This interval $[t,t+dt]$
is the one over which we investigate the fluctuations of the quantum heat
increment $\delta Q_{\gamma}$, and this ensemble average
$\langle\\!\langle\cdot\rangle\\!\rangle_{dt}$ corresponds to the one used in
the definitions of the quantum heat increment fluctuations, see Eq.(21). We
then have $\langle\\!\langle dW_{\gamma}(t)/dt\rangle\\!\rangle_{dt}=0$ in
Eq.(4) and

$\displaystyle\left\langle\\!\\!\\!\left\langle\frac{d\mathcal{C}_{\gamma}(t)}{dt}\right\rangle\\!\\!\\!\right\rangle_{dt}$
$\displaystyle=$ $\displaystyle 2\Gamma\frac{e^{-2\Gamma t}+e^{-2\Gamma
t}\cosh(4\Gamma tJ_{\gamma})}{(1+e^{-2\Gamma t}\cosh(4\Gamma
tJ_{\gamma}))^{2}}$ (26)
$\displaystyle-4\Gamma\,\frac{Q_{\gamma}}{\epsilon}\,\mathcal{C}_{\gamma}\langle\hat{\Phi}(t)\rangle_{\gamma}.$

The first term on the r.h.s. can be upper and lower bounded with
$\tilde{\sigma}_{\gamma}$ (see Append. B):

$\displaystyle\left\langle\\!\\!\\!\left\langle\frac{d\mathcal{C}_{\gamma}(t)}{dt}\right\rangle\\!\\!\\!\right\rangle_{dt}\geq
2\Gamma\left[\tilde{\sigma}_{\gamma}(t)-2\tilde{Q}^{2}_{\gamma}(t)\mathcal{C}_{\gamma}(t)\right]$
(27)
$\displaystyle\left\langle\\!\\!\\!\left\langle\frac{d\mathcal{C}_{\gamma}(t)}{dt}\right\rangle\\!\\!\\!\right\rangle_{dt}\leq
2\Gamma\left[2\tilde{\sigma}_{\gamma}(t)-2\tilde{Q}^{2}_{\gamma}(t)\mathcal{C}_{\gamma}(t)\right]$
(28)

Figure 4: Success rate of the energetic-based estimator $\mathcal{E}_{ss}$ for
different coarse-graining times, $\tau=0.1/\Gamma$ (a), $\tau=0.4/\Gamma$ (b),
$\tau=\Delta t/\Gamma$ (c), and for $\tau=\Delta t/\Gamma$ and different
values of the detection efficiency $\eta=1$ (a-b-c), $\eta=0.5$ (d),
$\eta=0.8$ (e), $\eta=0.9$ (f). The density plots show the success rate of
$\mathcal{E}_{ss}$ for all integration intervals $[t_{i},t_{i}+\Delta t]$,
$t_{i}$ and $\Delta t$ ranging from $0$ to $10$. In the simulation, we
considered as entangled the trajectories leading to a final concurrence
greater or equal to $0.8$. The success rate is defined as the proportion of
entangled trajectories for which the estimator takes a negative value over all
entangled trajectories. The white crosses on plots (c-f) indicate the points
of minimum $t_{i}$ reaching a success rate of $50\%$, i.e $\tau=\Delta
t=0.3/\Gamma$ and $t_{i}=3/\Gamma$, $6.4/\Gamma$, $3.8/\Gamma$ and
$3.4/\Gamma$, respectively, for plots c), d), e) and f).

Inequalities (27) and (28) constitute one of the main analytical results of
this work. The entanglement rate is exclusively upper and lower bounded by
energetic quantities, the quantum heat and quantum heat fluctuations defined
in Eq.(22). All quantities are defined over a (small) finite time interval
$dt$. Remarkably, we can further exploit Ineq. (27) towards the derivation of
an energetic-based estimator to assess the presence of entanglement at
intermediate times. Of interest towards single shot entanglement detection,
this will be done at the level of a unique quantum trajectory.

##  5 Single-shot energetic-based estimator for entanglement

Formally, a witness for entanglement is an observable that takes a negative
value when averaged with respect to a state that is entangled. If the witness
takes a positive value, no conclusions can be drawn, the state can either be
entangled or separable [39]. In the past years, few witnesses based on
temperature have been proposed, as a first attempt to exploit energetic
quantities to certify the presence of entanglement [40, 41]. Following the
spirit of assessing the presence of entanglement with some independent
quantities, we introduce in this section a _trajectory-based witness_ , i.e. a
quantity which takes negative value in presence of entanglement, just as usual
entanglement witnesses, but which takes as an input a single quantum
trajectory, or equivalently a weak measurement record, instead of a quantum
state. As a first step to build this witness, we use the property that a given
quantum trajectory $\gamma$ will drive the qubits onto an entangled state if,

$\displaystyle\left\langle\\!\\!\\!\left\langle\frac{d\mathcal{C}_{\gamma}(t^{\prime})}{dt}\right\rangle\\!\\!\\!\right\rangle_{dt}\geq
0\,,\quad\forall t^{\prime}\in[0,t].$ (29)

This condition is motivated by the idea that accumulating positive time-
derivatives would lead to a positive concurrence at final time $t$. While one
cannot certify the presence of entanglement along individual trajectory
$\gamma$ from Eq.(29), the integration over time of
$\frac{d\mathcal{C}_{\gamma}(t)}{dt}$ used to compute the concurrence at time
$t$ is expected to play the role of the ensemble average
$\left\langle\\!\\!\\!\left\langle\cdot\right\rangle\\!\\!\\!\right\rangle_{dt}$
for $\Gamma dt\ll 1$ and provided $t\gtrsim\Gamma^{-1}$. Therefore, condition
(29) implies an entangled state at time $t$ for a high fraction of the
considered trajectories.

Making use of inequality (27), this condition translates into:

$\displaystyle\tilde{\sigma}_{\gamma}(t)-2\tilde{Q}_{\gamma}^{2}(t)\mathcal{C}_{\gamma}(t)\geq
0\,,$ (30)

We can now use that the concurrence $\mathcal{C}_{\gamma}$ takes values within
$[0,1]$ to prove an inequality assessing the presence of entanglement, that
only depends on energetic quantities,
$\tilde{\sigma}_{\gamma}(t)-2\tilde{Q}_{\gamma}^{2}(t)\geq 0\,.$ We can now
directly use this condition to introduce our trajectory-based entanglement
witness $\mathcal{W}_{\gamma}$, defined as:

$\displaystyle\mathcal{W}_{\gamma}=\frac{1}{\Delta
t}\int_{t_{i}}^{t_{i}+\Delta
t}\left[2\tilde{Q}_{\gamma}^{2}(t)-\tilde{\sigma}_{\gamma}(t)\right]dt\,.$
(31)

The time-averaged over $\Delta t$ is meant to take into account a finite
acquisition time during the experiment. When $\mathcal{W}_{\gamma}$ is
negative, one can expect with high probability that the trajectory $\gamma$
leads to qubits in an entangled state. When it is positive, one can not draw a
definite conclusion. However, and as stated before, this quantity implies an
ensemble-average $\langle\\!\langle\cdot\rangle\\!\rangle_{dt}$ over all
trajectories occurring during the finite time interval $dt$, which is not yet
optimal for experimental purposes. We therefore define an estimator
$\mathcal{E}_{ss}$ (with the label $ss$ referring to single-shot), where the
ensemble average in the definition of $\sigma_{\gamma}(t)$ Eq.(21) is replaced
by a time average over an interval $\tau$. This procedure is similar to a
coarse-graining of the fluctuations along a single trajectory. The witness
(31) then transforms into an estimator, valid at the level of single
trajectory:

$\displaystyle\mathcal{E}_{ss}\\!\\!=\\!\\!\frac{1}{\Delta
t}\int_{t_{i}}^{t_{i}+\Delta
t}\left[2\tilde{Q}_{\gamma}(t)^{2}-\sqrt{\frac{1}{\tau}\int_{t}^{t+\tau}\delta\tilde{Q}_{\gamma}^{2}(t^{\prime})\,dt^{\prime}}\right]\\!dt\,.$

The performance of $\mathcal{E}_{ss}$ as an estimator to attest entanglement
between the qubits is analyzed through two figures of merit, its success rate
(ratio of detected vs. total number of trajectories leading to entangled
states) and its error rate (whenever $\mathcal{E}_{ss}$ takes a negative value
whereas the trajectory does not lead to entangled qubits). The error rate
takes the maximal value of $0.2\%$ for the coarse-graining times $\tau=0.1$
and $\tau=0.4$ and $1.2\%$ for $\tau=\Delta t$. Indeed, as the coarse-graining
time increases, the time average leads to trusty measurements, as expected
from the analytical bound derived with the ensemble-averaged. The finite error
rate forbids us to claim an energetic witness, but its small value
demonstrates the usefulness of our energetic-based estimator. As shown in
Fig.4, the success rate of $\mathcal{E}_{ss}$ does not strongly depend on the
overall integration window $\Delta t$, but rather on the initial time $t_{i}$
from which the averages are performed. At large $t_{i}$, the success rate
reaches 1 as expected from the convergence of each trajectory towards one of
the eigenstate of the measurement operator $\hat{\Phi}$. However, the success
rate exceeds 0.5 for $t_{i}$ as small as $3/\Gamma$ for $\tau=\Delta
t=0.3/\Gamma$, with a corresponding error rate of $0.4\%$, being promising for
future experiments towards single-shot energetic-based estimators to certify
the presence of entanglement. Finally, as a first step towards realistic
implementation, we have also investigated the robustness of our estimator
against finite detection efficiency. We have numerically simulated
trajectories when the half-parity detection channel has an efficiency
$\eta<1$, which leads to mixed state trajectories $\rho_{\gamma}(t)$ [29], see
App. C . As a consequence, the analytic justification for the estimator is not
valid anymore as the formula for the concurrence has to be modified for mixed
state[34] and there is no analytic expression for the state conditioned to a
given readout $J_{\gamma}$. However, a numerical treatment is possible, and it
is straightforward to extend the definition (5) to mixed state trajectories,
using the average heat increment along mixed state trajectory $\gamma$ defined
as $\delta Q_{\gamma}(t)=\text{Tr}\\{d\rho_{\gamma}(t)\hat{H}_{\cal S}\\}$.
The latter is related to the populations of the states
$|{\uparrow\uparrow}\rangle$ and $|{\downarrow\downarrow}\rangle$ and of the
even and odd subspaces in the same way as in Eq.(18), with an additional
overall factor $\sqrt{\eta}$. The generation of entanglement happens to be
quite robust to finite detection efficiency, as witnessed by the experimental
implementations [14, 15]. The success rate, computed from a sample of $1000$
numerically generated trajectories, is plotted for finite efficiency in Fig.
4d)-f) for different values of $\eta$, showing that our method is robust
against finite detection efficiency. High success rates (up to $80\%$ for
$\Delta t=10/\Gamma$) are predicted even for $\eta=0.5$ which is the order of
magnitude of the experimental conditions. The error rate remains smaller or of
the order of $1\%$.

##  6 Discussion

In this work, we investigate the energy fluctuations associated with
entanglement genesis during the paradigmatic half-parity measurement
procedure. Not only this measurement generates both product states and
entangled states, but it also constitutes an energy observable. Based on a
quantum-trajectory approach, and making use of the framework provided by
stochastic thermodynamics, we demonstrate that the generation of entanglement
is closely related to the presence of quantum heat fluctuations induced by the
stochasticity of the weak continuous measurement. We derive analytical upper
and lower bounds for the entanglement genesis rate. We then exploit the lower
bound to derive an estimator that is solely based on energetic observables and
their fluctuations. We show that this energetic-based estimator is indeed able
to attest the presence of entanglement at a finite time for a given quantum
trajectory, with a tunable probability that depends on the total time-averaged
window $\Delta t$. Remarkably, this single-shot estimator is valid at the
level of a single trajectory and so does not require any ensemble average.
Remarkably, the energetic-based estimator we proposed is robust to finite
detection efficiency, reaching a success rate close to 1 for suitable
integration times as discussed earlier.

These results do not aim at evaluating the energetic cost of creating quantum
correlations, see for instance Refs.[42, 43, 44], but rather at defining an
energetic signature associated to their generation. Whereas the quantum heat
is zero on average, the quantum fluctuations $\delta Q$ during a finite time
interval $dt$ contain relevant information to attest the presence of
entanglement.

Exact in the context of the half-parity measurement process which also plays
the role of an energetic filter, our work fixes theoretical tools for
thermodynamic analysis of the generation of quantum correlations and opens the
way to develop witnesses based exclusively on the measurements of energetic
quantities for quantum information purposes. This theoretical research is
additionally motivated by recent experimental achievements in the emergent
field of quantum caloritronics, aiming at controlling and measuring energetic
observables like the heat current in various quantum circuits [45, 46, 47, 48,
49, 50, 51].

Acknowledgements G.H. gratefully acknowledges discussions with J. Anders, A.
Jordan and I. Siddiqi. C.E. acknowledges the US Department of Energy grant No.
DE-SC0017890. This research was supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958. G.H. thanks the Swiss National
Science Foundation for support through the MHV grant 164466 and starting grant
PRIMA PR00P2$\\_$179748\. This research was supported in part by the National
Science Foundation under Grant No. NSF PHY17-48958.

##  Appendix A Long-time limit formula between the concurrence derivative and
infinitesimal heat fluctuations

The fluctuations of the heat increment associated to the lose of coherences
between the two parity subspaces are defined in a similar was as
$\sigma_{\gamma}$:

$\displaystyle\sigma_{\gamma}^{(eo)}(t)$ $\displaystyle=$
$\displaystyle\sqrt{\langle\\!\langle\delta
Q_{\gamma}^{(eo)2}(t)\rangle\\!\rangle_{dt}}\,.$ (33) $\displaystyle=$
$\displaystyle 2\epsilon\sqrt{\Gamma dt}\frac{e^{-2\Gamma t}\cosh{4\Gamma
tJ_{\gamma}}}{\left(1+e^{-2\Gamma t}\cosh{4\Gamma tJ_{\gamma}}\right)^{2}}\,.$
(34)

We can compare them to the derivative of the concurrence averaged over
realizations occurring during the time interval $dt$:

$\displaystyle\left\langle\\!\\!\\!\left\langle\frac{d\mathcal{C}_{\gamma}}{dt}\right\rangle\\!\\!\\!\right\rangle_{dt}$
$\displaystyle=$ $\displaystyle 2\Gamma\frac{e^{-2\Gamma t}+e^{-2\Gamma
t}\cosh(4\Gamma tJ_{\gamma})}{(1+e^{-2\Gamma t}\cosh(4\Gamma
tJ_{\gamma}))^{2}}$ (35)
$\displaystyle-4\Gamma\,\frac{Q_{\gamma}}{\epsilon}\,\mathcal{C}_{\gamma}\langle\hat{\Phi}(t)\rangle_{\gamma}.$

When $t\gg 1/\Gamma$, the probability distribution of the measurement outcome
$J_{\gamma}$ is narrowly peaked around the three values corresponding to the
eigenvalues of the half-parity measurement operator $\hat{\Phi}$ [29], i.e.
$\pm 1,0$. The average value of $\hat{\Phi}$ tends also to one of the three
eigenvalues. Hence, for trajectories corresponding to qubits in a maximally
entangled state at long times,
$J_{\gamma}=\langle\hat{\Phi}(t)\rangle_{\gamma}=0$ and the heat flow
$Q_{\gamma}=0$. Consequently, Eqs. (34) and (35) simplify to

$\displaystyle\left.{\sigma}_{\gamma}^{(eo)}\right|_{J_{\gamma}=0}$
$\displaystyle=$ $\displaystyle 2\epsilon\sqrt{\Gamma dt}\frac{e^{-2\Gamma
t}}{(1+e^{-2\Gamma t})^{2}}$ (36)
$\displaystyle\left.\left\langle\\!\\!\\!\left\langle\frac{d\mathcal{C}_{\gamma}}{dt}\right\rangle\\!\\!\\!\right\rangle_{dt}\right|_{J_{\gamma}=0}$
$\displaystyle=$ $\displaystyle 4\Gamma\frac{e^{-2\Gamma t}}{(1+e^{-2\Gamma
t})^{2}}\,.$ (37)

When time exceeds the measurement time, the following equality holds:

$\displaystyle\left.\left\langle\\!\\!\\!\left\langle\frac{d\mathcal{C}_{\gamma}}{dt}\right\rangle\\!\\!\\!\right\rangle_{dt}\right|_{J_{\gamma}=0}=\frac{2}{\epsilon}\sqrt{\frac{\Gamma}{dt}}\left.{\sigma}_{\gamma}^{(eo)}\right|_{J_{\gamma}=0}\,.$
(38)

Although only valid and meaningful at times longer than the measurement time,
this relation exemplifies the underlying fundamental role of infinitesimal
heat fluctuations for the generation of entanglement. As stated in the main
text, this relation is only exact in the context of the half-parity
measurement considered in this work and can not be exploited experimentally.
Indeed, one could not distinguish in an experiment infinitesimal heat
fluctuations originating in the loss of phase coherence between the two parity
subspaces ($\sigma_{\gamma}^{(eo)}$) from total infinitesimal heat
fluctuations ($\sigma_{\gamma}$).

##  Appendix B Energetic bounds for the entanglement genesis rate

The general expression of the concurrence derivative reads

$\displaystyle\frac{d\mathcal{C}_{\gamma}}{dt}$ $\displaystyle=$
$\displaystyle 2\Gamma\frac{e^{-2\Gamma t}+e^{-2\Gamma t}\cosh(4\Gamma
tJ_{\gamma})}{(1+e^{-2\Gamma t}\cosh(4\Gamma tJ_{\gamma}))^{2}}$
$\displaystyle-4\Gamma\frac{e^{-2\Gamma t}\sinh(4\Gamma
tJ_{\gamma})(1-e^{-2\Gamma t})d(J_{\gamma}t)/dt}{(1+e^{-2\Gamma
t}\cosh(4\Gamma tJ_{\gamma}))^{2}}$

Using Eqs. (6),(8) and (14), it can be rewritten as

$\displaystyle\frac{d\mathcal{C}_{\gamma}}{dt}$ $\displaystyle=$
$\displaystyle 2\Gamma\frac{e^{-2\Gamma t}+e^{-2\Gamma t}\cosh(4\Gamma
tJ_{\gamma})}{(1+e^{-2\Gamma t}\cosh(4\Gamma tJ_{\gamma}))^{2}}$ (40)
$\displaystyle-4\Gamma\,\frac{Q_{\gamma}}{\epsilon}\,\mathcal{C}_{\gamma}\left(\langle\hat{\Phi}(t)\rangle_{\gamma}+\dfrac{dW_{\gamma}(t)}{dt}\right).$

To enable the comparison with $\tilde{\sigma}_{\gamma}$, we perform the
ensemble average $\langle\\!\langle\cdot\rangle\\!\rangle_{dt}$ and obtain:

$\displaystyle\left\langle\\!\\!\\!\left\langle\frac{d\mathcal{C}_{\gamma}}{dt}\right\rangle\\!\\!\\!\right\rangle_{dt}$
$\displaystyle=$ $\displaystyle 2\Gamma\frac{e^{-2\Gamma t}+e^{-2\Gamma
t}\cosh(4\Gamma tJ_{\gamma})}{(1+e^{-2\Gamma t}\cosh(4\Gamma
tJ_{\gamma}))^{2}}$ (41)
$\displaystyle-4\Gamma\,\frac{Q_{\gamma}}{\epsilon}\,\mathcal{C}_{\gamma}\langle\hat{\Phi}(t)\rangle_{\gamma}\,,$

which corresponds to Eq.(26) in the main text.

Using the inequalities

$\displaystyle e^{-2\Gamma t}+e^{-2\Gamma t}\cosh(4\Gamma tJ_{\gamma})\leq
2e^{-2\Gamma t}\cosh(4\Gamma tJ_{\gamma})\,,$

and

$\displaystyle e^{-2\Gamma t}+e^{-2\Gamma t}\cosh(4\Gamma tJ_{\gamma})\geq
e^{-4\Gamma t}+e^{-2\Gamma t}\cosh(4\Gamma tJ_{\gamma})\,,$

we can now compare the r.h.s. of these two inequalities with the st. dev. of
the total heat fluctuations $\tilde{\sigma}_{\gamma}(t)$:

$\displaystyle\tilde{\sigma}_{\gamma}(t)=\frac{e^{-4\Gamma t}+e^{-2\Gamma
t}\cosh(4\Gamma tJ_{\gamma})}{(1+e^{-2\Gamma t}\cosh(4\Gamma
tJ_{\gamma}))^{2}}\,.$ (44)

These bounds directly lead to Eqs. (27) and (28) in the main text.

##  Appendix C Finite detection efficiency

When the efficiency of the detector takes a finite value $\eta$, the state of
the two qubits along a given trajectory $\gamma$ is a mixed state
$\rho_{\gamma}(t)$ which obeys [29]:

$\displaystyle d\rho_{\gamma}(t)=-idt[\hat{H}_{\cal S},\rho_{\gamma}(t)]$
$\displaystyle+\frac{\Gamma
dt}{2}\left(\hat{\Phi}\rho_{\gamma}(t)\hat{\Phi}-\tfrac{1}{2}\\{\hat{\Phi}^{2},\rho_{\gamma}(t)\\}\right)$
$\displaystyle+\sqrt{\eta\Gamma}dW_{\gamma}(t)\left(\\{\hat{\Phi},\rho_{\gamma}(t)\\}-2\langle\hat{\Phi}(t)\rangle\rho_{\gamma}(t\right),$
(45)

where $\\{A,B\\}=AB+BA$. The measurement record $I_{\gamma}(t)$ is now linked
to the Wiener increment $dW_{\gamma}(t)$ via:

$\displaystyle
I_{\gamma}(t)=\langle\hat{\Phi}(t)\rangle_{\gamma}+\frac{dW_{\gamma}(t)}{2\sqrt{\eta\Gamma}dt}.$
(46)

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