# Fractional Delay Alignment Modulation for Spatially Sparse Wireless
Communications††thanks: This work was supported by the National Natural
Science Foundation of China under Grant 62071114, by the ”Program for
Innovative Talents and Entrepreneur in Jiangsu” under grant number 1104000402,
and also by the Outstanding Projects of Overseas Returned Scholars of Nanjing
under grant 1104000396.

Zhiwen Zhou1, Zhiqiang Xiao12, and Yong Zeng12  1National Mobile
Communications Research Laboratory, Southeast University, Nanjing 210096,
China  2Purple Mountain Laboratories, Nanjing 211111, China  Email:
{220220790, zhiqiang xiao, yong<EMAIL_ADDRESS>
###### Abstract

Delay alignment modulation (DAM) is a novel transmission technique for
wireless systems with high spatial resolution by leveraging _delay
compensation_ and _path-based beamforming_ , to mitigate the inter-symbol
interference (ISI) without resorting to complex channel equalization or multi-
carrier transmission. However, most existing studies on DAM consider a
simplified scenario by assuming that the channel multi-path delays are integer
multiples of the signal sampling interval. This paper investigates DAM for the
more general and practical scenarios with fractional multi-path delays. We
first analyze the impact of fractional multi-path delays on the existing DAM
design, termed _integer DAM_ (iDAM), which can only achieve delay
compensations that are integer multiples of the sampling interval. It is
revealed that the existence of fractional multi-path delays renders iDAM no
longer possible to achieve perfect delay alignment. To address this issue, we
propose a more generic DAM design called _fractional DAM_ (fDAM), which
achieves fractional delay pre-compensation via upsampling and fractional delay
filtering. By leveraging the _Farrow_ filter structure, the proposed approach
can eliminate ISI without real-time computation of filter coefficients, as
typically required in traditional channel equalization techniques. Simulation
results demonstrate that the proposed fDAM outperforms the existing iDAM and
orthogonal frequency division multiplexing (OFDM) in terms of symbol error
rate (SER) and spectral efficiency, while maintaining a comparable peak-to-
average power ratio (PAPR) as iDAM, which is considerably lower than OFDM.

###### Index Terms:

Delay alignment modulation; fractional delay; single-carrier communication;
OFDM

##  I Introduction

Inter-symbol interference (ISI) is a critical issue for broadband
communications systems under time-dispersive multi-path channels. To resolve
this issue, numerous ISI mitigation techniques have been proposed, such as
channel equalization, RAKE receivers, and orthogonal frequency-division
multiplexing (OFDM). With the ever-increasing scale of antennas [1] and
carrier frequency[2][3], wireless channels are becoming increasingly sparse
[4][5]. Recently, by exploiting the channel spatial sparsity and high spatial
resolution of large antenna arrays, a novel transmission technique termed
delay alignment modulation (DAM) was proposed[6][7]. Different from
conventional methods such as channel equalization or OFDM, DAM leverages two
key techniques to eliminate the ISI, i.e., delay compensation and path-based
beamforming. Thus, DAM has the advantages of smaller guard interval overhead,
lower peak-to-average-power ratio (PAPR), and higher Doppler resilience
compared to OFDM [8].

However, most existing studies on DAM consider a simplified scenario where the
multi-path channel delays are assumed to be integer multiples of the signal
sampling interval [6][7]. In this case, to achieve perfect DAM so that all
multi-path signals arrive at the receiver simultaneously and constructively,
the transmitter only needs to deliberately introduce delay compensations that
are also integer multiples of the sampling interval. We term such existing DAM
designs [6][7] as _integer DAM_ (iDAM). However, for practical scenarios, the
multi-path delays are not necessarily integer multiples of the sampling
interval. Thus, if the existing iDAM design is directly applied, there may
exist residual fractional delays, whose length is non-negligible as compared
to the symbol duration in single-carrier systems. Specifically, the resulting
residual fractional delays after the conventional iDAM can lead to time
misalignment, causing residual ISI [9]. To tackle this issue, a tap-based DAM
design was proposed in [10], where delay compensation and beamforming are
performed based on the delay taps, with each tap containing multiple physical
paths. However, fractional multi-path delays can still cause the delay tap
power leakage issue [11], which may result in strong spatial correlation among
adjacent delay taps, rendering the tap-based zero-forcing (ZF) and maximum
ratio transmission (MRT) beamforming ineffective.

In this paper, we investigate DAM for a more general and practical wireless
communication system with fractional multi-path delays. First, we study the
impact of fractional multi-path delays on the existing iDAM designs[6][7]. The
analytic results demonstrate that in the presence of fractional delays,
although iDAM can still significantly reduce the channel delay spread, perfect
delay alignment is no longer possible. Thus, the resulting residual ISI after
iDAM may degrade the system performance. To address this issue, we propose a
more generic DAM design, termed _fractional DAM_ (fDAM), by noting that
perfect delay alignment is still doable even with fractional multi-path
delays, as long as fractional delay compensations can be achieved. To this
end, we employ the upsampling and fractional delay filtering techniques at the
transmitter side. By leveraging the _Farrow_ filter structure, the proposed
approach can achieve fractional delay compensation without real-time
computation of filter coefficients, as typically required in traditional
channel equalization techniques. Moreover, as compared to the use of high
upsampling ratios or dedicated hardware such as true time delay (TTD) lines,
the proposed approach achieves fractional delay compensation with a relatively
low hardware cost, as it only requires an upsampling ratio of 2. Extensive
simulation results demonstrate that the proposed fDAM outperforms the existing
iDAM and OFDM in terms of symbol error rate (SER) and spectrum efficiency for
scenarios with fractional multi-path delays. Furthermore, the proposed fDAM
can achieve a comparable peak-to-average power ratio (PAPR) as iDAM, which is
far lower than that of OFDM.

##  II System Model

Figure 1: A spatially sparse wireless communication system with fractional
multi-path delays.

As shown in Fig. 1, a spatially sparse multiple-input single-output (MISO)
wireless communication system is considered, where a base station (BS)
equipped with $M\gg 1$ transmit antennas wishes to serve a single-antenna
user. Let $L$ denote the number of physical channel paths, which is much
smaller than $M$ for spatially sparse systems. Thus, the channel impulse
response is written as

$\displaystyle\bm{h}^{H}\left(t\right)=\sum_{l=1}^{L}{\bm{h}_{l}^{H}\delta\left(t-\tau_{l}\right)},$
(1)

where $\bm{h}_{l}\in\mathbb{C}^{M\times 1}$ and $\tau_{l}\geq 0$ denote the
channel coefficient vector and the propagation delay of the $l$th path,
respectively.

Note that different from most existing works on DAM [6, 7, 8] which assume
that the multi-path delays are integer multiples of $T$, where $T$ is the
symbol duration, in this paper, we consider a more general and practical case
without making the above assumption. In this case, the delay of the $l$th path
$\tau_{l}$ can be expressed as

$\displaystyle\tau_{l}=n_{l}T+\tau_{f,l},$ (2)

where $n_{l}=\left.\lfloor\left.\frac{\tau_{l}}{T}\right.\rceil\right.$ and
$\tau_{f,l}=\left(\tau_{l}-n_{l}T\right)\in\left[\left.-\frac{T}{2},\frac{T}{2}\right)\right.$
are the integer and fractional parts of the delay, respectively, with
$\lfloor\cdot\rceil$ denoting round to the nearest integer.

Figure 2: Transmitter structure of existing iDAM [6].

As shown in Fig. 2, with the existing iDAM design proposed in [6], the
transmitted baseband discrete-time signal is

$\displaystyle\bm{x}\left[n\right]=\sum_{l=1}^{L}{\bm{f}_{l}s\left[n-\kappa_{l}\right]},$
(3)

where $\\{{s\left[n\right]}\\}_{n=-\infty}^{\infty}\ $ are the independent and
identically distributed (i.i.d) information-bearing symbols with normalized
power $\mathbb{E}\\{\left|s\left[n\right]\right|^{2}\\}=1$,
$\bm{f}_{l}\in\mathbb{C}^{M\times 1}$ and $\kappa_{l}\in\mathbb{C}^{M\times
1}$ are the path-based beamforming vector and the deliberately introduced
delay compensation for the $l$th path, respectively. Note that
$\sum_{l=1}^{L}{\left\|\bm{f}_{l}\right\|^{2}}\leq PT$, where $P$ is the
maximum transmit power. For the conventional iDAM,
$\\{\kappa_{l}\\}_{l=1}^{L}\ $ are non-negative integers. Let
$\varphi\left(t\right)$ denote the transmit pulse shaping filter. Then the
transmitted baseband continuous-time signal is

$\displaystyle\bm{x}\left(t\right)=\sum_{n=-\infty}^{\infty}\bm{x}\left[n\right]\varphi\left(t-nT\right).$
(4)

By substituting (3) into (4), we have

$\displaystyle\bm{x}\left(t\right)$
$\displaystyle=\sum_{n=-\infty}^{\infty}{\sum_{l=1}^{L}{\bm{f}_{l}s\left[n\right]}\varphi\left(t-nT-\kappa_{l}T\right)}$
(5) $\displaystyle=\sum_{l=1}^{L}\bm{f}_{l}s\left(t-\kappa_{l}T\right),$

where $s\left(t\right)$ is given by

$\displaystyle
s\left(t\right)\triangleq\sum_{n=-\infty}^{\infty}s\left[n\right]\varphi\left(t-nT\right).$
(6)

Thus, with the channel impulse response in (1), the signal received at the
receiver is

$\displaystyle y\left(t\right)$
$\displaystyle=\bm{h}^{H}\left(t\right)\ast\bm{x}\left(t\right)+z\left(t\right)$
(7)
$\displaystyle=\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}s\left(t-\kappa_{l}T-\tau_{l}\right)}$
$\displaystyle+\sum_{l=1}^{L}\sum_{l^{\prime}\neq
l}^{L}{\bm{h}_{l}^{H}\bm{f}_{l^{\prime}}s\left(t-\kappa_{l^{\prime}}T-\tau_{l}\right)}+z\left(t\right),$

where $z\left(t\right)$ is the AWGN with power spectral density $\sigma^{2}$.

##  III Impact of Fractional Delays on iDAM

For the existing iDAM design[6][7], it is assumed that the multi-path delays
are integer multiples of $T$, i.e., $\tau_{f,l}=0,\forall l$ in (2). In this
case, the deliberately introduced delay compensation in (3) can be set as
$\kappa_{l}=n_{\max}-n_{l}\geq 0$, with
$n_{\max}=\underset{l}{\max}\left\\{n_{l}\right\\},l=1,\ldots,L$. In this
section, we aim to study the impact of fractional multi-path delays if the
existing iDAM design is directly applied. In this case, by substituting
$\tau_{l}$ in (2) and the integer $\kappa_{l}$ design into (7), we have

$\displaystyle y\left(t\right)$
$\displaystyle=\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}s\left(t-n_{\max}T-\tau_{f,l}\right)$
(8) $\displaystyle+\sum_{l=1}^{L}{\sum_{l^{\prime}\neq
l}^{L}{\bm{h}_{l}^{H}\bm{f}_{l^{\prime}}s\left(t-n_{\max}T+n_{l^{\prime}}T-\tau_{l}\right)}}+z\left(t\right).$

With the path-based ZF beamforming for $\\{\bm{f}_{l}\\}_{l=1}^{L}$[6], which
ensures that $\bm{h}_{l}^{H}\bm{f}_{l^{\prime}}=0,\forall l\neq l^{\prime}$,
(8) reduces to

$\displaystyle y\left(t\right)$
$\displaystyle=\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}s\left(t-n_{\max}T-\tau_{f,l}\right)+z\left(t\right)$
(9)
$\displaystyle=\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\sum_{n=-\infty}^{\infty}{s}\left[n\right]\varphi\left(t-nT-\tau_{f,l}-n_{\max}T\right)$
$\displaystyle+z\left(t\right).$

By applying the receiving matched filter $\varphi\left(t\right)$ to $y(t)$, we
have

$\displaystyle r\left(t\right)$
$\displaystyle=y\left(t\right)\ast\varphi\left(t\right)$ (10)
$\displaystyle=\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\sum_{n=-\infty}^{\infty}s\left[n\right]\varrho\left(t-nT-\tau_{f,l}-n_{\max}T\right)$
$\displaystyle+v\left(t\right),$

where
$\varrho\left(t\right)\triangleq\varphi\left(t\right)*\varphi\left(t\right)$
denotes the matched filter response, and $v\left(t\right)\triangleq
z\left(t\right)*\varphi\left(t\right)$ is the resulting noise after receiving
matched filtering (MF).

For the receiving MF output $r\left(t\right)$, by sampling at
$t=kT+n_{\max}T+T_{\mathrm{peak}}$ with integer $k$, we have

$\displaystyle r\left[k\right]$
$\displaystyle=\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\sum_{n=-\infty}^{\infty}s\left[n\right]\psi\left(\left(k-n\right)T-\tau_{f,l}\right)+v\left[k\right]$
(11)
$\displaystyle=\underbrace{\left(\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\psi\left(-\tau_{f,l}\right)\right)s\left[k\right]}_{\mathrm{desired\,signal}}$
$\displaystyle+\underbrace{\sum_{n\neq
k}{\left(\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}\psi\left(\left(k-n\right)T-\tau_{f,l}\right)}\right)s\left[n\right]}}_{\mathrm{ISI}}+v\left[k\right],$

where $\left\\{v[k]\right\\}_{k=-\infty}^{\infty}$ are i.i.d complex normal
variables satisfying $v[k]\\!\sim\\!\mathcal{C}\mathcal{N}(0,\sigma^{2})$[12]
and
$\psi\left(t\right)\\!\triangleq\\!\varrho\left(t+T_{\mathrm{peak}}\right)$
with
$T_{\mathrm{peak}}\\!=\\!\underset{t}{\max}\left\\{\varrho\left(t\right)\right\\}$
being the time instant of the peak MF output. In practice,
$\varrho\left(t\right)$ must be causal, thus we have
$T_{\mathrm{peak}}\\!>\\!0$. The pulse $\psi\left(t\right)$ satisfies the
Nyquist ISI-free property $\psi\left(nT\right)\\!=\\!\delta\left[n\right]$,
where $\delta[\cdot]$ denotes the Kronecker delta function.

Note that different from [6, 7, 8], due to the existence of fractional delays
$\tau_{f,l}$, there exists residual ISI even after iDAM, as shown in the
second term of (11). By noting that $s[n]$ are i.i.d. with normalized power,
the output signal-to-interference-plus-noise ratio (SINR) is

$\displaystyle\gamma$
$\displaystyle=\frac{\left|\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\psi\left(-\tau_{f,l}\right)\right|^{2}}{\sum_{n\neq
k}\left|\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\psi\left(\left(k-n\right)T-\tau_{f,l}\right)\right|^{2}+\sigma^{2}}.$
(12)

If the multi-path delays were integer multiples of $T$, as assumed in [6, 7,
8], i.e. $\tau_{f,l}=0,\forall l$, then by using the property
$\psi\left(nT\right)=\delta\left[n\right]$, (12) reduces to

$\displaystyle\gamma=\frac{\left|\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\right|^{2}}{\sigma^{2}},$
(13)

which is the ideal signal-to-noise ratio (SNR) for DAM without ISI [6].

However, in practice, due to the existence of fractional multi-path delays,
$\tau_{f,l}\neq 0$, the ISI term is non-zero. Moreover, due to the deviated
sampling time $-\tau_{f,l}$, the desired signal power in the numerator of (12)
is also compromised. Thus, if the conventional iDAM is directly applied in
practical systems with fractional multi-path delays, it will result in
performance degradation, which will be verified via simulations in Section V.
To address this issue, in the following, we propose a novel DAM design called
fDAM.

##  IV Fractional DAM

Intuitively, in order to achieve perfect delay alignment even with fractional
multi-path delays, the transmitter needs to realize fractional delay
compensations. Thus, the transmitted signal in (5) should be designed as

$\displaystyle\bm{x}\left(t\right)$
$\displaystyle=\sum_{l=1}^{L}\bm{f}_{l}s\left(t-\rho_{l}\right),$ (14)

where $\rho_{l}=\tau_{\max}-\tau_{l}$ denotes the fractional delay
compensation, with
$\tau_{\max}\triangleq\underset{l}{\max}\left\\{\tau_{l}\right\\},l=1,\ldots,L$.
Note that different from (5), $\rho_{l}$ may not be integer multiples of $T$.
The received signal in (7) is then rewritten as

$\displaystyle y\left(t\right)$
$\displaystyle=\bm{h}^{H}\left(t\right)\ast\bm{x}\left(t\right)+z\left(t\right)$
(15)
$\displaystyle=\left(\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\right)s\left(t-\tau_{\max}\right)$
$\displaystyle+\sum_{l=1}^{L}\sum_{l^{\prime}\neq
l}^{L}{\bm{h}_{l}^{H}\bm{f}_{l^{\prime}}s\left(t-\tau_{\max}+\tau_{l^{\prime}}-\tau_{l}\right)}+z\left(t\right).$

Thus, with the path-based ZF beamforming such that
$\bm{h}_{l}^{H}\bm{f}_{l}^{\prime}=0,\forall l\neq l^{\prime}$, the resulting
received signal is

$\displaystyle
y\left(t\right)=\left(\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\right)s\left(t-\tau_{\max}\right)+z\left(t\right),$
(16)

where different from (9), the ISI is completely eliminated, thanks to the
fractional delay pre-compensation in (14). In this case, we would have the
same SNR expression as (13) even with fractional multi-path delays.

However, it is impossible to achieve the desired fractional delay compensation
with the existing iDAM design [6] shown in Fig.2, since the delay compensation
therein is performed in the digital domain, which must be integer multiples of
the symbol duration $T$. To achieve the desired fractional delay compensation,
there exist at least three potential solutions.

One possible solution is to increase the sampling rate at the transmitter, say
dozens or even hundreds times the symbol rate, so that the sampling interval
is only a small fraction of the symbol duration $T$. As a result, the
fractional delay shifts can be well approximated by the digital time shifts
with negligible errors. However, this method will drastically increase the
hardware requirements of the transmitter, because high-speed digital-to-analog
converters (DACs) are required.

Another potential solution is to move the delay compensation operation from
the digital domain to the analog domain. To this end, hardware components like
TTD lines could be used to achieve analog time shifts, but they will
significantly increase the cost since $L$ TTD lines are required for perfect
delay alignment.

Fortunately, by exploiting the fact that the transmitted signal is band-
limited, in this paper, we propose an efficient method for achieving
fractional delay compensation by the use of digital filters known as
fractional delay filters [13]. These filters are able to achieve fractional
time shifts in the digital domain with low complexity and hardware
requirements and avoid prohibitive upsampling factors required by the first
method mentioned above, or additional hardware components like TTD lines.

Figure 3: Transmitter structure of the proposed fDAM.

Fig. 3 shows the transmitter structure of the proposed fDAM design, where
fractional delay compensation is achieved by a combination of upsampling with
factor $Q$ and fractional delay filters. In this case, the transmitted
discrete-time signal is

$\displaystyle\bm{x}\left[n\right]=\sum_{l=1}^{L}{\bm{f}_{l}s_{u}\left[n\right]}\ast
d_{Q\zeta_{l}}\left[n\right],$ (17)

where $s_{u}\left[n\right]$ is the upsampled version of $s\left[n\right]$,
with an upsampling factor of $Q$, i.e.,

$\displaystyle s_{u}\left[n\right]=\left\\{\begin{aligned}
&s\left[n/Q\right],&&n\,\,\mathrm{mod}\,\,Q=0\\\ &0,&&\text{otherwise}\\\
\end{aligned}\right.,$ (18)

and $d_{Q\zeta_{l}}\left[n\right]$ is the fractional delay filter with desired
delay $Q\zeta_{l}T_{s}=\zeta_{l}T$, where $Q\zeta_{l}$ is in general non-
integer and $T_{s}=T/Q$ is the sampling interval after upsampling. The value
for desired $Q$ and $\zeta_{l}$ will become clear later. Farrow filter is one
of the most popular fractional delay filters, which can achieve delay shifts
in real-time [14]. Typically, $d_{Q\zeta_{l}}\left[n\right]$ is designed so
that its discrete-time Fourier transform (DTFT) approximates the ideal
fractional delay filter with delay $Q\zeta_{l}$, i.e.,

$\displaystyle D_{Q\zeta_{l}}\left(f\right)\approx e^{-j2\pi fQ\zeta_{l}},\ \
f\in\left[-\frac{1}{2}\right.,\left.\frac{1}{2}\right).$ (19)

Note that due to the periodic nature of DTFTs, $D_{Q\zeta_{l}}\left(f\right)$
can only be designed in one period, thus it is only required for (19) to hold
true for $f\in\left[-\frac{1}{2}\right.,\left.\frac{1}{2}\right)$. Taking into
account the approximation errors, it is appropriate to assume that for
$f\in\left[-0.4\right.,\left.0.4\right)$, the approximation in (19) is
accurate [13]. With the above designs, the transmitted baseband signal by fDAM
can be written as

$\displaystyle\bm{x}\left(t\right)=\sum_{n=-\infty}^{\infty}\bm{x}\left[n\right]\varphi\left(t-\frac{nT}{Q}\right).$
(20)

For ease of exposition, (20) can be equivalently written as

$\displaystyle\bm{x}\left(t\right)=\bm{x}_{i}\left(t\right)\ast\varphi\left(t\right),$
(21)

where

$\displaystyle\bm{x}_{i}\left(t\right)=\sum_{n=-\infty}^{\infty}\bm{x}\left[n\right]\delta\left(t-\frac{nT}{Q}\right).$
(22)

Denote by $f_{B}=1/T$ the symbol rate. The continuous Fourier transform (CFT)
of $\bm{x}_{i}\left(t\right)$ can be expressed as

$\displaystyle\bm{X}_{i,\mathrm{cft}}\left(f\right)$
$\displaystyle=\bm{X}_{i,\mathrm{dtft}}\left(\frac{fT}{Q}\right)=\bm{X}_{i,\mathrm{dtft}}\left(\frac{f}{Q{f}_{B}}\right)$
(23)
$\displaystyle=\sum_{l=1}^{L}{\bm{f}_{l}S_{u}\left(\frac{f}{Q{f}_{B}}\right)}D_{Q\zeta_{l}}\left(\frac{f}{Q{f}_{B}}\right)$
$\displaystyle=\sum_{l=1}^{L}{\bm{f}_{l}S\left(Q\frac{f}{Q{f}_{B}}\right)}D_{Q\zeta_{l}}\left(\frac{f}{Q{f}_{B}}\right)$
$\displaystyle\overset{(a)}{\approx}\sum_{l=1}^{L}{\bm{f}_{l}S\left(\frac{f}{f_{B}}\right)e^{-j2\pi
fT\zeta_{l}}},$ $\displaystyle
f\in\left[-\frac{Qf_{B}}{2}\right.,\left.\frac{Qf_{B}}{2}\right),$

where $S_{u}(f)$ and $S(f)$ are the DTFT of $s_{u}[n]$ and $s[n]$
respectively, and (a) holds due to (19). Then, the CFT of the transmitted
signal is

$\displaystyle\bm{X}\left(f\right)$
$\displaystyle=\bm{X}_{i,\mathrm{cft}}\left(f\right)\mathrm{\varPhi}\left(f\right)$
(24)
$\displaystyle\approx\sum_{l=1}^{L}{\bm{f}_{l}S\left(\frac{f}{f_{B}}\right)\mathrm{\varPhi}\left(f\right)e^{-j2\pi
fT\zeta_{l}}},$ $\displaystyle
f\in\left[-\frac{Qf_{B}}{2}\right.,\left.\frac{Qf_{B}}{2}\right),$

where $\mathrm{\varPhi}\left(f\right)$ is the CFT of $\varphi\left(t\right)$.
It is observed from (24) that the bandwidth of the transmitted signal
$\bm{X}\left(f\right)$ is limited by that of $\mathrm{\varPhi}\left(f\right)$.
As a result, even though upsampling is performed at the transmitter, the
bandwidth of the transmitted signal does not increase, which is desired. Note
that for $f\in\left[-0.4Qf_{B}\right.,\left.0.4Qf_{B}\right)$, the
approximation in (24) is accurate. The upsampling factor $Q$ can then be
chosen according to the bandwidth of $\mathrm{\varPhi}\left(f\right)$. Let
$B_{\mathrm{\varPhi}}$ denote the double-sided bandwidth of
$\mathrm{\varPhi}\left(f\right)$, then $Q$ should be chosen so that

$\displaystyle 0.8Qf_{B}\geqslant B_{\Phi}\,\text{ or }\,Q\geqslant
1.25\frac{B_{\Phi}}{f_{B}}.$ (25)

Typically, for the raised cosine pulse with roll-off factor $\beta$, we have
$B_{\Phi}=(1+\beta)f_{B}$. Thus, (25) can be further written as

$\displaystyle Q\geqslant 1.25\left(1+\beta\right).$ (26)

In particular, for $\beta\leq 0.6$, an upsampling factor $Q=2$ would suffice.
From (24), by using inverse continuous Fourier transform (ICFT), the time-
domain transmitted signal of the proposed fDAM is given by

$\displaystyle\bm{x}\left(t\right)$
$\displaystyle\approx\sum_{n=-\infty}^{\infty}\sum_{l=1}^{L}{\bm{f}_{l}s\left[n\right]\varphi\left(t-\zeta_{l}T-nT\right)}$
(27)
$\displaystyle=\sum_{l=1}^{L}{\bm{f}_{l}\sum_{n=-\infty}^{\infty}s\left[n\right]\varphi\left(t-nT-\zeta_{l}T\right)}$
$\displaystyle=\sum_{l=1}^{L}{\bm{f}_{l}s\left(t-\zeta_{l}T\right)},$

where $s\left(t\right)$ is the information-bearing symbols after pulse shaping
as defined in (6). Thus, with the channel impulse response given in (1), the
received signal is

$\displaystyle y\left(t\right)$
$\displaystyle=\bm{h}^{H}\left(t\right)\ast\bm{x}\left(t\right)+z\left(t\right)$
(28)
$\displaystyle\approx\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}s\left(t-\zeta_{l}T-\tau_{l}\right)}$
$\displaystyle+\sum_{l=1}^{L}\sum_{l^{\prime}\neq
l}^{L}{\bm{h}_{l}^{H}\bm{f}_{l^{\prime}}s\left(t-\zeta_{l^{\prime}}T-\tau_{l}\right)}+z\left(t\right).$

By letting $\zeta_{l}=\frac{\tau_{\max}-\tau_{l}}{T}$, we have

$\displaystyle y\left(t\right)$
$\displaystyle\approx\left(\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\right)s\left(t-\tau_{\max}\right)$
(29) $\displaystyle+\sum_{l=1}^{L}\sum_{l^{\prime}\neq
l}^{L}{\bm{h}_{l}^{H}\bm{f}_{l^{\prime}}s\left(t-\tau_{\max}+\tau_{l^{\prime}}-\tau_{l}\right)}+z\left(t\right).$

Thus, with the path-based ZF beamforming, the received signal can be expressed
as

$\displaystyle
y\left(t\right)\approx\left(\sum_{l=1}^{L}{\bm{h}_{l}^{H}\bm{f}_{l}}\right)s\left(t-\tau_{\max}\right)+z\left(t\right).$
(30)

It is observed from (30) that similar to the original iDAM design under the
assumption of integer multi-path delays, the proposed fDAM can transform the
multi-path channel with fractional delays to a simple AWGN channel with a
single delay $\tau_{\max}$. Besides, all the $L$ multi-path signal components
will contribute to the desired signal power.

##  V Simulation Results

Simulation results are provided to evaluate the performance of the proposed
fDAM. The number of transmit antennas is $M=64$, the carrier frequency is
$f_{c}=28\,\mathrm{GHz}$, and the symbol interval is $T=2.5\,\mathrm{ns}$.
Note that since no ISI exists at the receiver for the proposed fDAM, sinc
pulses can be used for maximum spectral efficiency. However, since sinc pulse
filters cannot be implemented precisely, a raised cosine pulse filter with a
roll-off factor of $\beta=0.05$ is used in the simulation. The bandwidth of
the signal is thus $B=(1+\beta)f_{B}=420\,\mathrm{MHz}$, where
$f_{B}=\frac{1}{T}=400\,\mathrm{MHz}$ is the symbol rate. The number of multi-
paths is $L=3$. For the scenarios with fractional multi-path delays, the
delays are chosen randomly from the continuous interval $[0,200T]$. On the
other hand, for the simplified scenarios by assuming integer multi-path
delays, the above randomly generated delays are rounded so that all the delays
are integer multiples of $T$.

For the benchmarking OFDM scheme, the number of subcarriers is $N_{sc}=1024$,
and the CP length is $N_{cp}T=200T$, which is no smaller than the maximum
possible delay spread. The channel coherence time is $T_{c}=1\,\mathrm{ms}$.
For fairness, the time-domain signal after OFDM modulation is passed through
the same transmit pulse shaping filter used in DAM. This ensures that both
systems transmit with the same bandwidth limit. 16-QAM constellation is used
for both systems. The received SNR varies from $0\mathrm{dB}$ to
$20\mathrm{dB}$. The following results are averaged over 500 channel
realizations and the symbol error rates (SERs) are calculated over 1024
symbols per channel realization.

Firstly, the impact of fractional delays on the existing iDAM is evaluated.
Fig. 4 shows the resulting SER of the iDAM design in [6][7]. It is observed
that, although iDAM gives rather good performance under integer multi-path
delays, the existence of fractional delays can cause severe performance
degradation, regardless of the beamforming schemes used. In particular, due to
the residual ISI caused by fractional delays, iDAM results in SER greater than
$10^{-1}$. Since the performance of ZF and MMSE are similar, in the following,
ZF beamforming is used for further comparison between iDAM and the proposed
fDAM.

Figure 4: SER of the existing iDAM design for scenarios with integer and
fractional multi-path delays. Figure 5: SER of iDAM, fDAM and OFDM for
channels with fractional multi-path delays.

Fig. 5 compares the SER of the proposed fDAM to the conventional iDAM and OFDM
for channels with fractional multi-path delays. It is observed that under the
more general and practical scenarios with fractional multi-path delays, fDAM
can significantly outperform iDAM, thanks to its effective elimination of
residual ISI brought by the fractional delays. Note that for OFDM, the
equivalent channel after the subcarrier-based maximal-ratio transmission (MRT)
beamforming can still be frequency selective even with $M=64$ antennas,
causing different SNRs among subcarriers. By contrast, the fDAM results in
constant SNR that is contributed by all the $L$ multi-path components, as can
be seen in (30). Additionally, the transmit and receive filters of OFDM can
cause attenuation for subcarriers at the edge of the frequency band. As a
result, the proposed fDAM may even slightly outperform OFDM. Note that the SER
overhead caused by CP [7] is not considered here, with which the advantage
would be greater.

Figure 6: Spectrum efficiency of iDAM, fDAM and OFDM for channels with
fractional multi-path delays.

The spectral efficiency of iDAM, fDAM and OFDM is shown in Fig. 6. Under
fractional multi-path delays, since fDAM can greatly improve SINR by
completely eliminating the ISI, the achievable rate of fDAM is considerably
larger than that of iDAM. Besides, the proposed fDAM also achieves
significantly larger spectrum efficiency than OFDM. This is mainly due to the
saving of guard interval overhead. Specifically, for fDAM, the overhead
introduced by the guard interval between different channel coherence blocks is
very small [8], and the overhead here is mainly determined by the roll-off
factor of the pulse shaping filter, which is around
$\frac{\beta}{(1+\beta)}=4.8\%$. This can be further lowered by reducing the
roll-off factor, and the limit is close to 0. For OFDM, the main overhead is
caused by CP, which can be calculated as
$\frac{N_{cp}}{N_{sc}+N_{cp}}=16.3\%$. Note that the overhead for OFDM can
theoretically be lowered to near 0 by increasing $N_{sc}$ so that
$(N_{sc}+N_{cp})T=T_{c}$, but this would mean $N_{sc}=399800$ subcarriers,
which is impractical since it would lead to extremely high PAPR and
vulnerability to frequency offsets.

Figure 7: PAPR of iDAM, fDAM and OFDM.

The PAPR of iDAM, fDAM and OFDM is compared in Fig. 7. Different from the PAPR
calculated in [8], here, the PAPR is calculated in the analog domain after
transmit filtering. As shown in Fig. 7, the PAPR of fDAM is slightly worse
than that of iDAM, but their upper limit is approximately the same. OFDM has
the worst PAPR among the three, which is expected since it is a multi-carrier
transmission technique. Note that the gap between OFDM and iDAM is smaller
compared to previous work [8]. This is due to the aforementioned transmit
filtering, without which the gap would be significantly larger.

##  VI Conclusion

This paper first analyzed the impact of fractional multi-path delays on
existing iDAM techniques, and then a novel method called fDAM was proposed to
achieve perfect delay alignment by leveraging upsampling with factor $2$ and
fractional delay filters. Simulation results demonstrated the superior
performance of the proposed fDAM technique over the existing iDAM design and
the conventional OFDM in terms of SER, spectral efficiency and PAPR.

## References

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# What is the most general class of quasiprobabilities of work?

Gianluca Francica<EMAIL_ADDRESS>Dipartimento di Fisica e
Astronomia “G. Galilei”, Università degli Studi di Padova, via Marzolo 8,
35131 Padova, Italy

###### Abstract

How to give a statistical description of thermodynamics in quantum systems is
an open fundamental question. Concerning the work, the presence of initial
quantum coherence in the energy basis can give rise to a quasiprobability of
work, which can take negative values. Our aim is to identify the most general
quasiprobability of work satisfying some fundamental conditions. By doing so,
we introduce a general notion of quasiprobability in analogy to the Gleason’s
theorem. Then, we use these quasiprobabilities to define the quasiprobability
of work, and finally we discuss the contextuality of the protocol.

##  I Introduction

Thermodynamics can be derived by using a statistical description of nature.
However, how to give such description in quantum systems still remains an open
problem of fundamental importance. In particular, several attempts have been
made to describe the work statistics, after the two-projective-measurement
scheme has been originally proposed talkner07 . For instance, among these,
work distributions have been defined in terms of a work operator deffner16 ,
gaussian measurements talkner16 , full-counting statistics solinas15 , weak
values allahverdyan14  and consistent histories miller17 . In particular, the
two quasiprobabilities of Refs. solinas15 ; allahverdyan14  can be viewed as
particular cases of a more general quasiprobability francica22 . To briefly
introduce the problem, we recall that if the work performed on a thermally
isolated quantum system is equal to the energy change of the system, then the
two-projective-measurement scheme fails to describe the work statistics in the
presence of initial quantum coherence in the energy basis. Basically, in this
invasive scheme, where two projective measurements of the energy are performed
at the initial and final times to infer the work statistics, the first
measurement destroys the initial coherence in the energy basis. A no-go
theorem perarnau-llobet17  states that there is no scheme having a probability
distribution of work, which is linear with respect to the initial state, such
that it reduces to the two-projective-measurement scheme for incoherent states
and the average work corresponds to the average energy change. Thus, if these
latter conditions are satisfied, we have to look for a quasiprobability
instead of a probability. As shown in Ref. lostaglio18 , this can be related
to the contextuality of the protocol. Now a fundamental question arises: what
is the quasiprobability of work which satisfies these conditions? Going in
this direction, a class of quasiprobabilities has been introduced in Ref.
francica22 . However, this does not exclude the possible existence of
different quasiprobabilities. Here, to give an answer we aim to deduce what
the work quasiprobability is, starting from some fundamental conditions. To do
this, in Sec. II we introduce a general notion of quasiprobability in analogy
to the well-known Gleason’s theorem gleason57 , starting from Ref. bush03 .
Thus, in Sec. III we use this notion to naturally define a quasiprobability of
work, and we determinate its form if some conditions need to be satisfied. To
identify the quantum features and signatures of the work statistics, we
discuss the contextuality of the protocol in Sec. IV. Finally, we summarize
and discuss our results in Sec. V.

##  II Quasiprobability

In general, experimental events (or facts) are represented as effects, which
are the positive operators which can occur in the range of a positive operator
valued measurement, i.e., an effect is a Hermitian operator $E$ acting on the
Hilbert space $\mathcal{H}$ such that $0\leq E\leq I$. The generalized
probability measures on the set of effects are functions $E\mapsto v(E)$ with
the properties

$\displaystyle 0\leq v(E)\leq 1\,,$ (1) $\displaystyle v(I)=1\,,$ (2)
$\displaystyle v(E+F+\cdots)=v(E)+v(F)+\cdots$ (3)

where $E+F+\cdots\leq I$. An analog of Gleason’s theorem bush03  states that,
if Eqs. (1)-(3) are satisfied, then the probability corresponding to the event
represented by $E$ is $v(E)=\text{Tr}\left\\{E\rho\right\\}$ for some density
matrix $\rho$. Here, we try to generalize this result in the case of multiple
events. For simplicity, let us focus on two events. We aim to define a
function $v(E,F)$ with the properties

$\displaystyle v(E,F)\in\mathbb{R}\,,$ (4) $\displaystyle
v(I,E)=v(E,I)=v(E)\,,$ (5) $\displaystyle
v(E+F+\cdots,G)=v(E,G)+v(F,G)+\cdots\,,$ $\displaystyle
v(G,E+F+\cdots)=v(G,E)+v(G,F)+\cdots$ (6)

where $E+F+\cdots\leq I$. In particular, we do not require that $v(E,F)$ is
non-negative, so that we can get a quasiprobability, as can be expected from
the Nelson theorem nelson ; breuer . If Eqs. (4)-(6) are satisfied, and if
$v(E,F)$ is sequentially continuous in its arguments, then the joint
quasiprobability corresponding to the events represented by $E$ and $F$ is a
bilinear function, in detail it is $v(E,F)=Re\text{Tr}\left\\{EF\rho\right\\}$
for some density matrix $\rho$.

In order to prove this result, let us focus on one argument. As in Ref. bush03
, we start to note that Eq. (6) implies $v(E,F)=nv(E/n,F)$, where $n$ is a
positive integer, since $v(E,F)=v(nE/n,F)=v(E/n+\cdots,F)=nv(E/n,F)$. Then,
$v(pE,F)=pv(E,F)$ for any rational $p$ such that $0\leq p\leq 1$, since given
$p=m/n$ we get $v(mE/n,F)=v(E/n+\cdots,F)=mv(E/n,F)=mv(E,F)/n$. To show that
$v(\alpha E,F)=\alpha v(E,F)$ for any real $\alpha$ such that $0\leq\alpha\leq
1$, it is enough to consider a sequence of rational numbers
$\alpha_{n}\in[0,1]$ such that $\alpha_{n}\to\alpha$ as $n\to\infty$. We note
that since $v(E,F)$ is not necessarily non-negative, we believe we cannot show
it without the continuity requirement as was done in Ref. bush03  for the case
of the probability $v(E)$. Then, by proceeding as in Ref. bush03 , it is
straightforward to show the linearity of $v$ as function of bounded Hermitian
operators. Thus, $v(E,F)$ is a bilinear function, there is an operator $X_{F}$
such that $v(E,F)=\text{Tr}\left\\{EX_{F}\right\\}$, and from Eq. (5) we get
$\text{Tr}\left\\{X_{F}\right\\}=\text{Tr}\left\\{F\rho\right\\}$, which
implies that $X_{F}$ is the convex combination $X_{F}=qF\rho+(1-q)\rho F$.
Then, from Eq. (4) we get $q=1/2$ and
$v(E,F)=Re\text{Tr}\left\\{EF\rho\right\\}$.

For more than two events, we have a quasiprobability $v(E,F,\cdots)$ with the
properties

$\displaystyle v(E,F,\cdots)\in\mathbb{R}\,,$ (7) $\displaystyle
v(I,E,F,\cdots)=v(E,I,F,\cdots)=\cdots=v(E,F,\cdots)\,,$ (8) $\displaystyle
v(E+F+\cdots,G,\cdots)=v(E,G,\cdots)+v(F,G,\cdots)+\cdots\,,$
$\displaystyle\cdots$ (9)

where $E+F+\cdots\leq I$. Analogously, if Eqs. (7)-(9) are satisfied, and if
$v(E,F,\cdots)$ is sequentially continuous in its arguments, then the joint
quasiprobability corresponding to the events represented by $E$, $F$, $\cdots$
is a multilinear function. In detail it is not fixed, and can be expressed as
an arbitrary convex combination of $Re\text{Tr}\left\\{X\rho\right\\}$ where
$X$ are all the possible products of the effects. In particular, for more than
two events, the quasiprobability is not fixed since depends on how the events
are grouped together. For instance, let us consider the case of three events
which are represented by the rank one projectors $E_{n}$, $E^{\prime}_{n}$ and
$E^{\prime\prime}_{n}$. If $E_{n}=E^{\prime}_{n}$, we have only two non-
trivial cases
$v(E_{i},E^{\prime}_{j},E^{\prime\prime}_{k})=Re\text{Tr}\left\\{E_{i}E^{\prime}_{j}E^{\prime\prime}_{k}\rho\right\\}=\delta_{ij}v(E_{i},E^{\prime\prime}_{k})$
and
$v(E_{i},E^{\prime\prime}_{k},E^{\prime}_{j})=Re\text{Tr}\left\\{E_{i}E^{\prime\prime}_{k}E^{\prime}_{j}\rho\right\\}$
corresponding, roughly speaking, to group the events $(E_{i},E^{\prime}_{j})$
with $E^{\prime\prime}_{k}$, and the events $(E_{i},E^{\prime\prime}_{k})$
with $E^{\prime}_{j}$. Basically, we have decomposed the proposition
$E_{i}\land E^{\prime}_{j}\land E^{\prime\prime}_{k}$ in two different ways
that are $E_{i}\land E^{\prime}_{j}$, $E^{\prime}_{j}\land
E^{\prime\prime}_{k}$ and $E_{i}\land E^{\prime\prime}_{k}$,
$E^{\prime\prime}_{k}\land E^{\prime}_{j}$. Thus, in general,
$Re\text{Tr}\left\\{EFG\cdots\rho\right\\}$ will be the quasiprobability
related to the joint events $E\land F$, $F\land G$, $G\land\cdots$.

It is interesting to see how a quasiprobability such defined is related to the
Wigner function wigner32 . For a one-dimensional system, if $x$ is the
position and $p$ is the momentum, the Wigner function is defined as

$W(x,p)=\frac{1}{\pi}\int e^{2ipy}\langle x-y|\rho|x+y\rangle dy\,.$ (10)

If we consider the three events which are represented by the position
projectors $|x\rangle\langle x|$, $|y\rangle\langle y|$ and the momentum
projector $|p\rangle\langle p|$, we get the quasiprobability

$v(x,p,y)=Re\langle x|p\rangle\langle p|y\rangle\langle y|\rho|x\rangle\,,$
(11)

which is intimately related to the Wigner function since

$\int g(x,p)W(x,p)dxdp=\int g\left(\frac{x+y}{2},p\right)v(x,p,y)dxdydp$ (12)

for any function $g(x,p)$. It is worth observing that when $W(x,p)\geq 0$ for
every $x$ and $p$, the quasiprobability $v(x,p,y)$ can be negative. For
instance, for a pure state $\rho=|\psi\rangle\langle\psi|$, the non-negativity
of $W(x,p)$ implies that $\langle x|\psi\rangle=\exp(-ax^{2}+bx+c)$ where
$a$,$b$ and $c$ are complex variables and $Rea>0$. However, it is easy to show
that by considering this state $|\psi\rangle$, $v(x,p,y)$ can be negative. In
particular, by considering Eq. (12), we get that $W(x,p)\geq 0$ if and only if
$\int v(x+y,p,x-y)dy\geq 0$.

To derive a general sufficient condition such that $v(E,F,\cdots)$ is non-
negative, we consider $E$, $F$, $\cdots$, as rank one projectors. If there are
only two events represented by $E_{i}$ and $F_{i}$, if $[E_{i},F_{j}]=0$, we
get $E_{i}F_{j}=\delta_{i,j}E_{i}$, then
$v(E_{i},F_{j})=\delta_{i,j}v(E_{i})\geq 0$. Conversely, if $[E_{i},\rho]=0$,
we get $E_{i}\rho=p_{i}E_{i}$, where $p_{i}\geq 0$, then
$v(E_{i},F_{j})=p_{i}\text{Tr}\left\\{E_{i}F_{j}\right\\}\geq 0$. We can
proceed analogously for more than two events, so that in general for $n$
events, $v(E,F,\cdots)$ is non-negative if there are at least $n-1$ different
commutators involving the operators $\rho$, $E$, $F$, $\cdots$, that are equal
to zero. This means that if $\rho$ is not incoherent with respect any
projector, i.e., $[E,\rho]\neq 0$, $[F,\rho]\neq 0$, $\cdots$, then
$v(E,F,\cdots)$ is non-negative if all the events are compatible, i.e., the
corresponding projectors commutate each other. On the other hand, if there are
$n-2$ couples of compatible events, a negative $v(E,F,\cdots)$ implies that
there is quantum coherence in the state $\rho$ with respect to the basis of at
least one event.

##  III Work statistics

We proceed by investigating the statistics of the work performed in a out-of-
equilibrium process. We consider a quantum coherent process generated through
a time-dependent Hamiltonian
$H(t)=\sum\epsilon_{k}(t)|\epsilon_{k}(t)\rangle\langle\epsilon_{k}(t)|$ where
$|\epsilon_{k}(t)\rangle$ is the eigenstate with eigenvalue $\epsilon_{k}(t)$
at the time $t$. The time evolution operator is
$U_{t,0}=\mathcal{T}e^{-i\int_{0}^{t}H(s)ds}$, where $\mathcal{T}$ is the time
order operator and the average work $\langle w\rangle$ done on the system in
the time interval $[0,\tau]$ can be identify with the average energy change

$\langle w\rangle=\text{Tr}\left\\{(H^{(H)}(\tau)-H(0))\rho_{0}\right\\}\,,$
(13)

where $\rho_{0}$ is the initial density matrix and given an operator $A(t)$ we
define the Heisenberg time evolved operator
$A^{(H)}(t)=U_{t,0}^{\dagger}A(t)U_{t,0}$. We introduce the projectors
$\Pi_{i}=|\epsilon_{i}\rangle\langle\epsilon_{i}|$ and
$\Pi^{\prime}_{k}=U_{\tau,0}^{\dagger}|\epsilon^{\prime}_{k}\rangle\langle\epsilon^{\prime}_{k}|U_{\tau,0}$,
which will represent our events, where $\epsilon_{i}=\epsilon_{i}(0)$ and
$\epsilon^{\prime}_{k}=\epsilon_{k}(\tau)$. Thus, we aim to find the most
general quasiprobability distribution of work $p(w)$ which (W1) reproduces the
two-projective-measurement scheme for an incoherent initial state, i.e., such
that

$p(w)=\sum_{i,k}\text{Tr}\left\\{\Pi_{i}\Pi^{\prime}_{k}\rho_{0}\right\\}\delta(w-\epsilon^{\prime}_{k}+\epsilon_{i})$
(14)

if $\rho_{0}$ is incoherent, i.e., $\Delta(\rho_{0})=\rho_{0}$ where $\Delta$
is the dephasing map defined as $\Delta(\rho)=\sum_{i}\Pi_{i}\rho\Pi_{i}$, so
that $[\rho_{0},\Pi_{i}]=0$. Furthermore, we require that (W2) the average
work given by Eq. (13) can be calculated as $\langle w\rangle=\int wp(w)dw$.
In particular, the nth work moment is given by $\langle w^{n}\rangle=\int
w^{n}p(w)dw$.

In general, the work will be represented in terms of the events $\Pi_{i}$,
$\Pi_{j}$, $\cdots$, $\Pi^{\prime}_{k}$, $\cdots$, thus has a quasiprobability
distribution of the form

$p(w)=\sum_{i,j,\cdots,k,\cdots}v(\Pi_{i},\Pi_{j},\cdots,\Pi^{\prime}_{k},\cdots)\delta(w-w(\epsilon_{i},\cdots))\,,$
(15)

where $w(\epsilon_{i},\cdots)$ is a certain function of the eigenvalues
corresponding to the events. From the condition (W1) of Eq. (14), there are
only two non-trivial cases:
$v(\Pi_{i},\Pi^{\prime}_{k})=Re\text{Tr}\left\\{\Pi^{\prime}_{k}\Pi_{i}\rho_{0}\right\\}$
with $w(\epsilon_{i},\cdots)=\epsilon^{\prime}_{k}-\epsilon_{i}$, and
$v(\Pi_{i},\Pi^{\prime}_{k},\Pi_{j})=Re\text{Tr}\left\\{\Pi_{i}\Pi^{\prime}_{k}\Pi_{j}\rho_{0}\right\\}$
with
$w(\epsilon_{i},\cdots)=\epsilon^{\prime}_{k}-q\epsilon_{i}-(1-q)\epsilon_{j}+f_{ijk}$,
where $f_{iik}=0$. In particular the first case can be obtained from the
second one by considering $q=0$ or $1$ and $f_{ijk}=0$. Concerning $f_{ijk}$,
the condition (W2) that the average obtained from $p(w)$ is equal to Eq. (13),
gives

$\sum_{i,j,k}Re\text{Tr}\left\\{\Pi_{j}\Pi^{\prime}_{k}\Pi_{i}\rho_{0}\right\\}f_{ijk}=0\,,$
(16)

which implies that $f_{ijk}=a_{ik}-a_{jk}+c_{ij}$ with $c_{ii}=0$. To prove
it, we note that $f_{ijk}=a_{ik}+b_{jk}+c_{ij}+d_{ijk}$ where $d_{ijk}$ is a
tensor. From Eq. (16), we get

$\displaystyle\sum_{i,k}Re\text{Tr}\left\\{\Pi^{\prime}_{k}\Pi_{i}\rho_{0}\right\\}(a_{ik}+b_{ik})+\sum_{i}\text{Tr}\left\\{\Pi_{i}\rho_{0}\right\\}c_{ii}$
$\displaystyle+\sum_{i,j,k}Re\text{Tr}\left\\{\Pi_{j}\Pi^{\prime}_{k}\Pi_{i}\rho_{0}\right\\}d_{ijk}=0\,,$
(17)

which needs to be satisfied for arbitrary $\Pi_{i}$ and $\Pi^{\prime}_{k}$,
then $a_{ik}+b_{ik}=0$, $c_{ii}=0$ and $d_{ijk}=0$. We note that if
$w(\epsilon_{i},\cdots)$ is a linear function of the eigenvalues, then
$f_{ijk}=0$. In particular, if it is linear then if $H(t)\to\lambda H(t)$ we
get $w(\epsilon_{i},\cdots)\to\lambda w(\epsilon_{i},\cdots)$ and so
$w\to\lambda w$. Furthermore, if $H(t)\to H(t)+e(t)$ where $e(t)$ is a
c-number, we have $w\to w+e(\tau)-e(0)$, which are reasonable features of the
work $w$. Anyway, if it is non-linear there are other possible choices, for
instance $a_{ik}\propto\epsilon_{i}^{2}$ and $c_{ij}=0$. It is clear that to
fix $f_{ijk}$ we have to consider a further condition. We find that
$f_{ijk}=0$ if we require that (W3) the work second moment is

$\langle
w^{2}\rangle=\text{Tr}\left\\{(H^{(H)}(\tau)-H(0))^{2}\rho_{0}\right\\}\,.$
(18)

If $f_{ijk}=0$, the most general quasiprobability reads

$p_{q}(w)=\sum_{k,j,i}Re\text{Tr}\left\\{\Pi_{i}\Pi^{\prime}_{k}\Pi_{j}\rho_{0}\right\\}\delta(w-\epsilon^{\prime}_{k}+q\epsilon_{i}+(1-q)\epsilon_{j})\,,$
(19)

since any other quasiprobability can be obtained as a convex combination of
quasiprobabilities $p_{q}(w)$ with different $q$. Thus, all these
quasiprobabilities form the most general class of quasiprobabilities having
the form of Eq. (15) and satisfying the conditions about the reproduction of
the two-projective-measurement scheme (W1), the average (W2) and the second
moment (W3). We note that the quasiprobability $p_{q}(w)$ has been introduced
in Ref. francica22 , where its relation with the initial quantum coherence has
been investigated, and fluctuation relations have been derived. In contrast,
if we do not require that the condition (W1) is satisfied, the number of
events in Eq. (15) is arbitrary, since always exists a function
$w(\epsilon_{i},\cdots)$ such that the condition (W2) is satisfied. For
instance, we can consider the linear function
$w(\epsilon_{i},\cdots)=q^{\prime}_{1}\epsilon^{\prime}_{k}+\cdots-
q_{1}\epsilon_{i}-\cdots$, where
$q_{1}+q_{2}+\cdots=q^{\prime}_{1}+q^{\prime}_{2}+\cdots=1$.

In the end, it is interesting to note that a quasiprobability having the form
of Eq. (15) is achieved from the condition (W2) by considering

$p(w)=\sum_{i,\cdots}\text{Tr}\left\\{X_{i\cdots}\rho_{0}\right\\}\delta(w-w(\epsilon_{i},\cdots))\,,$
(20)

where $X_{i\cdots}$ is a product of projectors, for instance
$X_{ik}=\Pi^{\prime}_{k}\Pi_{i}$ or $X_{iki}=\Pi_{i}\Pi^{\prime}_{k}\Pi_{i}$.
In the following we aim to prove that the condition (W2) prohibits repetitions
of the projectors in the product, e.g., we cannot have $X_{iki}$.

Let us start to consider the case where only two projectors $\Pi_{i}$ and
$\Pi^{\prime}_{k}$ appear in the product. A product $X_{i\cdots i}$ of the
form $\Pi_{i}\cdots\Pi_{i}$ does not satisfy the condition (W2) since in this
case $\text{Tr}\left\\{X_{i\cdots
i}\rho_{0}\right\\}=\text{Tr}\left\\{X_{i\cdots
i}\Pi_{i}\rho_{0}\Pi_{i}\right\\}$, then by considering an operator
$\rho_{0}=|\epsilon_{i}\rangle\langle\epsilon_{j}|$ with $j\neq i$, we get
$\text{Tr}\left\\{X_{i\cdots i}\rho_{0}\right\\}=0$ and $\int wp(w)dw=0$, but
$\langle
w\rangle=\sum_{k}\langle\epsilon_{j}|\Pi^{\prime}_{k}|\epsilon_{i}\rangle\epsilon^{\prime}_{k}$
so that the condition (W2) is not satisfied in general. Similarly, a product
$X_{k\cdots k}$ of the form $\Pi^{\prime}_{k}\cdots\Pi^{\prime}_{k}$ does not
satisfy the condition (W2) and we can only have products of the form
$\Pi_{i}\cdots\Pi^{\prime}_{k}$ or $\Pi^{\prime}_{k}\cdots\Pi_{i}$. For the
case $X_{i\cdots k}=\Pi_{i}\cdots\Pi^{\prime}_{k}$ we consider
$\rho_{0}=\Pi^{\prime}_{k}\Pi_{i}$, then condition (W2) reads
$\text{Tr}\left\\{X_{i\cdots
k}\right\\}w(\epsilon_{i},\cdots)=\text{Tr}\left\\{\Pi_{i}\Pi^{\prime}_{k}\right\\}(\epsilon^{\prime}_{k}-\epsilon_{i})$,
which implies that $X_{i\cdots k}=X_{ik}=\Pi_{i}\Pi^{\prime}_{k}$. Similarly,
the only product $X_{k\cdots i}$ of the form $\Pi^{\prime}_{k}\cdots\Pi_{i}$
which satisfies the condition (W2) is $X_{ki}=\Pi^{\prime}_{k}\Pi_{i}$. We
proceed by considering the case where only the three projectors $\Pi_{i}$,
$\Pi_{j}$ and $\Pi^{\prime}_{k}$ appear in the product. We cannot have
products of the form $\Pi_{i}\cdots\Pi_{i}$, $\Pi_{j}\cdots\Pi_{j}$ and
$\Pi^{\prime}_{k}\cdots\Pi^{\prime}_{k}$. We consider $X_{i\cdots k}$ of the
form $\Pi_{i}\cdots\Pi^{\prime}_{k}$, for $\rho_{0}=\Pi^{\prime}_{k}\Pi_{i}$
the condition (W2) reads $\sum_{j}\text{Tr}\left\\{X_{i\cdots
k}\right\\}w(\epsilon_{i},\cdots)=\text{Tr}\left\\{\Pi_{i}\Pi^{\prime}_{k}\right\\}(\epsilon^{\prime}_{k}-\epsilon_{i})$
which implies that $X_{i\cdots k}=X_{ijk}=\Pi_{i}\Pi_{j}\Pi^{\prime}_{k}$. For
$X_{i\cdots j}$ of the form $\Pi_{i}\cdots\Pi_{j}$, we consider
$\rho_{0}=|\epsilon_{j}\rangle\langle\epsilon_{i}|$, then the condition (W2)
reads $\sum_{k}\langle\epsilon_{i}|X_{i\cdots j}|\epsilon_{j}\rangle
w(\epsilon_{i},\cdots)=\sum_{k}\langle\epsilon_{i}|\Pi^{\prime}_{k}|\epsilon_{j}\rangle\epsilon^{\prime}_{k}$
which implies that $X_{i\cdots j}=X_{ikj}=\Pi_{i}\Pi^{\prime}_{k}\Pi_{j}$.
Thus, in general, the condition (W2) implies that the operator $X_{i\cdots}$
is a product of the projectors without repetitions, and a quasiprobability
having the form of Eq. (15) is achieved by taking the real part of the
quasiprobability of Eq. (20).

##  IV Contextuality

Having determined the form of the quasiprobability of work, now we can focus
on the problem if there is a non-contextual hidden variables model lostaglio18
which satisfies the conditions about the reproduction of the two-projective-
measurement scheme (W1), the average (W2) and the second moment (W3). We point
out that the negativity of
$Re\text{Tr}\left\\{\Pi^{\prime}_{k}\Pi_{i}\rho_{0}\right\\}$ or
$Re\text{Tr}\left\\{\Pi_{i}\Pi^{\prime}_{k}\Pi_{j}\rho_{0}\right\\}$ alone is
not sufficient to guarantee contextuality, and in principle we have to verify
if some quasiprobability of work is non-negative and so non-contextual. To
prove this we give a counterexample. We consider a one dimensional system,
with initial Hamiltonian $H(0)=\alpha x$ and final Hamiltonian $H(\tau)=\beta
p^{2}$. In this case we consider the projectors $\Pi_{x}=|x\rangle\langle x|$
and $\Pi^{\prime}_{p}=|p\rangle\langle p|$. It is easy to show that for
$q=1/2$ the work quasiprobability of Eq. (19) can be expressed in terms of the
Wigner function as

$p_{1/2}(w)=\int dxdpW(x,p)\delta(w-\beta p^{2}+\alpha x)\,.$ (21)

If we consider the initial wave function $\langle
x|\psi\rangle=\exp(-ax^{2}+bx+c)$, we get $W(x,p)\geq 0$, thus the protocol is
non-contextual. However, for this state it is easy to verify that the
quasiprobabilities
$Re\text{Tr}\left\\{\Pi^{\prime}_{p}\Pi_{x}\rho_{0}\right\\}$ and
$Re\text{Tr}\left\\{\Pi_{x}\Pi^{\prime}_{p}\Pi_{y}\rho_{0}\right\\}$ can take
negative values, so we cannot say in general if there is a protocol exhibiting
contextuality by only looking on the negativity of these quasiprobabilities.
We note that this situation is in contrast with the one examined in Ref.
lostaglio18 , where only the case $q=0$ and $1$ has been considered and the
negativity of $Re\text{Tr}\left\\{\Pi^{\prime}_{p}\Pi_{x}\rho_{0}\right\\}$
implies that the corresponding protocol is contextual. Basically, this is due
to the fact that, although can exist a unique couple of indexes $i$ and $k$
such that $\epsilon^{\prime}_{k}-\epsilon_{i}=w$, for $q\neq 0$ and $1$ we can
have different triples of indexes $i$, $j$ and $k$ such that
$\epsilon^{\prime}_{k}-q\epsilon_{i}-(1-q)\epsilon_{j}=w$, so that the
quasiprobability associated to this value $w$ is obtained by summing
$Re\text{Tr}\left\\{\Pi_{i}\Pi^{\prime}_{k}\Pi_{j}\rho_{0}\right\\}$ over
these triples, and of course the negativity of some quasiprobabilities
$Re\text{Tr}\left\\{\Pi_{i}\Pi^{\prime}_{k}\Pi_{j}\rho_{0}\right\\}$ does not
guarantee the negativity of the sum.

##  V Conclusions

Understanding how to describe nature from a statistical point of view is a
problem of fundamental importance. For instance, by characterizing the work
statistics, it is possible to derive some fluctuation theorems related to the
second law of thermodynamics jarzynski97 , which clarifies the importance of
this study. In quantum systems, by focusing on the work, this description is
less obvious than the classical case and it is still a matter of discussion
(see, e.g., Ref. book19 ). With the aim of helping to clarify what the work
statistics can be, we identified the form of the work quasiprobability which
is compatible with some fundamental conditions. In particular, we introduced a
general notion of quasiprobability by giving an argumentation analogous to the
Gleason’s theorem. Recently, the energy-change statistics in a generic open
quantum system has been characterized by means of the Kirkwood-Dirac
quasiprobability lostaglio22 . By following this approach, the condition that
the quasiprobability is real can be relaxed, however in our approach this
condition is important since ensures a unique quasiprobability related to
joint events. We note that these quasiprobabilities are experimentally
accessible, for instance, recently, the Margenau-Hill distribution of the work
(corresponding to $q=0$ or $1$) has been experimentally measured in a driven
three-level quantum system hernandez-gomez22 . Beyond their foundational
implications, we believe our findings can be useful to characterize the
quantum signatures in the work statistics. In particular, due to the initial
quantum coherence in the energy basis we can get (not necessarily) a
contextual protocol. However, we proved that for a given initial state, it is
not said that all work quasiprobabilities exhibit contextuality.

### Acknowledgements

The author acknowledges financial support from the project BIRD 2021
”Correlations, dynamics and topology in long-range quantum systems” of the
Department of Physics and Astronomy, University of Padova.

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