# Image of the phonon spectrum in the 1/f noise of topological insulators

M. Mihaila<EMAIL_ADDRESS>S. Dinulescu  P. Varasteanu  National
Institute of Research and Development in Microtechnologies - IMT Bucharest,
Erou Iancu Nicolae str. 126A, 077190, Bucharest, Romania

###### Abstract

As reported recently, the 1/f noise intensity in both (Bi,Sb)2Te3 [Islam et
al., Appl. Phys. Lett. 111, 062107 (2017)] and BiSbTeSe1.6 [Biswas et al.,
Appl. Phys. Lett. 115, 131601 (2019)] features noise peaks which develop at
some specific temperatures. We compared this noise structure with either
phonon density of states or Raman spectrum of each topological insulator (TI),
respectively. In (BiSb)2Te3, the comparison revealed that the noise peaks
track the van Hove singularities in the phonon density of states. It resulted
that bulk atomic oscillators are responsible for the noise peaks. The most
intense noise peak observed in (BiSb)2Te3 at 50 K is attributed to the thermal
motion of the Bi atoms. Other less intense noise peaks are assigned to either
a single phonon mode or multi-phonon combinations. We found that thermal
motions of Bi and Te2 atoms in different symmetry directions are involved in
most of the phonon combinations, which stand for the signature of the lattice
anharmonicity. The noise increase observed in both (Bi,Sb)2Te3 and BiSbTeSe1.6
above a specific temperature threshold is attributed to the anharmonicity-
induced strengthening of the carrier-phonon coupling. In the case of
BiSbTeSe1.6, we show that all noise singularities are mirrored in the Raman
spectrum of a structurally close TI (BiSbTeSe2) in the whole temperature
range. This indicates that although transport can be at the surface or in the
bulk or both of them, the carrier-phonon interaction is the only source of 1/f
fluctuations in TIs. Inherently, these results imply that microscopic origin
of 1/f noise in solid is in the perpetual thermal motion of the atoms.

For almost a century[1], 1/f noise is puzzling the scientific community with
its omnipresence in solid and solid-state devices[2, 3, 4, 5, 6, 7]. In the
last decade, it was observed[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] in
topological insulators (TI), which are quantum materials with metallic
conductivity at the surface, while their bulk is insulating[20, 21, 22, 23,
24, 25, 26, 27]. These properties are due to the presence of surface
electronic states with a Dirac cone energy dispersion located in the bulk band
gap[20, 21, 22, 23, 24, 25, 26, 27]. These surface states are protected by
inversion symmetry to backscattering, a property which opens fascinating
possibilities for TIs applications in spintronics and quantum topological
computing. However, the presence of the 1/f noise in the surface conduction is
a factor which can lead to certain technology roadblocks in applications such
as quantum computing, where it acts as source of decoherence[28, 29].
Therefore, finding its origin is of great practical interest. Hossain et
al.[9] suggested that the noise dominated by the surface states in TIs could
be very low because the time-reversal symmetry topologically protects the
surface states from backscattering. However, in Bi2Se3 samples wherein the
carrier transport was dominated by the surface, they found a high 1/f noise
parameter ($\alpha\approx 0.2$), therefore the noise would be due to a “mixed
volume-surface transport regime”[9]. Consequently, to reduce the noise level,
it would be necessary to protect the surface conduction from the bulk
influences. Nevertheless, in strong quaternary Bi1.6Sb0.4Te2Se, with
predominant surface conduction, Bhattacharyya et al.[11] found unexpected,
bulk-induced “sharp noise peaks at some characteristic”[11] temperatures,
while a similar structure was reported by both Islam et al.[14] in (Bi,Sb)2Te3
and Biswas et al.[17] in BiSbTeSe1.6 (BSTS). Peaks in the 1/f noise were also
reported in tunnel junctions having Bi2Te3 or Bi2Se3 as bottom electrode[10].
This situation is similar to the one reported long ago for GaAs Schottky
tunnel junction, wherein sharp 1/f noise peaks occurred at voltages
corresponding to the GaAs bulk phonon energies[30]. In fact, such a structure
in the 1/f noise intensity or $\alpha$ is quite common, for it has been so far
observed in many other solid-state physical systems, such as bipolar[31] and
MOS transistors[32], metallic point contacts[33, 34] , metal films[35, 36],
quartz crystals[37], carbon nanotubes[38], carbon soot [39, 40], metal
nanowires [41] or, as predicted [39], single molecule[42, 43]. As for its
microscopic origin, the structure in $\alpha$ was found to mirror the van Hove
singularities in either phonon density of states (PDOS), $\alpha\approx
F(\omega)$, or the Eliashberg function: $\alpha\approx
g^{2}(\omega)F(\omega)$, where $g^{2}(\omega)$ is the electron-phonon matrix
element and $\omega$ is the phonon frequency[36, 40, 41, 44]. In this work, we
show that the noise structure observed by Islam et al.[14] in (Bi,Sb)2Te3 and
Biswas et al.[17] in BiSbTeSe1.6 is the image of the phonon spectrum of these
materials. To this goal, the noise structure will be compared either with the
(Bi,Sb)2Te3 PDOS determined by inelastic neutron scattering (INS)[45, 46] or
with the Raman spectrum of BiSbTeSe2[47]. This comparison evidences that each
noise peak is associated with a specific phonon mode or multiphonon
combinations. The role of lattice anharmonicity in enhancing the noise is
brought to the fore. Finally, the significance of these results on the
microscopic origin of 1/f noise in solid is briefly discussed.

Figure 1 shows the dependence of the normalized noise intensity, SV/V2, on
temperature (T) for a 10 nm thick (Bi,Sb)2Te3 film, at f = 1 Hz, in a limited
temperature range, as reported by Islam et al.[14]. For comparison purposes,
the temperature scale was converted in energy (kBT, kB \- Boltzmann constant).
But, from Hooge’s[48] formula: $S_{V}/V^{2}=\alpha/fN$, where $S_{V}$ is the
noise spectral density of the fluctuating voltage ($V$) across the sample, at
the frequency ($f$) and $N$ is the total charge carrier number in the sample.
Hence, for f = 1 Hz, one gets $S_{V}/V^{2}=\alpha/N\approx F(\omega)$[36, 40,
44]. A strong noise peak was observed[14] at about 50 K (4.3 meV). Other less
intense peaks are visible at 25 K (2.15 meV), 82 K (7.06 meV), 110 K (9.47
meV), 140 K (12.06 meV) and 178 K (15.33 meV).

Figure 1:  Comparison between the temperature dependence of $S_{V}/V^{2}$ in
(Bi,Sb)2Te3 (green dots - reprinted from Ref. [14], S. Islam et al., Appl.
Phys. Lett. 111, 062107 (2017), with permission of AIP Publishing) and PDOS of
(Bi0.26Sb0.74)(2)Te3 (blue and red dots, - reproduced (adapted) with
permission from Ref. [45]\- B. Klobes et al. Phys. Stat. Sol. 9, 57, 2015;
copyright 2015 John Wiley and Sons); the numbers denote noise peak energies;
dotted line is guide to the eye.

In a preliminary investigation, the noise peak energies were compared with the
phonon energies of both Bi2Te3 and Sb2Te3 binary compounds. It was found that
the noise peak at 25 K (2.15 meV) corresponds to a 0.51 THz (2.11 meV)
transverse acoustic phonon ($E_{u}^{1}$) in Bi2Te3, measured by INS at 77 K,
at the Z point of symmetry[49, 50]. At 25 K, its frequency, which decreases
with temperature[49] with the slope $-3.139\times 10^{-4}$ THz/K (linear
approximation), becomes 0.526 THz, namely 2.17 meV, an energy which reasonably
fits the noise one. Surprisingly, the frequency of this ($E_{u}^{1}$) phonon
at 50 K is 0.518 THz (2.145 meV), so that, fortuitous or not, the most intense
noise peak at 4.3 meV seems to be associated with the energy (4.29 mV) of its
second ($2\times 2.145$ meV) overtone. Also, the average energy (4.3 meV) of
the $E_{g}^{1}$ Raman modes in Bi2Te3 and Sb2Te3 layers, observed at 4.26 meV
(34.4 cm-1)[51] and 4.35 meV (35.1 cm-1)[52], respectively, is in agreement
with the noise data. The small noise peak at 7.06 meV (82 K) is close to the
6.95 meV (56 cm-1, $E_{u}^{1}$(TO) infrared phonon observed in Sb2Te3 at 80
K[53].The noise peak at 9.47 meV (110 K) is close to the $E_{u}^{1}$
transversal mode at 2.25 THz (9.31 meV) at 77 K, measured in Bi2Te3 at the Z
point of symmetry [49]. At 110 K, its frequency, which increases with
temperature with the slope $7.623\times 10^{-4}$THz/K [49], is 2.275 THz (9.42
meV), a value consistent with 9.47 meV from noise measurement. Also, a 9.49
meV (76.5 cm-1) phonon mode was reported[53] at 80 K in Sb2Te2 for the
$E_{u}^{1}$(LO) infrared mode. A sharp $E_{u}^{2}$ phonon peak was found in
the Eliashberg function of Bi2Te3 by Inelastic Electron Tunneling
Spectroscopy(IETS)[54] at 12 meV. Although measured at 4.2 K, it is strikingly
close to the noise peak at 12.06 meV (140 K). However, from Richter et
al.[55], the $E_{u}^{2}$ phonon frequency at 140 K in Bi2Te3, which shifts
with -0.0105 cm-1/K, is 96.69 cm-1 or 11.99 meV, a value close to 12.06 meV
from noise data. As for the enlarged noise band with the maximum at 15.33 meV
(178 K), it corresponds to the 3.7 THz (15.32 meV) $A_{2u}^{2}$ phonon at the
$\Gamma$ point, measured[49] in Bi2Te3 at 77 K or to 15.4 meV infrared
$E_{u}^{2}$ mode in Sb2Te3 at 80 K[53]. All these comparisons are
synthetically presented in the Table I.

The foregoing analysis showed that the temperatures where the noise peaks
develop are closely related to the phonon energies of the binary compounds.
The noise data were further compared with PDOS of Bi2Te3 [56], Sb2Te3[56],
(Bi0.26Sb0.74)2Te3[45] and (Bi,Sb)2Te3[46] compounds, all obtained by INS. The
comparison between the noise structure and the bulk PDOS of the as-cast
(Bi0.26Sb0.74)2Te3, obtained by Klobes et al. [45] at 295 K, is shown in
Figure. 1. According to these authors [45], the phonon spectrum of this
material is similar to (Bi,Sb)2Te3. The comparison revealed that the first
noise peak at 2.15 meV corresponds to a shoulder in PDOS. Strikingly, the
dominant noise peak at 4.3 meV fits the first PDOS peak. Next, the PDOS part
located in the (3.5-14.2) meV energy range was downward shifted and scaled
down till its second peak fitted the intensity of the 7.06 meV noise peak (red
dots in Figure. 1). This procedure unraveled a good fit between the noise
peaks and PDOS structure. It also revealed that the third major PDOS peak is
well mirrored in the noise peak at 12.06 meV. The strongly scattered PDOS
points in the (14-18) meV energy range are surprisingly well enveloped by the
large noise band with the maximum at 15.33 meV. Although the second as
intensity, the noise peak at 9.47 meV corresponds to a clear but less intense
PDOS peak. Figure. 1 shows that the dominant PDOS peaks feature almost the
same intensity, while the intensity of their noise counterparts is quite
different. This indicates that besides $F(\omega)$, there is another factor
involved in the noise intensity control, which, as stated previously[36, 40,
41, 44], could be the carrier-phonon coupling matrix element. This interaction
should depend on the nature of the atoms the carriers interact with. Since at
low energies the phonon spectrum is dominated by thermal vibration of the
heaviest element (Bi)[56], one would expect that the noise peaks at lower
energies to be related to the group V element (Bi), while those at higher
energies can be due to the modes associated with the group VI element
(Te)[56]. This statement is supported by the existence of an intense peak
around 4.3 meV in the Bi partial density of states in Bi2Te3 [45] Also, using
neutron diffraction, Serrano-Sanchez et al. [57] found that in-plane motion of
the Bi atoms in the direction [$1\overline{1}0$], which is the major, most
favored axis of their anisotropic displacements in Bi0.35Sb1.65Te3,
corresponds to a 4.4 meV Einstein oscillator. On these reasons, we tentatively
attribute the strongest noise peak at 4.3 meV to the interaction of the
carriers with the thermal motion of the Bi atoms (Table I). Since the Bi atoms
move in a strong anharmonic potential [49, 50, 58, 59], a stronger carrier-
phonon coupling is expected, therefore more noise [36, 40, 44]. A 6.9 meV
Einstein oscillator has been reported [57] for the out-of-plane motion of the
Te2 atoms in the direction [001], which is close to the 7.06 meV noise peak
energy. Using the same data [57], values of 9.5 meV or 9.4 meV, resulting from
combinations 2$\times$Te1[001]-Bi[001] and
3$\times$Te2[$1\overline{1}0$]-Bi[$1\overline{1}0$], respectively, are close
to the 9.47 meV noise peak energy, while the 12.06 meV noise peak corresponds
to a combination 4.6 + 7.4=12 meV of the Te2+Te1 motion in the directions
[$1\overline{1}0$] and [001], respectively. Although the 15.33 meV noise band
is located in a PDOS region dominated by both Sb and Te thermal motion [45,
56], it also fits the combination 3$\times$7.4-6.9 = 15.3 meV, where 7.4 meV
and 6.9 meV are the energies of the Te1 and Te2 out-of plane atomic motion
([001], parallel to the c-axis), respectively[57]. All these data complete the
Table I.

Table 1: Comparison between noise peak energies in (BiSb)2Te3[14] and phonon energies in Bi2Te3, Sb2Te3 and Bi0.35Sb1.65Te3 determined by INS, Raman (R), infrared (IR) and neutron diffraction (ND) spectroscopy. Te1 and Te2 denote the two Te atoms in the Te1-Bi-Te2-Bi-Te2 quintuple layer. Peak | T (K) |  $\bm{k_{B}T}$ from noise (meV) | Phonon energy (meV) | Phonon mode or combinations, TI | Ref./Method
---|---|---|---|---|---
1 | 25 | 2.15 | 2.17 |  $E_{u}^{1}$(25 K),Bi2Te2 |  [49, 50]/INS
2 | 50 | 4.3 |  | $4.29=2\times 2.145$
---
$4.3=(4.26+4.35)/2$
$4.4$
|  $2\times E_{u}^{1}$ (50 K), Bi2Te3
---
$E_{g}^{1}$, (Bi2Te3 \+ Sb2Te3)/2
Bi$[1\overline{1}0]$, Bi0.35Sb1.65Te3
|  [49, 50]/INS
---
[51, 52]/R
[57]/ND
3 | 82 | 7.06 |  | 6.95
---
6.9
|  $E_{u}^{1}$(TO) (80 K), Sb2Te3
---
Te2[001],Bi0.35Sb1.65Te3
|  [53]/IR
---
[57]/ND
4 | 110 | 9.47 |  | 9.42
---
9.49
$9.5=2\times 7.4-5.3$
$9.4=3\times 4.6-4.4$
|  $E_{u}^{1}$(110 K), Bi2Te3
---
$E_{u}^{1}$(LO)(80 K), Sb2Te3
$2\times$Te1[001] - Bi[001], Bi0.35Sb1.65Te3
$3\times$ Te2$[1\overline{1}0]$ \- Bi$[1\overline{1}0]$, Bi0.35Sb1.65Te3
|  [49]/INS
---
[53]/IR
[57]/ND
[57]/ND
5 | 140 | 12.06 |  | 12
---
11.99
12 = 4.6 + 7.4
|  $A_{2u}^{2}$ (77 K), Bi2Te3
---
$E_{u}^{2}$ (80 K), Sb2Te3
$3\times$Te1$[1\overline{1}0]$ \+ Te2[001], Bi0.35Sb1.65Te3
|  [54]/IETS
---
[55]/IR
[57]/ND
6 | 178 | 15.33 |  | 15.32
---
15.4
15.3 = $3\times 7.4-6.9$
|  $A_{2u}^{2}$ (77 K), Bi2Te3
---
$E_{u}^{2}$ (80 K), Sb2Te2
$3\times$Te1[001] - Te2[001], Bi0.35Sb1.65Te3
|  [49]/INS
---
[53]/IR
[57]/ND
7 | 240 | 20.68 |  | 20.7 = $3\times 6.9$
---
20.7 = $3\times 4.6+6.9$
$20.7=4.9+13.8$
|  $3\times$Te2[001], Bi0.35Sb1.65Te3
---
$3\times$ Te2$[1\overline{1}0]$ \+ Te2[110], Bi0.35Sb1.65Te3
Te2[001] + Te2[110], Bi0.35Sb1.65Te3
|  [57]/ND
---
[57]/ND
[57]/ND
8 | 290 | 24.98 |  | $24.9=2\times 5.3+7.4+6.9$
---
$24.9=4.4+6.7+2\times 6.9$
$24.9=4.4+6.7+3\times 4.6$
|  $2\times$Bi[001] + Te1[001] + Te2[001]
---
Bi$[1\overline{1}0]$ \+ Te1$[1\overline{1}0]+2\times$Te2[001]
Bi$[1\overline{1}0]$ \+ Te1$[1\overline{1}0]+3\times$Te2$[1\overline{1}0]$
all in Bi0.35Sb1.65Te3
|  [57]/ND
---
[57]/ND
[57]/ND

The role of the electron-phonon coupling in the mechanism of noise appears
even more pregnant from Figure. 2, where the complete noise data of Islam et.
al. [14] are compared with the (Bi,Sb)2Te3 phonon spectrum obtained by Xie et
al.[46] by INS, at 300 K. In this case, a small difference, due to
anharmonicity, between the noise peaks and the phonon ones exists. One also
notes that the intensity of the dominant peaks in the phonon spectrum is quite
different. It can be seen that while the phonon density increases, the noise
intensity decreases and vice-versa for temperature higher than about 200K: the
PDOS slowly decreases, while the noise intensity increases. In this
temperature range, two new noise peaks develop at 240 K (20.68 meV) and 290 K
(24.98 meV) (Table I), which reasonably fit some bulges in the phonon
spectrum.

Figure 2:  Comparison between temperature dependence of $S_{V}/V^{2}$ in
(Bi,Sb)2Te3 (green dots - reprinted from Ref. [14], S. Islam et al., Appl.
Phys. Lett. 111, 062107 (2017), with permission of AIP Publishing) and
(Bi,Sb)2Te3 (P/G)DOS (black dots, reprinted/adapted with permission from Ref.
[46] \- W. Xie et al., Nano Lett. 10, 3283 (2010); copyright (2010) American
Chemical Society); red, dashed lines indicate the correspondence noise
structure - phonon spectrum, while numbers denote noise peak energies.

In terms of Einstein oscillators [57], both the 3$\times$Te2[001] overtone
(3$\times$6.9=20.7 meV) and combinations such as
3$\times$Te2[$1\overline{1}0$]+Te2[001] or Te2[001]+Te2[110] gave the same
result (20.7 meV, Table I), close to the noise peak at 20.68 meV. The noise
peak at 24.98 meV (290 K) is well fitted by 24.9 meV, which results from the
four-phonon 2$\times$Bi[001]+Te2[001] + Te1[001],
Bi[$1\overline{1}0$]+Te1[$1\overline{1}0$]+2$\times$Te2[001] or
Bi[$1\overline{1}0$]+3$\times$Te2[$1\overline{1}0$]+Te1[$1\overline{1}0$][57]
five-phonon combinations (Table I). Such multi-phonon combinations are the
signatures of lattice anharmonicity. One observes that single phonons and
their overtones associated with the thermal motion of Bi and, especially, Te2
atoms are involved in most of these multi-phonon combinations. This seems to
be the manifestation of lattice anharmonicity, because, as shown by Serrano-
Sanchez et al.[57], Bi and Te2 are the most anharmonic oscillators in
Bi0.35Sb1.65Te3. One can thus attribute the noise increase starting around 200
K to an enhanced carrier-phonon coupling induced by the stronger anharmonicity
installed in the sample at this temperature. A similar noise increase was
observed in Bi1.6Sb0.4Te2Se at the same temperature [11], while in BiSbTeSe1.6
Biswas et al. [17] showed that the noise starts increasing slowly at about 160
K [15] (Figure. 3), followed by a sudden increase at a kink located around
186K and culminated with a noise peak at 230 K.

Figure 3:  Comparison between the temperature dependence of $<$$\delta
R^{2}$$>$/$<$$R^{2}$$>$ (red circles) in BiSbTeSe1.6 (Reprinted from Ref.
[17], Biswas et al., Appl. Phys. Lett. 115, 131601 (2019), with permission of
AIP Publishing) and Raman spectrum of BiSbTeSe2 (black circles, reprinted with
permission from Ref. [47] \- R. German et al., Phys. Rev. Mat. 3, 054204
(2019); copyright 2019 by the American Physical Society); the correspondence
noise structure-Raman spectrum is indicated by discontinuous lines, while
numbers denote the noise peak energies in equivalent wavenumbers (cm-1).

The relative resistance (R) noise of Biswas et al.[17] is compared in Figure.
3 with the Raman spectrum of BiSbTeSe2 collected by German et al. [47] in
parallel polarization, c(aa)c geometry, at 5 K. The equivalent wavenumber
(24.3 cm-1, 1 K = 0.6945 cm-1) of the first noise peak at 35 K (Un in the
Table II) has no correspondent in the Raman spectrum of Ref. [47]. The noise
peak at 55 K (38.2 cm-1) is associated with the Eg phonon [47] which is 39.8
cm-1 at 55 K. The prominent noise peak at 105 K (72.92 cm-1) is well mirrored
in the Raman spectrum by a 72.7 cm-1 peak at 5 K [47], which shifts to about
71.5 cm-1 at 105 K (from Figure. 3a of Ref. [47]). A weak but clear shoulder
in the noise data at about 170 K (118 cm-1) corresponds to the 5th phonon mode
of German et al.[47], with a frequency of 118 cm-1 at 170 K (from Figure. 3b
of Ref. [47]). The noise kink at 186 K(129.2 cm-1) corresponds to the 6th BSTS
Raman phonon of German et al.[47], for its frequency at 186 K (slope
-2.56$\times$10-2 cm-1/K) is 129.5 cm-1. The strongest noise peak at 230 K
(159.7 cm-1) is close to 160.2 cm-1, an $A_{1g}$ phonon in BSTS [47], which
shows the highest full width at the halve maximum (FWHM) among all Raman peaks
of Ref. [47]. Since the FWHM is a measure of the carrier-phonon coupling, the
enhanced noise can be explained by a stronger coupling. In addition, the
anharmonicity involvement in noise generation is supported by the observation
that the third noise peak is very well described by the third overtone
3$\times$24.3 = 72.9 cm-1 of the first noise peak. Also, the six phonons
combination (5$\times$24.3 + 38.2) of the first two noise peak energies gives
a similar value (159.7 cm-1) for the 6th noise peak. All these correlations
are presented in the Table II.

Table 2: Comparison between the noise peak energies in BiSbTeSe1.6 [17] and phonon energies of BiSbTeSe2 determined by Raman spectroscopy Peak | T (K) |  $\bm{k_{B}T}$ from noise (cm-1) | Phonon from Raman spectroscopy (cm-1) | Phonon mode | Ref.
---|---|---|---|---|---
1 | 35 | 24.3 | - | Un | -
2 | 55 | 38.2 | 39.8 | $E_{g}^{1}$ | [47]
3 | 105 | 72.9 |  | 71.5 (at 105 K)
---
$72.9=3\times 24.3$
| $A_{1g}$
---
3Un
| [47]
---
[17]
4 | 170 | 118 | 118 (at 170 K) | $E_{g}$ | [47]
5 | 186 | 129.2 | 129.2 (at 186 K) | $A_{1g}$ | [47]
6 | 230 | 159.7 |  | 160.2 (294 K)
---
$159.7=5\times 24.3+38.2$
| $A_{1g}^{2}$
---
5Un \+ $E_{g}^{1}$
| [47]
---
[17]

Carrier hopping in the bulk was the transport mechanism found by Biswas et
al.[17] in BSTS at temperature higher than 150 K. Voss[60], who found hopping
in a MOSFET inversion layer at 4.2 K, concluded that 1/f noise “is an
intrinsic transport property of the hopping conduction”[60]. As shown
elsewhere [32], Voss’ statement implies phonons contribution because “hopping
can only occur if phonons can be emitted or absorbed to compensate for the
change in energy of the electron”[61]. Therefore, 1/f noise in BSTS sample
could be generated by a phonon-assisted hopping mechanism. However, as shown
by authors [17], this noise source can be obscured by the presence of the
random telegraph signal noise. It is thus not clear whether the noise
structure in this temperature domain is exclusively a 1/f noise-related
property. However, for T $<$ 150 K, the normalized noise $fS_{R}/R^{2}$ is
(almost) flat [17], hence 1/f source is dominant. In this temperature domain,
a surface conduction channel is active [17], therefore a surface 1/f noise
source may exist, which, according to Biswas et al.[17], is due to “coupling
of surface transport with defect dynamics in bulk”[17]. Similarly, Islam et
al.[14] concluded that “the internal bulk chalcogenide defects are mainly
responsible for resistance fluctuations for both surface and bulk electronic
states”[14]. Since the noise structure correlates with the bulk phonon
energies in the whole temperature range, it results that wherever the current
flows, the carriers interaction with the bulk phonons is the unique
microscopic source of noise in both TIs. For decades, it has been shown that
1/f noise in solid is generated at surface [62, 63, 64, 65, 66] or in bulk
[48, 67, 68], while other works proved that the noise source can be shifted
from the surface to the bulk [69] or vice-versa [70]. Phonons appear to be
involved in both cases, for surface and bulk phonons have been observed in the
noise of the same physical system [71, 72], while specific phonon
properties[73] were exploited to control interface noise[74]. Also,
fluctuations in the number of both surface [65, 66] and bulk phonons[75] were
proposed as 1/f noise sources. The analysis of the noise data in TIs revealed
a new situation in solid: the fluctuation of the surface conductivity is
controlled by the bulk phonons. The association of the noise peaks in TIs with
the thermal motion of different atomic oscillators supports the earlier
findings which considered that microscopic origin of 1/f noise in solid is in
the “perpetual equilibrium atomic motion”[36, 41]. This mechanism can explain
1/f noise in any solid-state system, in both equilibrium [76] and
nonequilibrium [40].

In conclusion, the 1/f noise structure observed by Islam et al.[14] in
(Bi,Sb)2Te3 was correlated with the van Hove singularities in the bulk phonon
spectrum of this TI. Thermal motion of the Bi atoms was identified as the
source of the most intense noise peak observed in this material at 50 K. Other
noise peaks were assigned to multi-phonon combinations, which are the
signatures of (Bi,Sb)2Te3 lattice anharmonicity. We found that the 1/f noise
structure reported by Biswas et al.[17] in BiSbTeSe1.6 tracks the BiSbTeSe2
Raman spectrum, each noise singularity being assigned to a single- or multi-
phonon energy. Stronger carrier-phonon coupling due to anharmonicity was
invoked as explanation of the noise increase observed above some specific
temperature thresholds in both materials. The observation of the phonon
structure in the whole temperature range strongly supports the carrier-phonon
interaction as the unique microscopic source of 1/f noise at the surface or in
the bulk. Our investigation revealed that the thermal motion of different bulk
atoms is involved in the mechanism of 1/f noise in topological insulators. The
noise data in TI supports the thermal motion of the atoms as microscopic
source of 1/f noise in solid.

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# Higher-order incompatibility improves distinguishability of causal quantum
networks

Nidhin Sudarsanan Ragini  Institute of Physics, Slovak Academy of Sciences,
Dubravska cesta 9, Bratislava 845 11, Slovakia  Sk Sazim  Center for
Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668
Warsaw, Poland

(September 3, 2024)

###### Abstract

We show that incompatible “higher-order quantum theoretic testing procedures”,
known as quantum testers, perform better in distinguishing causal quantum
networks than its compatible counterparts. To demonstrate our finding, we
consider a specific two-party quantum comb discrimination (QCD) game based on
the task in which a receiver gets a black-box that implements a quantum comb
and they attempt to determine which comb it implements based on prior
information. We show that the receiver does better at winning the game when
they have exclusive access to incompatible testers over compatible ones. Our
investigation generalises the earlier finding that the incompatibility of
POVMs is a bona fide resource in quantum state distinguishability. We also
show that any quantum resource in testers is useful in comb exclusion tasks
using the resource measure, convex weight.

##  I Introduction

Historically, one of the perplexing avenues of quantum theory studies has been
the notion of non-commuting observables and their uncertainty relations,
manifesting joint-measurability as well as error-disturbance trade-offs [1].
In the current age of active development and understanding of quantum
information processing and quantum computation [2], this avenue has
transformed into studies on joint measurability beyond non-commutativity (See
[3] for a recent colloquium and [4] for an overview of related aspects) and,
subsequently extending this notion to joint implementation of other quantum
devices beyond quantum measurements [5, 6, 7, 4]. This inadmissibility of
joint implementation of certain collections of quantum devices is now referred
to as quantum incompatibility of those collections [3, 4]. Moreover, the
incompatibility of quantum observables can be understood as a bona fide
resource that is consumed in certain quantum protocols, for example, in
enhancing secure key rates in quantum cryptographic protocols [8].

Recently, along the above lines, it has been found that for a particular
quantum state discrimination game, all incompatible collections of observables
result in an advantage over compatible collections [9, 10]. This result is
similar to those that found all entangled states improve performance of
channel distinguishability tasks [11, 12]. The aforementioned result on
incompatible observables has an important operational significance: A given
set of measurement observables are incompatible if and if they provide
advantage in such discrimination games over compatible ones [9, 10].
Subsequently, these results garnered much attention and sparked studies within
the the resource theoretic perspective of quantum objects [13, 14, 15, 16, 17,
18].

In those works which are discussed in this passage, a lot of focus was put on
finding the physical implications of resource contained within the quantum
objects, specifically quantum states, its transformations, quantum channels
and the measurement observables. It has been shown that not only the resources
within measurement observables are advantageous in state discrimination tasks,
the resources within quantum states play auxiliary roles in discrimination
tasks involving of their transformations, specifically, quantum channels and
measurement observables [13]. Moreover, we can observe that three distinctive
research directions emerge based on these studies: 1) Identifying and
characterising the exact quantum features of measurement observables which are
advantageous in quantum state discrimination tasks [9, 16], and 2) Studies
involving usage of resources present in quantum states for improving
discrimination of quantum observables [13, 19]. Furthermore, 3) Studies
identifying the resourcefulness present in general quantum processes,
including channels and higher-order transformations, those result in
advantages in quantum state discrimination tasks [20]. With regard to the
third direction, in [20] the authors consider higher-order transformations
[21] as part of the strategies in their state discrimination games with fixed
measurement observables and study the advantages.

A common feature among all of these studies is that all of them consider
resourcefulness of a single quantum object class, among the different classes
those construct the quantum information processing task. However, we can
notice that for discrimination tasks of quantum channels, resources present in
both probe state and probe observable can, in synergy, result in advantages.
In fact, in Ref.[22], the authors have shown that the advantages in such tasks
are ramified as the product of the individual advantages arising from each of
the test objects. In addition, such channel discrimination tasks can also be
strategised to incorporate intermediate, auxiliary probe transformations
within the testing network (these are referred to as adaptive strategies in
the literature). When testing networks those implement such strategies are
used, the resources present in those transformations can also result in
advantages, and should be accounted for as well. Unlike in [22], when we are
interested in the collective advantage of a particular testing network
furnish, rather than the individual contributions of the component probe
objects, we can resort to describe the network as a single quantum object.
Such descriptions are given by quantum testers [23, 24, 25], which are unified
descriptions of the associated probe objects of a testing network. In fact,
the framework of quantum testers have been developed to describe not only
testing networks for discriminating quantum channels with single time-step,
but also for general multiple time-step quantum causal processes those which
appear themselves as incomplete circuits or networks, also referred to higher-
order quantum processes or quantum combs [26, 24, 21]; These combs describe
the most general transformations of channels, super-channels and so on. As
such, within the premise of quantum discrimination tasks, the central objects
to be distinguished can be lifted to be considered as quantum combs and
quantum testers describing the associated discrimination networks. This
outlook is all-encompassing since quantum states and observables can be
considered as special cases of quantum channels with single time steps, which
are themselves a special case of quantum combs.

As partially hinted earlier, the notion of quantum incompatibility is not
exclusive to observables. In our work, we consider the incompatibility of
quantum testers, which we refer to as higher-order incompatibility [7], and
study the resourcefulness of this incompatibility in the quantum comb
discrimination game that we present. Subsequently, we find that incompatible
testers are advantageous over compatible collections of testers.
Alternatively, our results can be understood as providing the following
operational meaning to tester incompatibility. A set of quantum testers are
incompatible if and only if they provide advantage over compatible ones in our
quantum comb discrimination task. Moreover, this advantage is quantified in
terms of the robustness of incompatibility of testers. Thus, in contrast to
the works discussed earlier, we can infer that possible non-trivial individual
contributions, from the resources in probe states, transformations and
observables, to the quantum advantage is collectivley reflected in the
robustness. Moreover, we also study the advantageous roles of tester resources
in comb exclusion games using the resource-theoretic measure convex weight,
which also implies that tester incompatibility being a resource here as well.

In Section II, the framework of higher-order quantum theory is sketched, where
mathematical characterisations of quantum combs and testers are furnished.
Within this section, the notion of incompatibility of testers and an
associated characterisation of it is also presented. In Section III, we
present the particular discrimination game and our results regarding the
advantage furnished by higher-order incompatibility. In Section IV, we
characterise the advantage of the incompatibility of testers (or, more
generally any quantum resource in it) in comb exclusion games. Finally, we
conclude with a discussion in Section V.

##  II Higher-order quantum theory: Incomplete quantum networks

###  II.1 Quantum Combs

Quantum channels, modeled by completely-positive trace-preserving maps,
describe all quantum state transformations admitted by quantum theory. We can
then generalise this notion of transformations by replacing states with
quantum processes. These input and output processes can be channels, finite
collections of time-ordered channels, superchannels or, most generally and
equivalently, memory channels [27]. The transformations are described by
quantum circuits with open slots. Such incomplete circuits are described by
quantum causal networks or quantum combs [26, 24, 21], where the number of
open slots characterise the combs. For example, see Figure 1, for an
illustration for the action of a 2-slot comb. It has also been identified that
a $n$-slot quantum comb are equivalent to $(n+1)$ channels with memory [23].
It should also be noted that a comb describes all physical network
implementations which result in identical transformations.

Figure 1: A 2-slot quantum comb (comprised of channels $\mathcal{C}_{1}$,
$\mathcal{C}_{2}$, $\mathcal{C}_{3}$; Note that the red/light greyed out wires
are quantum memories) transforming the two channels, $\mathcal{N}_{1}$ and
$\mathcal{N}_{2}$, to a channel $\mathcal{R}$. In fact, those two slots can be
plugged in with 1-slot combs. Here, $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$
can be seen as a single 1-slot comb without any shared memory.

As reflected in the figure, following the Pavia indexing convention [21], for
each $n$-slot comb, the input systems are indexed even $2k$ and output systems
indexed odd $(2k+1)$, with $k\in\\{0,\dots,n\\}$. Now, we can write down the
characterisation for an $n$-slot quantum comb $\mathcal{C}^{(n)}$, with its
Choi-Jamiołkowski operator $\mathsf{C}^{(n)}$. The following linear
constraints on the Choi-Jamiołkowski operator reflects the causal structure of
the comb, as well as the fact that the positive operators $\mathsf{C}^{(k)}$,
for all values of $k$, are themselves Choi-Jamiołkowski operators of combs
with lesser and lesser number of open slots, on respective spaces.

$\displaystyle\text{Tr}_{2k+1}[\mathsf{C}^{(k)}]=\mathds{1}_{2k}\otimes\mathsf{C}^{(k-1)}\quad;\forall
k\in\\{0,\dots,n\\}.$ (1)

Now, from Figure 1, we can notice that a 2-slot comb can transform a 1-slot
comb to a channel (even though not depicted, it can also transform a 0-slot
comb to a 1-slot comb). In general, we can see that an $n$-slot comb can
transform $(k-1)$-slot combs to appropriate $(n-k-1)$-slot combs. These
transformations generate hierarchical orders of transformations, referred to
as higher-order transformations [21], enabling us to refer to this framework
as higher-order quantum theory.

###  II.2 Quantum Testers

The notion of quantum testers, introduced in [23, 24, 25], describe
measurements of quantum combs. That is, they describe measurement networks
which send quantum combs to probability distributions. It should be noted that
different physical implementations of measurement networks which result in
identical probability distributions are described by the same quantum tester.

Similar to combs, testers are also characterised by the number of open slots
they have. See Figure 2, for the action of a 3-slot tester. An $n$-slot tester
measures $(n-1)$-slot combs. As such, we can see that 1-slot testers test
0-slot combs, which are nothing but single time-step quantum channels. Now,
given an finite, discrete mesurement outcomes set $\Omega$, an $n$-slot tester
is a collection of positive operators
$\mathsf{T}^{(n)}\equiv\\{\mathsf{T}_{x}^{(n)}\\}$, which satisfies the
following normalisation conditions [23].

$\displaystyle\sum_{x\in\Omega}\mathsf{T}_{x}^{(n)}$ $\displaystyle=$
$\displaystyle\mathds{1}_{2n-1}\otimes\Xi^{(n)}$ (2)
$\displaystyle\text{Tr}_{2k-2}[\Xi^{(n)}]$ $\displaystyle=$
$\displaystyle\mathds{1}_{2k-3}\otimes\Xi^{(k-1)}\quad\forall
k\in\\{2,\dots,n\\}$ (3) $\displaystyle\text{Tr}[\Xi^{(1)}]$ $\displaystyle=$
$\displaystyle 1.$ (4)

The operators forming the collection $\\{\Xi^{(k)}\\}$ are referred to as the
normalization objects of the tester, and are themselves combs.

Figure 2: A 3-slot quantum tester (comprised of probe state $\xi$, channels
$\mathcal{E}_{1}$, $\mathcal{E}_{2}$, and probe POVM $\mathsf{M}$) measuring
the 2-slot comb depicted in Figure 1.

Implementing a tester $\\{\mathsf{T}_{k}\\}$ on given a quantum comb
$\mathcal{C}$ with Choi operator $\mathsf{C}$, results in a probability
distribution prescribed by the generalized Born rule

$p(k|\mathcal{C})=\text{Tr}[\mathsf{T}_{k}\mathsf{C}].$ (5)

This Born rule, using the normalization of the tester
$\sum_{k}\mathsf{T}_{k}=\mathds{1}_{2n-1}\otimes\Xi$, can always be uniquely
rewritten as to have a quantum state and a POVM operator inside the trace
functional. This POVM, which is in one-to-one correspondence with the tester,
is referred to as its canonical POVM [23]. It is given by

$\mathsf{P}_{j}=(\mathds{1}_{2n-1}\otimes\Xi^{-\frac{1}{2}})[\mathsf{T}_{j}](\mathds{1}_{2n-1}\otimes\Xi^{-\frac{1}{2}}).$
(6)

###  II.3 Incompatibility of Testers

Similar to how compatible POVMs are jointly measurable, compatible testers are
jointly implementable. That is, there should exist atleast one joint tester
such that the compatible ones can be arrived at by some coarse-graining. Let
$\mathsf{T}^{(n)}$ and $\mathsf{S}^{(n)}$ be two $n$-slot testers,
respectively, with normalisations $\\{\Xi^{(k)}\\}$ and $\\{\Theta^{(k)}\\}$,
and canonical POVMs ${\mathsf{P}}$ and ${\mathsf{Q}}$. Then,
$\mathsf{T}^{(n)}$ and $\mathsf{S}^{(n)}$ are compatible when their
corresponding normalisations coincide, $\Xi^{(k)}=\Theta^{(k)}$ for all $k$,
and their canonical POVMs are compatible themselves [7]. This notion of
incompatibility can be extended to finite collections of $n$-slot testers.

Now, given an incompatible collection of $n$-slot testers
$\\{\mathsf{T}_{\alpha}^{(n)}\\}$, physical mixing of appropriate noise
$n$-slot testers $\\{\mathsf{N}_{\alpha}^{(n)}\\}$ can render them compatible
(Only testers with identical sequences of input and output spaces can be
mixed; The outcome sets should also be same even though we can always make
them same when not). This notion serves as the basis for robustness, which was
developed for collections of observables [28, 29, 30]. Based on this notion
and inspired from the one introduced in [16], we have the robustness of
incompatibility for a collection of $n$-slot testers,

$\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})=\min
r:\left\\{\frac{1}{1+r}(\mathsf{T}_{\alpha}^{(n)}+r\mathsf{N}_{\alpha}^{(n)})\right\\}\text{
is compatible.}$ (7)

The incompatible set of testers $\\{\mathsf{T}_{\alpha}\\}$ can be rendered
compatible by mixing it with another appropriate set of noise testers
$\\{\mathsf{N}_{\alpha}\\}$. Here, the noise testers are restricted to be non-
identical to each other [7]. See appendix for a detailed definition of the
robustness, exposing the notion of compatibility as well as the constraints on
the quantum objects involved.

Similar to the robustness of incompatible of POVMs, the
$\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})$ possesses certain desirable
properties [7]:

  1. 1.

Faithfulness, i.e., $\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})=0$ for compatible
testers and non-zero otherwise.

  2. 2.

Monotonic under tester simulation (convex combination + post processing): Lets
say, $\\{\mathsf{T}^{\prime}_{\beta}\\}_{\beta}$ can be simulated from
$\\{\mathsf{T}_{\alpha}\\}_{\alpha}$ such that

$\displaystyle\mathsf{T}^{\prime}_{b|\beta}=\sum
p(\lambda)p(\alpha|\beta,\lambda)p(b|a,\beta,\lambda)\mathsf{T}_{a|\alpha},$
(8)

where $\\{p(\cdot)\\}$ are arbitrary sets of probability distribution then
following relation holds

$\displaystyle\mathcal{R}(\\{\mathsf{T}^{\prime}_{\beta}\\})\leq\mathcal{R}(\\{\mathsf{T}_{\alpha}\\}).$
(9)

One might wonder whether second property holds in general. We showed in the
following section that the property $2$ is true as the relation Eq. (13)
holds.

##  III Advantage in Quantum comb discrimination game (QCD)

###  III.1 QCD

We denote an ensemble of $m$ $N$-slot quantum combs as
$\mathcal{X}=\\{\mathcal{C}_{b},w(b)\\}_{b=1}^{m}$, where $w(b)$ is the
probability with which the comb $\mathcal{C}_{b}$ is chosen from the ensemble.
The following multi-round discrimination game is played between two parties,
Bob and Alice. Bob has access to $s$ different ensembles
$\\{\mathcal{X}_{\beta}\\}_{\beta=1}^{s}$ of quantum combs with
$\mathcal{X}_{\beta}=\\{\mathcal{C}_{b|\beta},w(b|\beta)\\}_{b=1}^{k}$. In
each round of the protocol, Bob chooses one out of the $s$ ensembles with
probability $w(\beta)$ and, from that ensemble, one out of the $m$ networks
$\mathcal{C}_{b|\beta}$ with conditional probability $w(b|\beta)$. Then he
sends Alice the classical information $\beta$ (regarding which ensemble he
chose) and a black box that implements $\mathcal{C}_{b|\beta}$. Now, Alice’s
task is to guess which network the received black box will implement or,
equivalently, to identify the value of $b$. For this task, she chooses and
implements a tester from a finite fixed collection of appropriate testers
$\\{\mathsf{T}_{\alpha}\\}$ available to her. Then, a measurement outcome $a$
occurs with the conditional probability $p(a|b)={\rm
Tr}[\mathsf{C}_{b|\beta}\mathsf{T}_{a|\alpha}]$.

###  III.2 Main result: Tester incompatibility as a resource in QCD

To arrive at how incompatible testers are advantageous in the above game, we
can look at the following two complementary scenarios. Scenario 1: The
collection $\\{\mathsf{T}_{\alpha}\\}_{\alpha}$ is incompatible. In this
gameplay, Alice’s access is restricted to incompatible collections of testers
$\\{\mathsf{T}_{\alpha}\\}_{\alpha}$. After receiving the value of $\beta$ and
the black box, by having access to a random variable $\lambda$, Alice chooses
and implements the tester $\mathsf{T}_{\alpha}$ with probability
$p(\alpha|\beta,\lambda)$. From the measurement outcome $a$, she guesses the
value of $b$ as prescribed by $p(g|a,\beta,\lambda)$. She can then evaluate
the optimal average probability of correct guessing $g$ by optimizing over all
possible strategies,
$\mathcal{S}=\\{p(\lambda),p(\alpha|\beta,\lambda),p(g|a,\beta,\lambda)\\}$,

$\displaystyle
p(\\{\mathcal{X}_{\beta}\\},\\{\mathsf{T}_{\alpha}\\})=\max_{\mathcal{S}}\sum_{b,\beta,a,\alpha,g,\lambda}w(b,\beta)p(\lambda)p(\alpha|\beta,\lambda)\text{Tr}[\mathsf{C}_{b|\beta}\mathsf{T}_{a|\alpha}]p(g|a,\beta,\lambda)\delta_{g,b}.$
(10)

Scenario 2: The collection of available testers is compatible. In this
gameplay, Alice can implement only a fixed tester $\mathsf{Q}_{\nu}$ per a
whole run of the game. Each run of the game she can choose $\mathsf{Q}_{\nu}$
based on the random variable $\nu$. Moreover, after receiving the classical
side information $\beta$, she can only post-process the fixed tester based on
it. Similar to how post-processings of a certain observable leads to a
collection of compatible observables, post-processings of a fixed tester
result in a collection of compatible testers, since the normalisation states
stay invariant and identical and, the canonical observables are post-
processings of the parent one [10]. Thus, this is equivalent to Alice having
access to compatible collections of observables whose joint observable is
$\mathsf{Q}_{\nu}$. For this scenario, the optimal guessing success
probability is given by

$\displaystyle
p(\\{\mathcal{X}_{\beta}\\})=\max_{\mathcal{S}_{c}}\sum_{b,\beta,a,g,v}w(b,\beta)p(v)\text{Tr}[\mathsf{C}_{b|\beta}\mathsf{Q}_{a|v}]p(g|a,\beta,v)\delta_{g,b}.$
(11)

Here, optimization is done over all strategies,
$\mathcal{S}_{c}=\\{p(v),\mathsf{Q}_{v},p(g|a,\beta,v)\\}$.

When maximised over all possible ensembles, the maximum value of the relative
increase in the guessing probability attainable by the given set of
incompatible testers $\\{\mathsf{T}_{\alpha}\\}_{\alpha}$ with respect to that
is attainable only by single parent testers is given by

$\max_{\\{\mathcal{X}_{\beta}\\}}\frac{p(\\{\mathcal{X}_{\beta}\\},\\{\mathsf{T}_{\alpha}\\})}{p(\\{\mathcal{X}_{\beta}\\})}.$
(12)

We find that this maximum advantage attainable by Alice if she has access to
incompatible testers over the compatible ones is exactly related to the
robustness of incompatibility in the following way

$\displaystyle\max_{\\{\mathcal{X}_{\beta}\\}}\frac{p(\\{\mathcal{X}_{\beta}\\},\\{\mathsf{T}_{\alpha}\\})}{p(\\{\mathcal{X}_{\beta}\\})}=1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\}).$
(13)

Our proof of the above result follows closely to that of in Ref. [9]; however
our investigation generalizes their findings to higher-order incompatibility.
We revert enthusiastic readers to Appendix A for a detailed proof of the above
expression using semi-definite programming arguments.

Now, it is straightforward to conclude that the advantage using incompatible
testers in our QCD game is non-increasing under tester simulation discussed
above. This can be understood in following way: Under tester simulation, one
will not be able to elevate the compatible set of testers to incompatible
ones. Moreover, under such action, the robustness of tester incompatibility
decreases. Therefore, the gain in QCD game will also monotonically decrease
under tester simulation. For completeness, we refer our readers to the Ref.
[9] where it is discussed in detail.

##  IV The “convex weight” for quantum testers, and advantages in exclusion
games of comb

In [31], the authors have found that, using the quantum resource theoretic
measure convex weight and auxiliary mathematical programming arguments, any
quantum resource improves performance of quantum exclusion tasks/games of
zeroth order quantum devices, including states, POVMS, channels and
instruments. These observations has been bolstered by the subsequent work
[32]. In general, since the result hold for quantum instruments, it can be
recast for the rest of the devices mentioned here. Since higher-order quantum
devices also admit equivalent analyses with minor changes, we would expect
that for exclusion games of higher-order quantum devices, any resource would
result in improvement. This suggests that incompatibility is a resource, not
just for distinguishability games but also for exclusion or anti-
distinguishability games.

Convex weight for an arbitrary quantum device is defined in Ref. [31]. The
convex weight fulfills all the desirable properties to be a useful resource
quantifiers similar to robustness of quantum devices. For quantum tester
assemblage $\mathcal{T}_{\alpha}:=\\{\mathsf{T}_{a|\alpha}\\}$, we can define
its convex weight in terms of conic program as

$\displaystyle 1-\mathcal{W}(\mathcal{T}_{\alpha})=\max\frac{1}{\alpha D}{\rm
Tr}\left[\sum_{a,\alpha}\bar{\mathsf{Q}}_{a|\alpha}\right]$ (14)
$\displaystyle{\rm
s.t.}\quad\mathsf{T}_{a|\alpha}\geq\bar{\mathsf{Q}}_{a|\alpha}\quad\forall(a,\alpha),$
$\displaystyle\\{\bar{\mathsf{Q}}_{a|\alpha}\\}\in\mathcal{T}_{F},$

where $D=\prod_{j=0}^{n}d_{2n-(2j+1)}$ where $d_{x}$ is the dimension of the
system indexed $x$ and $\alpha$ is the number of testers. Note that the set,
$\mathcal{T}_{F}$, forms a convex and compact set of devices which can be used
multiple times to produce the devices in $\mathcal{T}_{\alpha}$. Using the
similar approach from earlier section, the dual of the above conic program
reads as

$\displaystyle 1-\mathcal{W}(\mathcal{T}_{\alpha})=\min_{Y_{a|\alpha}}{\rm
Tr}\left[\sum_{a,\alpha}\mathsf{T}_{a|\alpha}Y_{a|\alpha}\right],$ (15)
$\displaystyle{\rm s.t.}\quad Y_{a|\alpha}\geq 0\quad\forall(a,\alpha),$
$\displaystyle{\rm
Tr}\left[\sum_{a,\alpha}\bar{\mathsf{Q}}_{a|\alpha}Y_{a|\alpha}\right]\geq
1,\quad\\{\bar{\mathsf{Q}}_{a|\alpha}\\}\in\mathcal{T}_{F},$

where $Y_{a|\alpha}$ can be decomposed as such that it satisfies the Eq. (62)
with $Y_{a|\alpha}$ replacing $\omega_{a\alpha}^{*}$.

Now, in a comb exclusion game, a referee sends a black box to the player who
implements one of the comb from the set $\mathcal{X}_{\beta}$ with probability
$w(b|\beta)$. The player succeed at the game if they guess a comb which is not
sent to them. Given the structure of the game, the best classical strategy for
player is to choose the testers from set $\mathcal{T}_{F}$ always, i.e., the
minimum error probability is given by

$\displaystyle\bar{p}(\mathcal{X}_{\beta})=\min_{\bar{\mathsf{Q}}\in\mathcal{T}_{F}}\sum_{a,b}w(b|\beta){\rm
Tr}\left[\bar{\mathsf{Q}}_{a|\alpha}\mathsf{C}_{b|\beta}\right]\delta_{a,b}.$
(16)

Can we do better with quantum strategies? Given the objects
$\mathcal{X}_{\beta}$ and the set of testers $\mathcal{T}_{\alpha}$, the
minimum error probability in an arbitrary quantum strategy is given by using
the formulation of Ref. [31],

$\displaystyle\bar{p}(\mathcal{X}_{\beta},\mathcal{T}_{\alpha})=\min_{\mathsf{T}}\sum_{a,b}w(b|\beta){\rm
Tr}\left[\mathsf{T}_{a|\alpha}\mathsf{C}_{b|\beta}\right]\delta_{a,b}.$ (17)

In general, the player will have less error with quantum strategies than its
classical counterpart, i.e.,
$\bar{p}(\mathcal{X}_{\beta},\mathcal{T}_{\alpha})/\bar{p}(\mathcal{X}_{\beta})\leq
1$. In fact, we can prove the following result, using the similar techniques
deployed in the Ref. [31] (see also Appendix B),

$\displaystyle\min_{\mathcal{X}_{\beta}}\frac{\bar{p}(\mathcal{X}_{\beta},\mathcal{T}_{\alpha})}{\bar{p}(\mathcal{X}_{\beta})}=1-\mathcal{W}(\mathcal{T}_{\alpha}).$
(18)

The above result implies that there exists a quantum strategy where the
decrease in the minimum error probability is given by the convex weight of the
quantum device available to the player.

##  V Conclusion

The notion of quantum testers was introduced to describe the testing or
measurements of multiple time-step quantum causal networks, now popularised as
quantum combs. Since these testers are ramified quantum measurements, the
operational notion of quantum incompatibility also has been introduced and
developed for them, where we study collections of testers those seldom admit
joint testers. In the wake of all of these developments, in this work, we
investigated the quality of resourcefulness of incompatible testers.
Consequently, we showed that the incompatible testers result in advantages,
over compatible ones, in a specific game in which quantum combs are
distinguished. This heralds that the incompatibility, as a quantum feature,
present in general testing procedures is a resource in discriminating higher-
order quantum objects. Moreover, this generalises the earlier finding that
‘POVM incompatibility is a resource for distinguishability of states’, which
was discovered in Ref.[9]. One of the main differences from our work is that
the aforementioned work was based on the discrimination of quantum objects of
static nature, namely, quantum states to demonstrate the resourcefulness of
incompatible quantum testing procedures for states; In our work, where central
objects of interest are quantum combs, they describe networks composed of
quantum objects of both nature: static as well as dynamical. In this context,
the result of [9], as qualitatively powerful as it is, can be viewed as a
special case of our generalisation when the associated combs are restricted to
be states. Another peculiarity of our result is that it encompasses all
distinguishability or discrimination schemes possible for any basic quantum
object (including states, single time-step operations and POVMs): irrespective
of whether it is single-shot or multiple-shot and whether it is parallel or
adaptive. Moreover, it encompasses distinguishability of quantum channels with
memory channels[27].

Moreover, we investigated quantum comb exclusion games and discovered that any
quantum resource present in quantum testers result in performance improvement
in such games. Our result generalizes the result found in [31] which dealt
with resourcefulness of POVMs in state exclusion games. Quantum resources of
general testers are contributed from the component quantum objects of the
network as well as potentially from the quantum memories they possess. As a
consequence of our result, quantum incompatibility of testers, which arise in
the same way, can then be identified as a resource in quantum comb exclusion
games. In conclusion, we have discovered that the more intricate notion of
higher-order incompatibility is a resource in quantum hypothesis-testing tasks
involving combs, including distinguishability and exclusion.

###### Acknowledgements.

NSR acknowledges Michal Sedlák for clarifying certain concepts regarding
quantum combs. NSR acknowledges the support of projects APVV-22-0570 (DEQHOST)
and VEGA 2/0183/21 (DESCOM). SS acknowledges funding through PASIFIC program
call 2 (Agreement No. PAN.BFB.S.BDN.460.022 with the Polish Academy of
Sciences). This project has received funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 847639 and from the Ministry of Education and Science of Poland.

## Author declarations

The authors have no conflicts to disclose.

## Data Availability Statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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##  Appendix A Tester incompatibility results in advantages in QCD

###  A.1 Noise robustness

As we have seen, Eqs(2-4) prescribe $n$-slot testers. We consider a finite
incompatible collection of such principal testers $\\{\mathsf{T}_{\alpha}\\}$.
The the robustness of this collection $\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})$
is defined as the follows, from which we can identify a compatible collection
of testers, corresponding to parent observables $\mathsf{Q}_{\mu}$ and post-
processing probability distribution $p(a|\alpha,\mu)$ as appearing below.

$\displaystyle\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})$ $\displaystyle=$
$\displaystyle\min\>\>\>r\>\>$ (20) $\displaystyle\text{s.t.
}\frac{1}{(1+r)}(\mathsf{T}_{a|\alpha}+r\mathsf{N}_{a|\alpha})=\sum_{\mu}p(a|\alpha,\mu)\mathsf{Q}_{\mu}$
[principal testers] $\displaystyle\mathsf{T}_{a|\alpha}\geq
0,\sum_{a}\mathsf{T}_{a|\alpha}=\mathds{1}_{(2n-1)}\otimes\Theta^{(n)}_{\alpha}$
(24)
$\displaystyle\text{Tr}_{(2k-2)}\\{\Theta^{(n)}_{\alpha}\\}=\mathds{1}_{(2k-3)}\otimes\Theta^{(k-1)}_{\alpha};k\in\\{2,\dots,n\\}$
$\displaystyle\text{Tr}\\{\Theta_{\alpha}^{(1)}\\}=1.$ [noise testers]
$\displaystyle\mathsf{N}_{a|\alpha}\geq
0,\sum_{a}\mathsf{N}_{a|\alpha}=\mathds{1}_{(2n-1)}\otimes\Gamma^{(n)}_{\alpha}$
(28)
$\displaystyle\text{Tr}_{(2k-2)}\\{\Gamma^{(n)}_{\alpha}\\}=\mathds{1}_{(2k-3)}\otimes\Gamma^{(k-1)};k\in\\{2,\dots,n\\}$
$\displaystyle\text{Tr}\\{\Gamma_{\alpha}^{(1)}\\}=1.$ [probability
distribution] $\displaystyle p(a|\alpha,\mu)\geq
0,\sum_{\mu}p(a|\alpha,\mu)=1$ (30) [joint tester]
$\displaystyle\mathsf{Q}_{\mu}\geq
0,\sum_{\mu}\mathsf{Q}_{\mu}=\mathds{1}\otimes\Theta^{(n)}$ (34)
$\displaystyle\text{Tr}_{(2k-2)}\\{\Theta^{(n)}\\}=\mathds{1}_{(2k-3)}\otimes\Theta^{(n-1)};k\in\\{2,\dots,n\\}$
$\displaystyle\text{Tr}\\{\Theta^{(1)}\\}=1.$ [normalisation objects]
$\displaystyle\Theta^{(n)}=\frac{1}{(1+r)}(\Theta^{(n)}_{\alpha}+r\Gamma^{(n)}_{\alpha})\quad\forall\alpha\>\forall
n$ (37) $\displaystyle\Theta^{(n)}_{\alpha}\geq 0\;\text{ and
}\;\Gamma^{(n)}_{\alpha}\geq 0\quad\forall\alpha\>\forall n.$

###  A.2 Primal SDP

See Chapter 6 of [33] on measurement incompatibility, as well as the work [9],
where the arguments that we use in the formulation of the following SDP can be
found. From (A2), the process effects of the noise testers are given by,

$\mathsf{N}_{a|\alpha}=\frac{1}{r}\\{(1+r)\sum_{\mu}p(a|\alpha,\mu)\mathsf{Q}_{\mu}-\mathsf{T}_{a|\alpha}\\}.$
(38)

Then, with $s=(1+r)$, the positive semidefiniteness of the process effects
results in,

$s\sum_{\mu}p(a|\alpha,\mu)\mathsf{Q}_{\mu}\geq\mathsf{T}_{a|\alpha}.$ (39)

Now, let $a_{k}$ be the outcome label for the $k^{\text{th}}$ tester. As such,
we can collectively write all the $m$ outcome labels into a vector
$\vec{a}=(a_{1},\dots,a_{m})$. Since, each of the $m$ testers has $o$
outcomes, that is $a_{k}\in\\{1,\dots,o\\}$ for all $k$, there are $o^{m}$
possible vectors. Given the result $\mu$ of the joint tester, the probability
of producing a specific vector $\vec{a}$ is given by
$p(\vec{a}|\mu)=p(a_{1}|\alpha=1,\mu)p(a_{2}|\alpha=2,\mu)\cdots
p(a_{m}|\alpha=m,\mu)=\prod_{\alpha}p(a_{\alpha}|\alpha,\mu)$. Now, the
probability of the $\alpha^{\text{th}}$ entry being equal to $a$, that is
$a_{\alpha}=a$, is given by
$p(a|\alpha,\mu)=\sum_{\vec{a}}\delta_{a_{\alpha},a}p(\vec{a}|\mu)$. Then,
conditional probability distribution can be decomposed using the basis of
deterministic conditional probabilities,
$\\{d_{\vec{a}}(a|\alpha)=\delta_{a,\vec{a}_{x}}\\}$ as
$p(a|\alpha,\mu)=\sum_{a}d_{\vec{a}}(a|\alpha)p(\vec{a}|\mu)$, where

$\sum_{\mu}p(a|\alpha,\mu)\mathsf{Q}_{\mu}=\sum_{\vec{a}}d_{\vec{a}}(a|\alpha)\mathsf{Q}_{\vec{a}},$
(40)

while $\mathsf{Q}_{\vec{a}}=\sum_{\mu}p(\vec{a}|\mu)\mathsf{Q}_{\mu}$. Since
$\mathsf{Q}_{\vec{a}}\geq 0$, we can define
$\tilde{\mathsf{Q}}_{\vec{a}}:=s\mathsf{Q}_{\vec{a}}$. Then,

$\displaystyle\tilde{\mathsf{Q}}_{\vec{a}}\geq 0$ (41)
$\displaystyle\sum_{\vec{a}}\tilde{\mathsf{Q}}_{\vec{a}}=s\mathds{1}\otimes\Theta^{(n)}$
(42)
$\displaystyle\text{Tr}_{(2k-2)}\\{\Theta^{(k)}\\}=\mathds{1}_{(2k-3)}\otimes\Theta^{(k-1)};\quad
k\in\\{2,\dots,n\\}$ (43)

This means we essentially find the parent tester which has the following
normalisation,

$\displaystyle\Theta^{(k)}=\frac{1}{s}(\Theta^{(k)}_{\alpha}+r\Gamma^{(k)}_{\alpha})\quad\forall\alpha\>\forall
k.$ (44)

In fact, the above conditions are automatically satisfied since post-
processings preserve normalisations and as such does not appear in the SDP.
Now, we can formulate the SDP in the following form.

$\displaystyle 1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})$ $\displaystyle=$
$\displaystyle\min\>\>\>s$ (48) $\displaystyle\text{s.t.
}\sum_{\vec{a}}d_{\vec{a}}(a|\alpha)\tilde{\mathsf{Q}}_{\vec{a}}\geq\mathsf{T}_{a|\alpha}$
$\displaystyle\tilde{\mathsf{Q}}_{\vec{a}}\geq
0,\;\sum_{\vec{a}}\tilde{\mathsf{Q}}_{\vec{a}}=\mathds{1}\otimes\tilde{\Theta}^{(n)}$
$\displaystyle\text{Tr}_{(2k-2)}\\{\tilde{\Theta}^{(k)}\\}=\mathds{1}_{(2k-3)}\otimes\tilde{\Theta}^{(k-1)};\>\>\forall\quad
k\in\\{2,\dots,n\\}.$

###  A.3 Lower bound

We can use the primal SDP to have an upper bound on the advantage in the QCD
game. In order to achieve this, let us start by assuming that one of the
solutions to our SDP shall be achieved by the triple
$(N_{a|\alpha}^{*},Q_{\mu}^{*},p^{*}(a|\alpha,\mu))$. Then, we have the
following

$\displaystyle\frac{1}{1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})}\left(\mathsf{T}_{a|\alpha}+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})N_{a|\alpha}^{*}\right)=\sum_{\mu}p^{*}(a|\alpha,\mu)Q_{\mu}^{*}.$
(49)

Since $\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})\geq 0$ and $N_{a|\alpha}^{*}\geq
0$, the above equation results in the inequality,

$\displaystyle[1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})]\sum_{\mu}p^{*}(a|\alpha,\mu)Q_{\mu}^{*}\geq\mathsf{T}_{a|\alpha},\>\>\forall
a,\alpha.$ (50)

Now multiplying both sides of the inequality with $\mathsf{C}_{b|\beta}$, the
probabilities $w(b,\beta),p(\nu),p(\alpha|\beta,\nu),p(g|a,\beta,\nu)$ and
$\delta_{g,b}$ and then summing over the corresponding indices and taking
trace, we arrive at the inequality,

$\displaystyle[1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})]\sum_{\mu,g,a,b,\alpha,\beta,\nu}w(b,\beta)p(\nu)p(\alpha|\beta,\nu)p^{*}(a|\alpha,\mu)\text{Tr}[\mathsf{C}_{b|\beta}Q_{\mu}^{*}]p(g|a,\beta,\nu)\delta_{g,b}$
$\displaystyle\geq\sum_{\mu,g,a,b,\alpha,\beta,\nu}w(b,\beta)p(\nu)p(\alpha|\beta,\nu)\text{Tr}[\mathsf{C}_{b|\beta}\mathsf{T}_{a|\alpha}]p(g|a,\beta,\nu)\delta_{g,b},$
(51)

Let us define the following relation,

$\displaystyle
p(g|\mu,\beta,\nu)=\sum_{a,\alpha}p^{*}(a|\alpha,\mu)p(\alpha|\beta,\nu)p(g|a,\beta,\nu).$
(52)

With this definition, the sum on the L.H.S. of the Ineq.(A.3) simplifies to
$\sum_{\mu
gb\beta\nu}w(b,\beta)p(\nu)\text{Tr}[\mathsf{C}_{b|\beta}Q_{\mu}^{*}]p(g|\mu,\beta,\nu)\delta_{g,b}$.
Notice that this expression is similar to $p(\\{\mathcal{X}_{\beta}\\})$.
However, it is not in the most general form as $Q_{\mu}^{*}$ does not depend
on $\nu$. Hence, the sum is never larger than the optimal success probability
in QCD game scenario played with access to single testers,
$p(\\{\mathcal{X}_{\beta}\\})$. Therefore, we have

$\displaystyle[1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})]p(\\{\mathcal{X}_{\beta}\\})\geq\sum_{\mu,g,a,b,\alpha,\beta,\nu}w(b,\beta)p(\nu)p(\alpha|\beta,\nu)\text{Tr}[\mathsf{C}_{b|\beta}\mathsf{T}_{a|\alpha}]p(g|a,\beta,\nu)\delta_{g,b}.$
(53)

The above inequality is valid for all the probabilities denoted as
$p(\nu),p(\alpha|\beta,\nu),p(g|a,\beta,\nu)$. Therefore, the inequality will
remain valid when we take maximization over all such probabilities. With this
maximization, R.H.S. of Ineq.(53) is equal to the optimal success probability
in QCD game with the incompatible testers,
$p(\\{\mathcal{X}_{\beta}\\},\\{\mathsf{T}_{\alpha}\\})$. Eventually, we are
left with the following bound,

$1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})\geq\max_{\\{\mathcal{X}_{\beta}\\}}\frac{p(\\{\mathcal{X}_{\beta}\\},\\{\mathsf{T}_{\alpha}\\})}{p(\\{\mathcal{X}_{\beta}\\})}.$
(54)

Hence, we find that $1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})$ is always
greater than or equal to the maximal relative performance between the two
strategies, while maximised over all possible ensembles.

###  A.4 Dual SDP

The Lagrangian associated with our primal SDP is the following.

$\displaystyle\mathcal{L}=$ $\displaystyle s+\sum_{a,\alpha}{\rm
Tr}\Big{[}\omega_{a\alpha}\left(\mathsf{T}_{a|\alpha}-\sum_{\vec{a}}d_{\vec{a}}(a|\alpha)\tilde{\mathsf{Q}}_{\vec{a}}\right)\Big{]}-{\rm
Tr}\Big{[}X\left(\mathds{1}\otimes\tilde{\Theta}^{(n)}-\sum_{\vec{a}}\tilde{\mathsf{Q}}_{\vec{a}}\right)\Big{]}$
$\displaystyle-{\rm
Tr}\sum_{\vec{a}}x_{\vec{a}}\tilde{\mathsf{Q}}_{\vec{a}}-\text{Tr}[y^{(k)}(\mathds{1}_{(2k-3)}\otimes\tilde{\Theta}^{(k-1)}-\text{Tr}_{(2k-2)}[\tilde{\Theta}^{(k)}])]\Big{)}.$
(55)

where we have introduced the associated dual variables. By regrouping, we can
write

$\displaystyle\mathcal{L}=$ $\displaystyle
s(1-\text{Tr}[X(\mathds{1}\otimes\Theta^{(n)})]-\sum_{k=2}^{n}\text{Tr}[y^{(k)}(\mathds{1}_{(2k-3)}\otimes\Theta^{(k-1)}-\mathds{1}_{(2k-2)}\otimes\Theta^{(k)})])$
(56) $\displaystyle+{\rm
Tr}\sum_{a,\alpha}\omega_{a\alpha}\mathsf{T}_{a|\alpha}+{\rm
Tr}\sum_{\vec{a}}\Big{[}X-\sum_{a,\alpha}\omega_{a\alpha}d_{\vec{a}}(a|\alpha)-x_{\vec{a}}\Big{]}\tilde{\mathsf{Q}}_{\vec{a}}.$
(57)

The Lagrangian is made independent of the primal variables when
$\text{Tr}[X(\mathds{1}\otimes\Theta^{(n)})]+\sum_{k=2}^{n}\text{Tr}[y^{(k)}(\mathds{1}_{(2k-3)}\otimes\Theta^{(k-1)}-\mathds{1}_{(2k-2)}\otimes\Theta^{(k)})]=1$,
$X=\sum_{a,\alpha}\omega_{a\alpha}d_{\vec{a}}(a|\alpha)+x_{\vec{a}}$
$\forall\vec{a}$. This enables us to write down the dual SDP as,

$\displaystyle 1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})=$
$\displaystyle\max_{y,X,\\{\omega_{a\alpha}\\}}{\rm
Tr}\left[\sum_{a,\alpha}\omega_{a\alpha}\mathsf{T}_{a|\alpha}\right]$ (58)
$\displaystyle\text{s.t.}\>\>X\geq\sum_{a,\alpha}\omega_{a\alpha}d_{\vec{a}}(a|\alpha),\>\>\omega_{a\alpha}\geq
0$
$\displaystyle\text{Tr}[X(\mathds{1}\otimes\Theta^{(n)})]+\sum_{k=2}^{n}\text{Tr}[y^{(k)}(\mathds{1}_{(2k-3)}\otimes\Theta^{(k-1)}-\mathds{1}_{(2k-2)}\otimes\Theta^{(k)})]=1.$
(59)

We can find solutions for $y^{(k)}$s for the relation in Eq. (59) those result
in satisfying $\text{Tr}[X(\mathds{1}\otimes\Theta^{(n)})]\leq 1$. This is
always satisfied if we write $\text{Tr}[X]\leq 1/\bar{D}$, where
$D=\prod_{j=0}^{n}d_{2n-(2j+1)}$ and $d_{x}$ is the dimension of the system
labeled “$x$”. Therefore, the final dual SDP is of the form,

$\displaystyle 1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})=$
$\displaystyle\max_{X,\\{\omega_{a\alpha}\\}}{\rm
Tr}\left[\sum_{a,\alpha}\omega_{a\alpha}\mathsf{T}_{a|\alpha}\right]$ (60)
$\displaystyle\text{s.t.}\>\>X\geq\sum_{a,\alpha}\omega_{a\alpha}d_{\vec{a}}(a|\alpha)$
$\displaystyle\omega_{a\alpha}\geq 0,\>\>\text{Tr}[X]\leq 1/\bar{D}.$

Equivalence between the primal and dual happens if strong duality exists,
i.e., a strictly feasible solution exists for a dual problem. One such
solution is $X=(\mathds{1}_{2n-1}\otimes\mathds{1})/(D\bar{D})$, and
$\omega_{a\alpha}=\gamma\mathds{1}_{2n-1}\otimes\mathds{1}$ for $\gamma$ such
that $1/(mD\bar{D})>\gamma>0$ for $m$ $\in\mathds{R}^{+}$, where
$\bar{D}=\prod_{j=1}^{2n}d_{2n-j}$.

###  A.5 Upper bound

Consider that the optimal dual variables $y^{*}$, and $\omega_{a\alpha}^{*}$,
then we have

$\displaystyle 1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})=$ $\displaystyle{\rm
Tr}\left[\sum_{a,\alpha}\omega_{a\alpha}^{*}\mathsf{T}_{a|\alpha}\right]$ (61)
$\displaystyle\text{s.t.}\>\>X^{*}\geq\sum_{a,\alpha}\omega_{a\alpha}^{*}d_{\vec{a}}(a|\alpha)$
$\displaystyle\omega^{*}_{a\alpha}\geq 0,\>\>\text{Tr}[X^{*}]\leq 1/\bar{D}.$

Let us introduce three auxiliary variables to have a connection with QCD game,

$\displaystyle\ell^{*}={\rm
Tr}\left[\sum_{a,\alpha}\omega_{a\alpha}^{*}\right],\>\>q^{*}(a,\alpha)=\frac{{\rm
Tr}\left[\omega_{a\alpha}^{*}\right]}{\ell^{*}}\>\>{\rm
and}\>\>\mathsf{C}_{a|\alpha}^{*}=\frac{\omega_{a\alpha}^{*}}{\ell^{*}q^{*}(a,\alpha)},$
(62)

where $\\{q^{*}(a,\alpha)\\}$ is normalized probability distributions and
$\ell^{*}$ is introduced such that ${\rm Tr}[\mathsf{C}_{a|\alpha}^{*}]=D$.
Then, Eq.(61), can be rewritten as

$\displaystyle
1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})=\ell^{*}\sum_{a,\alpha}q^{*}(a,\alpha){\rm
Tr}[\mathsf{C}_{a|\alpha}^{*}\mathsf{T}_{a|\alpha}]$ (63)

Let us assume that Bob will play the QCD game with following strategy. After
receiving the ensemble of channels $\\{\mathcal{X}^{*}_{\beta}\\}_{\beta}$
with
$\mathcal{X}_{\beta}^{*}=\\{\mathcal{C}_{b|\beta}^{*},w^{*}(b|\beta)\\}_{b}$,
he plays the game with $p(v)=\delta_{v,0}$ (i.e., sets $v=0$),
$p(\alpha|\beta,v=0)=\delta_{\alpha,\beta}$ (i.e., applies $T_{\beta}$ given
$\beta$) and $p(g|a,\beta,v=0)=\delta_{g,b}$ (i.e., guesses $b=a=g$ after
getting outcome $a$). Choosing this sub-optimal strategies, one gets,

$\displaystyle p(\\{\mathcal{X}^{*}_{\beta}\\},\\{\mathsf{T}_{\alpha}\\})$
$\displaystyle\geq\sum
q^{*}(b,\beta)\delta_{\lambda,0}\delta_{\alpha,\beta}{\rm
Tr}[\mathsf{C}_{b|\beta}^{*}\mathsf{T}_{a|\alpha}]\delta_{g,b}\delta_{a,g},$
$\displaystyle=\sum_{a,\alpha}q^{*}(a,\alpha){\rm
Tr}[\mathsf{C}_{a|\alpha}^{*}\mathsf{T}_{a|\alpha}]$
$\displaystyle=(1/\ell^{*})[1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})].$ (64)

Now, from the constraint of above SDP, we find following relation for the
current game that

$\displaystyle
X^{*}\geq\sum_{b,\beta}d_{\vec{b}}(b|\beta)\ell^{*}q^{*}(b,\beta)\mathsf{C}_{b|\beta}^{*}.$
(65)

Then, multiplying both sides by an arbitrary Tester $\tilde{Q}_{\vec{b}}$,
summing over $\vec{b}$ and taking trace, we get

$\displaystyle{\rm
Tr}\left[X^{*}\sum_{\vec{b}}\tilde{Q}_{\vec{b}}\right]\geq\sum_{b,\beta,\vec{b}}d_{\vec{b}}(g|\beta)\delta_{b,g}\ell^{*}q^{*}(b,\beta){\rm
Tr}\left[\mathsf{C}_{b|\beta}^{*}\tilde{Q}_{\vec{b}}\right].$ (66)

Notice that due to normalization of an $n$-slot tester, the lhs is exactly
equal to unity. Also, the above equation should hold if we maximize over all
$\tilde{Q}_{\vec{b}}$. This implies that

$\displaystyle\frac{1}{\ell^{*}}\geq\max_{\tilde{Q}_{\vec{b}}}\sum_{b,\beta,\vec{b}}q^{*}(b,\beta){\rm
Tr}\left[\mathsf{C}_{b|\beta}^{*}\tilde{Q}_{\vec{b}}\right]d_{\vec{b}}(g|\beta)\delta_{b,g}.$
(67)

One can see the resemblance of rhs of above equation with the success
probability in QCD game with only single measurement. In particular, we prove
below in (A.5.1) that $(1/\ell^{*})\geq p(\\{\mathcal{X}_{\beta}^{*}\\})$.
Therefore, it implies

$1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})\leq\frac{p(\\{\mathcal{X}_{\beta}^{*}\\},\\{\mathsf{T}_{\alpha}\\})}{p(\\{\mathcal{X}_{\beta}^{*}\\})}.$
(68)

Hence, we arrive at the concluding from both upper and lower bound of
$1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\})$ that

$\max_{\\{\mathcal{X}_{\beta}\\}}\frac{p(\\{\mathcal{X}_{\beta}\\},\\{\mathsf{T}_{\alpha}\\})}{p(\\{\mathcal{X}_{\beta}\\})}=1+\mathcal{R}(\\{\mathsf{T}_{\alpha}\\}).$
(69)

####  A.5.1 Proof of $(1/\ell^{*})\geq p(\\{\mathcal{X}_{\beta}^{*}\\})$

Let us recall the success probability defined below,

$\displaystyle
p(\\{\mathcal{X}_{\beta}\\})=\max_{\mathcal{S}_{c}}\sum_{b,\beta,a,g,v}w(b,\beta)p(v)\text{Tr}[\mathsf{C}_{b|\beta}\mathsf{Q}_{a|v}]p(g|a,\beta,v)\delta_{g,b}.$
(70)

We already know that it is possible to write
$p(g|a,\beta,v)=\sum_{\vec{b}}p(\vec{b}|a,v)d_{\vec{b}}(g|\beta)$, then we can
rewrite the below expression as

$\displaystyle\sum_{b,\beta,a,g,v}w(b,\beta)p(v)\text{Tr}[\mathsf{C}_{b|\beta}\mathsf{Q}_{a|v}]p(g|a,\beta,v)\delta_{g,b}=\sum_{b,\beta,\vec{b}}w(b,\beta)\text{Tr}[\mathsf{C}_{b|\beta}\tilde{Q}_{\vec{b}}]d_{\vec{b}}(g|\beta)\delta_{g,b},$
(71)

where we introduce a new object defined as

$\displaystyle\tilde{Q}_{\vec{b}}=\sum_{a,v}p(v)\mathsf{Q}_{a|v}p(\vec{b}|a,v).$
(72)

Notice that for all values of $\vec{b}$, $\tilde{Q}_{\vec{b}}\geq 0$ and also
satisfies following

$\displaystyle\sum_{\vec{b}}\tilde{Q}_{\vec{b}}=\sum_{a,v,\vec{b}}p(v)\mathsf{Q}_{a|v}p(\vec{b}|a,v)=\sum_{a,v}p(v)\mathsf{Q}_{a|v}\sum_{\vec{b}}p(\vec{b}|a,v)$
$\displaystyle=\sum_{v}p(v)\sum_{a}\mathsf{Q}_{a|v}=\sum_{v}p(v)\mathds{1}_{2n-1}\otimes\Xi^{(n)}_{v}$
$\displaystyle=\mathds{1}_{2n-1}\otimes\Xi^{(n)}$ (73)

It is evident now that this strategy with $\tilde{Q}_{\vec{b}}$ is as good as
the strategy with $\mathsf{Q}_{a|v}$. Therefore, we write equivalently,

$\displaystyle
p(\\{\mathcal{X}_{\beta}\\}):=\max_{\tilde{Q}_{\vec{b}}}\sum_{b,\beta,\vec{b}}w(b,\beta){\rm
Tr}\left[\mathsf{C}_{b|\beta}\tilde{Q}_{\vec{b}}\right]d_{\vec{b}}(g|\beta)\delta_{b,g}.$
(74)

Using this in Eq. (67), we arrive at the conclusion.

##  Appendix B Proof concept for convex weight

It is easy to see that we can convert the Eq. (14) to an SDP. Using the
definition of convex weight of tester assemblage
$\mathcal{T}_{\alpha}:=\\{\mathsf{T}_{a|\alpha}\\}$, we get

$\displaystyle\mathsf{T}_{a|\alpha}-\mathcal{W}(\mathcal{T}_{\alpha})N_{a|\alpha}$
$\displaystyle=[1-\mathcal{W}(\mathcal{T}_{\alpha})]\sum_{\mu}p(a|\alpha,\mu)\mathsf{Q}_{\mu},$
$\displaystyle=\bar{\mathsf{Q}}_{a|\alpha},$ (75)

where we have identified
$[1-\mathcal{W}(\mathcal{T}_{\alpha})]\sum_{\mu}p(a|\alpha,\mu)\mathsf{Q}_{\mu}:=\bar{\mathsf{Q}}_{a|\alpha}$.
Now considering $s_{w}=1-\mathcal{W}(\mathcal{T}_{\alpha})$, we reach to an
equivalent SDP formulation below for convex weight as

$\displaystyle 1-\mathcal{W}(\mathcal{T}_{\alpha})$ $\displaystyle=\max\quad
s_{w}$ (76) $\displaystyle{\rm
s.t.}\quad\frac{1}{s_{w}}\left(\mathsf{T}_{a|\alpha}-[1-s_{w}]N_{a|\alpha}\right)=\sum_{\mu}p(a|\alpha,\mu)\mathsf{Q}_{\mu}.$
(77)

Given this formulation, we can reach to the nicer form of SDP by similar logic
from Appendix A and using the fact that
$[1/(1-s_{w})]\bar{\mathsf{Q}}_{a|\alpha}$ are free testers. Therefore, the
prove of Eq. (18) will follow similar to Appendix A.