# Observable-enriched entanglement

Joe H. Winter  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer
Straße 40, 01187 Dresden, Germany  Max Planck Institute for the Physics of
Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany  SUPA, School of
Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews
KY16 9SS, UK  Reyhan Ay  Max Planck Institute for Chemical Physics of Solids,
Nöthnitzer Straße 40, 01187 Dresden, Germany  Max Planck Institute for the
Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
Izmir Institute of Technology, Gülbahçe Kampüsü, 35430 Urla Izmir, Türkiye
Bernd Braunecker  SUPA, School of Physics and Astronomy, University of St.
Andrews, North Haugh, St. Andrews KY16 9SS, UK  A. M. Cook  Max Planck
Institute for Chemical Physics of Solids, Nöthnitzer Straße 40, 01187 Dresden,
Germany  Max Planck Institute for the Physics of Complex Systems, Nöthnitzer
Straße 38, 01187 Dresden, Germany

###### Abstract

We introduce methods of characterizing entanglement, in which entanglement
measures are enriched by the matrix representations of operators for
observables. These observable operator matrix representations can enrich the
partial trace over subsets of a system’s degrees of freedom, yielding reduced
density matrices useful in computing various measures of entanglement, which
also preserve the observable expectation value. We focus here on applying
these methods to compute observable-enriched entanglement spectra, unveiling
new bulk-boundary correspondences of canonical four-band models for
topological skyrmion phases and their connection to simpler forms of bulk-
boundary correspondence. Given the fundamental roles entanglement signatures
and observables play in study of quantum many body systems, observable-
enriched entanglement is broadly applicable to myriad problems of quantum
mechanics.

Entanglement is essential in characterising quantum many body systems given
its role in quantum information theory [1, 2, 3], with various measures of
entanglement applied for characterising topological states [4, 5, 6], quantum
critical phenomena [7, 8, 9], phase transitions [10, 11, 12], and dynamics
[13, 14, 15]. Integral to entanglement characterisation is the partial trace
operation [16, 17, 18]: considerable information about quantum systems derives
from partial trace applied to the density matrix over myriad subsets of
Hilbert space [19, 20, 21, 22].

Such entanglement measures provide information about the connection between
the Hilbert spaces of physical subsystems that goes beyond the decomposition
into joint basis states. Given the on-going research into the meaning of these
measures, we show that similar information can be encoded in observable
representations. To this end, we introduce a generalized, observable enriched
(OE) partial trace (OEPT) $\tilde{\mathrm{Tr}}$ with respect to an observable
$S$. When applied on the full density matrix $\rho$, the OEPT produces an
auxiliary density matrix $\rho_{s}=\tilde{\mathrm{Tr}}[\rho]$, which is
constructed entirely from $\langle S\rangle$ and captures all entanglement and
topological information by the requirement that $\text{Tr}[\rho
S]=\text{Tr}[\rho_{s}\tilde{S}]$, where Tr is the trace operation. That is,
the expectation value of the observable is preserved by
$\tilde{\text{T}}\text{r}$.

We demonstrate the power of our methods by studying topological Skyrmion
phases of matter [23], which are lattice counterparts of the quantum skyrmion
Hall effect [24]. Our methods reveal essential features of these topological
states, which generalise those [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38] within the framework of the quantum Hall effect (QHE) [39, 40].
Notably, we find first evidence of the generalised bulk-boundary
correspondence of topological Skyrmion states [24] in the particularly simple
four-band Hamiltonians [41] capturing this physics, and we utilise OE
entanglement to reveal it as a generalisation of bulk-boundary correspondence
in the QHE framework.

Topological Skyrmion phases of matter—We first briefly introduce key concepts
of topological Skyrmion phases of matter and the quantum Skyrmion Hall effect
to later introduce our methods. The first known topological Skyrmion phases
are $2+1$-dimensional topological states possible in effectively non-
interacting systems [23, 41], which are characterized by the topological
invariant $\mathcal{Q}$, the topological charge of winding in the spin texture
of occupied states over the Brillouin zone defined by momentum components
$k_{x}$ and $k_{y}$, or

$\mathcal{Q}={1\over
4\pi}\int_{BZ}d\boldsymbol{k}\left[\langle\boldsymbol{\hat{S}}(\boldsymbol{k})\rangle\cdot\left(\partial_{k_{x}}\langle\boldsymbol{\hat{S}}(\boldsymbol{k})\rangle\times\partial_{k_{y}}\langle\boldsymbol{\hat{S}}(\boldsymbol{k})\rangle\right)\right],$
(1)

where $\boldsymbol{S}=\left(S_{1},S_{2},S_{3}\right)$ is the spin
representation and
$\langle\boldsymbol{\hat{S}}(\boldsymbol{k})\rangle=\langle\boldsymbol{S}(\boldsymbol{k})\rangle/|\boldsymbol{S}|$
is the normalized expectation value of the spin for occupied states. Here
$\langle S_{i}(\boldsymbol{k})\rangle=\sum_{n\in\mathrm{occ}}\langle
n,\boldsymbol{k}|S_{i}|n,\boldsymbol{k}\rangle$, with $i\in\\{1,2,3\\}$ and
$|n,\boldsymbol{k}\rangle$ the Bloch state associated with the $n$th band. For
systems with only a spin degree of freedom (DoF), $\mathcal{Q}$ is the total
Chern number, but decouples from the total Chern number in systems with
multiple DoFs, to characterize a topological state distinct from the Chern
insulator [23, 41, 42].

Defining the full Hamiltonian of the system as $\mathcal{H}$, where
$\mathcal{H}=\sum_{\boldsymbol{k}}\Psi^{\dagger}_{\boldsymbol{k},\alpha,\beta}H(\boldsymbol{k})\Psi_{\boldsymbol{k},\alpha,\beta}$
in terms of Bloch Hamiltonian $H(\boldsymbol{k})$, we choose the basis
$\Psi_{\boldsymbol{k}}=(c_{\boldsymbol{k},+}c_{\boldsymbol{k},-},c^{\dagger}_{\boldsymbol{k},-},c^{\dagger}_{\boldsymbol{k},+})^{T}$,
where $c_{\boldsymbol{k},\alpha}$ annihilates a fermion with momentum
$\boldsymbol{k}$ and $\alpha\in\\{+,-\\}$ defines a two-fold (pseudo)spin DoF.
For the purposes of this discussion, it is sufficient to consider $\alpha$ as
a spin-$1/2$ DoF.

The Bloch Hamiltonian $H(\boldsymbol{k})$ is then a generalized Bogoliubov de
Gennes (BdG) Hamiltonian, consisting of a generic two band normal state
Hamiltonian $H_{N}(\boldsymbol{k})$ and pairing term $\Delta(\boldsymbol{k})$
as

$H_{BdG}(\boldsymbol{k})=\begin{bmatrix}H_{N}(\boldsymbol{k})&\Delta(\boldsymbol{k})\\\
\Delta^{\dagger}(\boldsymbol{k})&-H^{T}_{N}(\boldsymbol{k})\end{bmatrix}.$ (2)

We consider
$H_{N}(\boldsymbol{k})=h_{0}(\boldsymbol{k})\mathbf{I}+\boldsymbol{h}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$
and
$\Delta(\boldsymbol{k})=i\Delta_{0}(d_{0}(\boldsymbol{k})+\boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma})\sigma_{y}$.
In these expressions, $\Delta_{0}$ represents a constant;
$h_{0}(\boldsymbol{k})$ and $d_{0}(\boldsymbol{k})$ are real scalar functions;
$\boldsymbol{h}(\boldsymbol{k})$ and $\boldsymbol{d}(\boldsymbol{k})$ are real
vector functions and finally,
$\boldsymbol{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$, where
$\sigma_{\mu}$ is the $\mu$th Pauli matrix.

Observable enriched partial trace— We now introduce the observable enriched
partial trace (OEPT) by mapping the ground state (GS) of $H(k)$, represented
by the density matrix $\rho_{GS}$, onto an auxiliary system that reproduces
the same expectation values of $S$. To this end we define a two-level density
matrix $\rho_{s}$, for some state of an auxiliary system with Bloch
Hamiltonian $h(\boldsymbol{k})$ and basis
$\psi_{\boldsymbol{k}}=(c_{\boldsymbol{k},+},c_{\boldsymbol{k},-})^{T}$. That
is, $\psi$ possesses only the spin-$1/2$ DoF for each momentum
$\boldsymbol{k}$. Subsequently, we may enforce the following relation

$\text{Tr}[\rho_{\mathrm{GS}}(\boldsymbol{k})S_{\mu}]=\text{Tr}[\rho_{s}(\boldsymbol{k})\sigma_{\mu}],$
(3)

which implicitly defines a $\rho_{s}(\boldsymbol{k})$ yielding the same spin
expectation value as $\rho_{\mathrm{GS}}(\boldsymbol{k})$, despite their
different corresponding spin representations.

Using $SU(2)$ commutation relations and trace properties, eq. 3 yields
$\rho_{s}\left(\boldsymbol{k}\right)$ of the form:

$\rho_{s}(\boldsymbol{k})=\left(\mathbf{I}_{2}+\braket{\boldsymbol{S}(\boldsymbol{k})}\cdot\boldsymbol{\sigma}\right)/2,$
(4)

where $\mathbf{I}_{2}$ is the $2\times 2$ identity matrix.

Now consider the existence of a unitary transformation which maps spin
representation $\\{S^{\mu}\\}_{\mu\in 1,2,3}$ onto
$\\{\mathbf{I}_{2}\otimes\sigma_{\mu}\\}_{\mu\in x,y,z}$. Then the spin, $S$,
is completely decoupled from the non-spin DoFs, $\overline{S}$, and
consequently the map from $\rho_{GS}\to\rho_{s}$ reduces to performing a
partial trace over the non-spin degrees of freedom, $\bar{S}$, as:

$\hat{O}_{S}=\mathbf{I}_{\overline{S}}\otimes\hat{o}_{S}\implies\text{Tr}\left[\text{Tr}_{\overline{S}}[\rho]\
\hat{o}_{S}\right]=\text{Tr}[\rho\hat{O}_{S}].$ (5)

We interpret our system as consisting of a spin DoF coupled to a bath of the
non-spin DoFs which are traced out during the OEPT. The
$\rho_{s}\left(\boldsymbol{k}\right)$ is then interpreted as a reduced density
operator useful in computing various entanglement properties [16, 17, 18].
Notably, even if the spin and particle-hole DoFs are not separable,
$\rho_{s}(\boldsymbol{k})$ remains a valid density matrix by construction and
faithfully reproduces the GS spin expectation values over the Brillouin zone
(BZ). This exemplifies the utility of this technique.

To proceed, we note that the class of Hamiltonians is
$\mathscr{C}^{\prime}$-symmetric, which is a generalised charge conjugation
symmetry defined by
$\mathscr{C}^{\prime-1}H(\boldsymbol{k})\mathscr{C}^{\prime}=-H(\boldsymbol{k})^{T}$.
Furthermore, the symmetry suggests a spin representation of the form
$S_{\mu}=\mathrm{diag}\left(\sigma_{\mu},-\sigma^{*}_{\mu}\right)$. As
$\sigma^{\dagger}_{y}\sigma_{\mu}\sigma_{y}=-\sigma^{*}_{\mu}$, we may define
a unitary operator $U=\mathbf{I}\oplus\sigma_{y}$ to rotate $S_{\mu}$ to
$\mathbf{I}\otimes\sigma_{\mu}$ for each $\mu$. Thus, for this class of
Hamiltonians, the spin Hilbert space is separable, and the effective spin GS
map is:

$\rho_{s}=\tilde{\text{Tr}}[\rho_{GS}]=\text{Tr}_{\bar{S}}[U^{\dagger}\rho_{GS}U].$
(6)

Here, $\bar{S}$ denotes the non-spin degrees of freedom following the basis
transformation achieved with $U$ rather than representing the actual non-spin
degrees of freedom.

Figure 1: Comparison of GS spin expectation value
$\langle\boldsymbol{S}\rangle$ textures over the bulk Brillouin zone for
$\boldsymbol{h}=\boldsymbol{h}_{QWZ}$ in panels a), b) and
$\boldsymbol{h}=\boldsymbol{h}_{stic}$ in c), d). Panels a),c) show
$\langle\boldsymbol{S}\rangle$ computed from $\rho_{GS}$; panels b),d) the
corresponding $\langle\boldsymbol{S}\rangle$ obrained from $\rho_{s}$. Arrow
direction indicates the expectation values in the $S_{x},S_{y}$ plane; the
colour map represents the magnitude $S_{z}$. The arrow length is proportional
to $|\langle\boldsymbol{S}\rangle|$

.

This class Hamiltonians, with real $\boldsymbol{d}$ vector, commute with
operator $\tau_{y}\otimes\sigma_{y}$, where $\tau_{\mu}$ are the Pauli
matrices in the particle-hole Hilbert space [43]. They therefore block-
diagonalize to the form:

$H_{BdG}(\boldsymbol{k})=\begin{bmatrix}\left[\boldsymbol{h}(\boldsymbol{k})+\boldsymbol{d}(\boldsymbol{k})\right]\cdot\boldsymbol{\sigma}&0\\\
0&\left[\boldsymbol{h}(\boldsymbol{k})-\boldsymbol{d}(\boldsymbol{k})\right]\cdot\boldsymbol{\sigma}\end{bmatrix}.$
(7)

We compute the Skyrmion number $\mathcal{Q}$ as in Eq. 1 presented in the SM
[44].

Numerical results—To demonstrate our method through characterization of the
topological skyrmion phases, we first compare the texture over the BZ of the
spin expectation value $\langle\boldsymbol{S}(\boldsymbol{k})\rangle$ of the
GS for Eq. 2, with the skyrmionic texture over the BZ of
$\rho_{s}(\boldsymbol{k})$ as shown in fig. 1 for two different choices of
$\boldsymbol{h}(\boldsymbol{k})$. In fig. 1 a) and fig. 1 b), we take the
normal state Hamiltonian to be that of a QWZ two band Chern insulator [45],
which is a two level system with $\boldsymbol{h}$-vector
$\boldsymbol{h}_{QWZ}\left(\boldsymbol{k}\right)=(\beta\sin(k_{x}),\beta\sin(k_{y}),\mu-
t_{q}\cos(k_{x})-t_{q}\cos(k_{y}))$. Here, $\mu$ defines a staggered onsite
potential; the $t_{q}$ are nearest-neighbour hopping integrals; and $\beta$ a
pseudo-spin orbit coupling strength. In fig. 1 c) and fig. 1 d), we instead
take the $\boldsymbol{h}$-vector to be that of Sticlet  _et al._ [46], or
$\boldsymbol{h}_{stic}\left(\boldsymbol{k}\right)=(\alpha\cos(k_{x}),\alpha\cos(k_{y}),t_{s}\cos(k_{x}+k_{y}))$.
Here, $t_{s}$ denotes a diagonal hopping integral over the square lattice; and
the $\alpha$ another pseudo-spin orbit coupling.

Fig. 1 confirms that the skyrmionic textures computed with the full (panels a)
and c) for the two models) and with the auxiliary spin systems (corresponding
panels b) and d), respectively) are identical. By construction, this
equivalence is expected, but the data of fig. 1 serves as the basis for our
further analysis.

Observable-enriched entanglement spectrum—Even more intriguingly, the
auxiliary system produces further more intricate features, including an
additional bulk-boundary correspondence arising from the spin topology. To
begin the analysis, we first perform a Fourier transform on the auxiliary spin
system while also noting that the spin operators are mutually diagonal in
momenta and in position.

$\rho_{s}(r-r^{\prime})=\sum_{r,r^{\prime}}\frac{1}{2}(\mathbf{I}_{2}+\text{Tr}[\rho_{r,r^{\prime}}\boldsymbol{S}]\cdot\boldsymbol{\sigma})\
\ket{r}\bra{r^{\prime}}.$ (8)

Here, $\rho_{r,r^{\prime}}$ are the matrix elements
$\braket{r}{\rho}{r^{\prime}}$ with $r,r^{\prime}$ denoting real space
coordinates. Inspecting the Fourier transform, it is clear that the procedure
to reduce to the auxiliary system is to perform the standard $N\times N\to
2\times 2$ reduction to each $r,r^{\prime}$ block within the GS projector.

We begin by partitioning our system in the $\hat{x}$-direction, while keeping
$k_{y}$ as a good quantum number, into subsystem $A$ and subsystem $B$. We
then choose the partition such that subsystem $A$ includes layers of index
from $1$ to $\frac{N_{x}}{2}$, and subsystem $B$ includes layers of index from
$\frac{N_{x}}{2}+1$ up to $N_{x}$. The partial trace over the many body
groundstate corresponds to calculating the equal time one body correlators by
the method of [47]—an approach that, in the context of free fermion systems,
reduces to projecting from the GS onto the $A$ subsystem [19]. We then perform
an additional OEPT over $\bar{S}$ DoFs. We present the slab energy
eigenspectrum, $E$; eigenspectrum, $\xi$, of the system $A$ density matrix
known as entanglement spectrum and eigenspectrum of the system $A$ density
matrix after OEPT, $\xi_{S}$, which we denote Observable Enriched Entanglement
Spectrum OEES. We compute these quantities for each of
$\boldsymbol{h}_{QWZ}(\boldsymbol{k})$ and
$\boldsymbol{h}_{stic}(\boldsymbol{k})$, respectively, for representative
parameter sets in fig. 2.

Figure 2: Slab energy spectra (a, b), entanglement spectra (c, d), and
observable-enriched entanglement spectra (e, f) are shown for
$\boldsymbol{h}_{QWZ}(\boldsymbol{k})$ (a,c,e) and
$\boldsymbol{h}_{stic}(\boldsymbol{k})$ (b,d,f), respectively, for OBC in the
$\hat{x}$-direction. Parameters of $\boldsymbol{h}_{QWZ}(\boldsymbol{k})$ are
$\mu/t_{q}=0.5,\ \Delta_{0}=1,\ \beta=1$, corresponding to $\mathcal{C}=2$ and
$\mathcal{Q}=-1$. Parameters of $\boldsymbol{h}_{stic}(\boldsymbol{k})$ are
$\alpha/t_{S}=1\ \Delta_{0}=0.1$, corresponding to $\mathcal{C}=-4$ and
$\mathcal{Q}=2$. Dashed lines in a), b), c), and d) indicate two-fold
degeneracy.

For half-filling and $\mu/t_{q}=0.5,\ \Delta_{0}=1,\ \beta=1$, the total Chern
number $\mathcal{C}$ and Skyrmion number $\mathcal{Q}$ of
$\boldsymbol{h}_{QWZ}(\boldsymbol{k})$ are $2$ and $-1$, respectively. We find
the slab spectrum of $\boldsymbol{h}_{QWZ}(\boldsymbol{k})$ depicted in fig. 2
a) exhibits two-fold degenerate chiral modes on each edge, as expected from
$\mathcal{C}$. The corresponding entanglement spectrum of the ground state
after tracing out system $B$ shown in fig. 2 c) also exhibits two-fold
degenerate chiral modes as expected: chiral mode(s) in $\xi$, which connect
bands at $0$ and $1$ when tuning $k_{y}$, are signature(s) of non-trivial
topology [6]. However, the OEES shown in fig. 2 e) also depicts a single
chiral mode in correspondence with $\mathcal{Q}$.

For half-filling and $\alpha/t_{S}=1\ \Delta_{0}=0.1$, the $\mathcal{C}$ and
$\mathcal{Q}$ of $\boldsymbol{h}_{stic}(\boldsymbol{k})$ are instead -4 and 2,
respectively. We find the slab spectrum of
$\boldsymbol{h}_{stic}(\boldsymbol{k})$ depicted in fig. 2 b) exhibits a pair
of two-fold degenerate chiral modes on each edge, totaling to four chiral
modes per edge, as expected from $\mathcal{C}$. The corresponding entanglement
spectrum of the GS after tracing out system $B$ shown in fig. 2 d) also
exhibits a pair of two-fold degenerate chiral modes as expected. The OEES,
however, shown in fig. 2 f), instead exhibits $\mathcal{Q}$—rather than
$\mathcal{C}$—chiral mode(s).

Bulk-boundary correspondence for topological skyrmion phase with zero Chern
number— The power of OEPT, however, does not lie in reproducing such results
with less DoFs. We now show how it can unveil Skyrmion topology where the
conventional entanglement spectrum classification suggests triviality. To this
end, we break the $\tau_{y}\otimes\sigma_{y}$ symmetry by adding constant term
$H^{\prime}=\Delta_{0}\tau_{x}\otimes\mathbf{I}_{2}$ to the Hamiltonian Eq. 2,
making $d_{y}$ complex. Such a term respects $\mathscr{C}^{\prime}$ but, as
shown in the SM [44], allows for further decoupling of $\mathcal{Q}$ and
$\mathcal{C}$ yielding regions of phase space with trivial $\mathcal{C}$ and
non-trivial $\mathcal{Q}$.

Figure 3: a) Slab energy spectrum for OBC in the $\hat{x}$-direction for
$\mathcal{C}=0$, $\mathcal{Q}=-1$, at $m/t_{q}=0.2,\Delta_{0}=1,\beta=1$, b)
corresponding entanglement spectrum for cylindrical geometry, c) corresponding
observable-enriched entanglement spectrum for cylindrical geometry, and d) log
of probability density vs. layer index $x\in A$ for in-gap bands highlighted
in red and yellow, in c). $x=0$ corresponds to the real edge of the system;
$x=100$ corresponds to the virtual edge separating $A$ and $B$. The system is
$200$ sites long.

The slab energy spectrum for Hamiltonian Eq. 2 including the term
$H^{\prime}$, with OBCs in the $\hat{x}$-direction, is shown in fig. 3 a) for
a region of phase space with total Chern number $\mathcal{C}=0$ and Skyrmion
number $\mathcal{Q}=-1$. Although the Chern number is zero and the system is
insulating in the bulk, there are in-gap Bloch-states localized at the edge.
(Localization of these edge states in the slab energy spectrum is shown in the
SM [44].) As shown in fig. 3, these in-gap bands do not extend from the bulk
valence to conduction bands or vice versa, but rather extend from the bulk
valence (conduction) states into the gap, and return to the bulk valence
(conduction) states. However, any value for the Fermi level that lies within
the bulk gap intersects edge bands, due to the overlap of these bands in
energy.

The entanglement spectrum, $\xi$, for a virtual cut in real-space is shown in
fig. 3 b). As expected for trivial total Chern number, this entanglement
spectrum is trivial [48], in contrary to fig. 2, where there is spectral
connectivity between $\xi=0$ and $\xi=1$. However, the OEES, $\xi_{s}$, with a
virtual cut in real-space, shown in fig. 3 c), exhibits a chiral mode in
correspondence with the non-trivial Skyrmion number $\mathcal{Q}$, localized
on the virtual edge as shown in fig. 3 d). Thus, the OEES correctly captures
the non-trivial Skyrmion topology, where the conventional entanglement
spectrum and Bloch topology fails. Furthermore, changing the sign of
$\mathcal{Q}$ changes chirality of the edge state in the OEES [44].

In addition, we observe a further discontinous in-gap state, red in fig. 3,
localized on the real edge of the system, such a feature is absent in ES with
OEPT in full periodic boundary conditions as well as with OEES with a spatial
virtual cut in this geometry, which only has the expected, orange in fig. 3,
chiral modes. Consequently, this implies an additional bulk-boundary
correspondence, independent of the spectral contribution from Bloch states, on
top of the effects seen from purely spatial cuts. This anomalous state also
changes chirality with change in sign of $\mathcal{Q}$, further indicating
that it is a consequence of the non-trivial bulk spin topology.

Supplementary to the OEES, upon opening boundary conditions we see a real
space chiral spin texture on the boundary of the system as shown in fig. 4.
Change in sign of $\mathcal{Q}$ also corresponds to change in handedness of
$\langle S_{z}\rangle$ of the spin texture in the full model, as shown in fig.
4. Such effects show how the non trivial OEES manifests itself on the real
boundary of the system.

Figure 4: Comparison of real-space spin expectation value
$\langle\boldsymbol{S}\rangle$ texture, similar to fig. 1, for Hamiltonian Eq.
S1 with additional $H^{\prime}$ term, for OBCs in each of the $x$\- and
$y$-directions. Left panel: $\mathcal{Q}=1$ and $\mathcal{C}=0$ at
$m/t_{q}=-0.2,\Delta_{0}=1,\beta=1$. Right panel: $\mathcal{Q}=-1$ and
$\mathcal{C}=0$ at $m/t_{q}=0.2,\Delta_{0}=1,\beta=1$

Discussion and conclusion—We have introduced the concept of observable
enriched entanglement. This is based on a generalisation of the partial trace
operation to obtain an entanglement spectrum. The result is an enriched
partial trace which preserves the values of the selected observables. Applied
to the ground state (GS) this produces an auxiliary density matrix,
$\rho_{s}$, with DoF only associated with the observable. Remarkably, we find
that the $\rho_{s}$ reproduces the entanglement spectra corresponding to non-
trivial spin topology of the full GS with non-zero Chern number. In addition,
however, we could reveal that $\rho_{s}$ contains further information on bulk-
boundary correspondence of the observable even for zero Chern number, where
the standard entanglement spectrum is inconclusive. We demonstrated these
features on different models with Chern and Skyrmion numbers in the context of
revealing edge polarisation textures bound to the Skyrmion number. Future work
will more broadly characterise other entanglement measures with established
expressions in terms of reduced density matrices (e.g., von Neumann entropy
[49]), but computed instead from OE reduced density matrices.

###### Acknowledgements.

Acknowledgements—We thank R. F. Calderon, G. F. Lange, S. Liu, M. Pacholski,
A. Pal, and D. Varjas for fruitful discussions. This research was supported in
part by the National Science Foundation under Grants No.NSF PHY-1748958 and
PHY-2309135, and undertaken in part at Aspen Center for Physics, which is
supported by National Science Foundation grant PHY-2210452. The work presented
in this paper is theoretical. No data were produced, and supporting research
data are not required.

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Supplemental material for “Observable Enriched Entanglement”

Supplemental material for “Observable-enriched entanglement’
Joe H. Winter1,2,3, Reyhan Ay1,2,4, Bernd Braunecker3 and Ashley M. Cook1,2,∗

1Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40,
01187 Dresden, Germany
2Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse
38, 01187 Dresden, Germany
3SUPA, School of Physics and Astronomy, University of St. Andrews, North
Haugh, St. Andrews KY16 9SS, UK
4Izmir Institute of Technology, Gülbahçe Kampüsü, 35430 Urla Izmir, Türkiye
∗Electronic address<EMAIL_ADDRESS>
(Dated: )

##  S1 Analytic Calculation of Topological Invariants

In this section, we extend the discussion of the bulk topological invariants
for the Hamiltonians presented in the main text. We begin by utilising the
$\tau_{y}\otimes\sigma_{y}$ symmetry, where $\tau_{\mu}$ are the Pauli
matrices in the particle-hole Hilbert space. This allows reduction of the
Hamiltonian to a simple block diagonalization:

$H_{BdG}(\boldsymbol{k})=\begin{bmatrix}(\boldsymbol{h}+\boldsymbol{d})(\boldsymbol{k})\cdot\boldsymbol{\sigma}&0\\\
0&(\boldsymbol{h}-\boldsymbol{d})(\boldsymbol{k})\cdot\boldsymbol{\sigma}\end{bmatrix}.$
(S1)

Furthermore, and crucially, the spins still remain separable with some
directions modified:

$S_{x}\to\boldsymbol{I}\otimes-\sigma_{x},\
S_{y}\to\boldsymbol{I}\otimes-\sigma_{y},\
S_{z}\to\boldsymbol{I}\otimes\sigma_{z}.$ (S2)

The modification is effectively a change in handedness of the Bloch sphere, so
any winding calculated with the tuple $(-\sigma_{x},-\sigma_{y},\sigma_{z})$
has opposite sign to our original set of Pauli matrices.

With these lemmas, we can now perform a topological band theoretical analysis
of these models. Firstly, we utilise the block diagonal form in eq. S1 to
calculate the ground-state topology of this Hamiltonian. Given each block
possesses two bands and is, individually, $\mathscr{C}^{\prime}$ symmetric,
the total Chern number of the four-band system is the sum of the Chern numbers
of the two-band blocks. This therefore reduces to the sum of Skyrmion numbers
of the vectors $\boldsymbol{h}+\boldsymbol{d},\boldsymbol{h}-\boldsymbol{d}$.

$\mathcal{C}=\mathcal{Q}[\boldsymbol{h}+\boldsymbol{d}]+\mathcal{Q}[\boldsymbol{h}-\boldsymbol{d}],$
(S3)

where $\mathcal{Q}[\boldsymbol{v(k)}]$ denotes the skyrmion number computed as
the winding of vector $\boldsymbol{v}$ in the Brillouin zone. We take the
partial trace in this block-diagonal basis, noting we have a different handed
winding, which gives:

$\rho_{S}=\rho^{+}_{GS}+\rho^{-}_{GS},$ (S4)

where $\rho^{\pm}_{GS}$ denotes the rank $1$ ground-state projector of
$(\boldsymbol{h}\pm\boldsymbol{d})\cdot\sigma$, respectively. This now allows
us to very simply calculate the Skyrmion number for our four-band system,
which is the Chern number/Skyrmion number of this effective two-band system.
As such, taking into account our rotation of spin operators where
$\braket{\sigma_{x,y}}_{S}=-\braket{S_{x,y}}$, we find that the four-band
Skyrmion number is equivelent to :

$\mathcal{Q}_{tot}=-\mathcal{Q}\left[\frac{(\boldsymbol{h}+\boldsymbol{d})}{|\boldsymbol{h}+\boldsymbol{d}|}+\frac{(\boldsymbol{h}-\boldsymbol{d})}{|\boldsymbol{h}-\boldsymbol{d}|}\right].$
(S5)

Now, the integrand for the Skyrmion number is non-linear and, typically, we
cannot write a simple formula in terms of a linear combination of winding
numbers of the individual $\boldsymbol{h}\pm\boldsymbol{d}$ vectors. However,
we are saved by the fact that we are looking a topological quantity. First,
assume we compute the winding number of:

$\mathcal{Q}_{tot}(\alpha)=-\mathcal{Q}\left[\frac{(\boldsymbol{h}+\boldsymbol{d})}{|\boldsymbol{h}+\boldsymbol{d}|}+\alpha\frac{(\boldsymbol{h}-\boldsymbol{d})}{|\boldsymbol{h}-\boldsymbol{d}|}\right],\
\ \alpha\in[0,1].$ (S6)

If we first assume $\alpha=0$, this is simply the winding of the normalised
$\boldsymbol{h}+\boldsymbol{d}$ vector. Now, provided $\alpha\in[0,1)$, the
vector $\boldsymbol{h}-\boldsymbol{d}$ cannot cancel the vector
$\boldsymbol{h}+\boldsymbol{d}$, as its magnitude is always less than one.
Consequently, we remain in the same topological class. Therefore, when
$\alpha=1$, we either remain in the same topological class or the total vector
passes through zero in magnitude and the integral is ill-defined. Therefore,
employing this effective spin ground state, we have proved the statement
previously verified only numerically:

$\mathcal{C}=-2\mathcal{Q}.$ (S7)

$\mathcal{Q}[\boldsymbol{h}+\boldsymbol{d}]=\mathcal{Q}[\boldsymbol{h}-\boldsymbol{d}]$
for the case where we have $\tau_{y}\otimes\sigma_{y}$ symmetry.

##  S2 Phase Diagram for Complex $\boldsymbol{d}$-vector

Here, we present phase diagrams for the Hamiltonian Eq. 2 with additional
$H^{\prime}$ term in fig. S5. As shown in fig. S5 a), regions with non-trivial
total Chern number for half-filling narrow with increasing magnitude of
pairing strength $\Delta_{0}$, while regions with non-trivial skyrmion number
are independent of $\Delta_{0}$ as shown in fig. S5 b). As result, finite
$\Delta_{0}$ yields regions of phase space with trivial total Chern number and
non-trivial skyrmion number. In addition, type-II topological phase
transitions are realized for $\mu=0$ and $\pm 2$, across which the skyrmion
number changes from one integer value to another without the closing of the
minimum direct bulk energy gap. Instead, the skyrmion number changes as the
minimum magnitude of the ground state spin expectation value over the
Brillouin zone becomes zero.

Figure S5: Topological phase diagrams for Hamiltonian Eq. 3 with additional
$H^{\prime}$ term corresponding to results for Figs. 3 and 4 in the main text:
a) total Chern number $\mathcal{C}$ and b) skyrmion number $\mathcal{Q}$, for
half-filling, vs. normal state mass parameter $\mu/t_{q}$ and pairing strength
$\Delta_{0}$ with pseudo-spin orbit coupling $\beta=1$.

##  S3 OBC Characterisation for Complex $\boldsymbol{d}$-vector with
$\mathcal{Q}=+1$

Figure S6: Probability density vs. layer index for the two in-gap edge states
shown in fig. 3 a) (highlighted in red) at a) $k_{y}=-\pi/2$ and b)
$k_{y}=\pi/2$, respectively. Figure S7: a) Slab energy spectrum for OBC in the
$\hat{x}$-direction for $\mathcal{Q}=1$, $\mathcal{C}=0$ and parameter set
$\mu/t_{q}=-0.2,\Delta_{0}=1,\beta=1$, b) corresponding entanglement spectrum
for cylindrical geometry and c) observable-enriched entanglement spectrum for
cylindrical geometry, and d) log of probability density vs. layer index $x$
for in-gap bands highlighted in red and yellow, in c).

In fig. S7, we present the complement to fig. 3 but for $\mathcal{Q}=1$ and
$\mathcal{C}=0$, and parameter set $\mu/t_{q}=-0.2,\Delta_{0}=1,\beta=1$. We
see that there is a difference in chirality in panel c) of both the expected
chiral state, in orange, and the additional state living on the real edge,
red, relative to the corresponding in-gap states in fig. 3 c) in the main
text. This is further evidence that these features are in correspondence with
the Skyrmion number of the bulk.

##  S4 Periodic Boundary Conditions OEES for Complex $\boldsymbol{d}$-vector
with $\mathcal{Q}=-1$

Figure S8: Various spectra calculated for the model with Complex
$\boldsymbol{d}$ vector at parameters $\mu/t_{q}=-0.2,\Delta_{0}=1,\beta=1$,
$\mathcal{Q}=-1$, $\mathcal{C}=0$ in full periodic boundary conditions. Here
panel a) is the ground state entanglement spectrum $\xi$; b) the ground state
entanglement spectrum with virtual cut on the torus of sites $x=101$ to
$x=200$ inclusive, $\xi_{A}$; c) ground state entanglment spectrum after OEPT,
$\xi_{S}$; d) ground state density matrix with both OEPT and virtual cut on
torus, $\xi_{S,A}$.

Fig. S8 displays results on various entanglement spectra in a cylindrical
geometry – with open boundary conditions in one direction and periodic
boundary conditions in the other – to a toroidal geometry – with full periodic
boundary conditions. We see in panel a) the eigenspectrum, $\xi$ of the
groundstate density matrix, which is composed of pure states only. Panel b)
shows the spectrum $\xi_{A}$, which is the eigenspectrum after a virtual cut
on the last $100$ sites of the torus, forming two entanglement edges. We see
the states are trivial as expected, as they do not connect the spectrum at
$\xi_{A}=0$ and $\xi_{A}=1$. Panel c) shows the ground state with only the
OEPT applied to it, $\xi_{S}$: edge states are absent, in contrast to the
entanglement spectrum presented in fig. 3 c). Finally, d) shows the $\xi_{S}$
spectrum after a real space virtual cut of the type in b), $\xi_{S,A}$. We see
two crossing chiral states of the type presented in orange in fig. 3 d).


# On large sieve inequalities involving $p$th powers of trigonometric
polynomials

S. NORVIDAS

(Vilnius University Institute of Mathematics and Informatics, Akademijos 4,
Vilnius 08663, Lithuania
e-mail<EMAIL_ADDRESS>
Abstract. In this paper, we extend the large sieve type estimates to sums
involving $p$th powers of trigonometric polynomials. An approach to such
estimates that does not rely on the usual $L^{2}$-technique is given. Our
method is based on comparing the norm and the spectral radius of convolution
operators on a normed space of trigonometric polynomials.

##  1  Introduction

We denote by $T_{N}$ the set of trigonometric polynomials of degree at most
$N$

$s(x)=\sum_{k=-N}^{N}c_{k}e^{ikx}$

with a positive integer $N$ and $c_{k}\in\mathbb{C}$, $k=-N,\dots,N$. Suppose
that $x_{1}<x_{2}<\dots<x_{r}$, $r>1$, is any sequence in
$\mathbb{T}=(-\pi,\pi]$ such that

(1) $\min\Bigl{\\{}x_{j+1}-x_{j},j=1,\dots,r-1;\
2\pi-(x_{r}-x_{1})\Bigr{\\}}=\delta>0.$

The usual large sieve inequality states that

(2)
$\sum_{j=1}^{r}|s(x_{j})|^{2}\leq\Bigl{(}\frac{N}{2\pi}+\frac{1}{\delta}\Bigr{)}\int_{\mathbb{T}}|s(x)|^{2}\,dx.$

See Selberg [8, p. 221], but note the different notation. Gallagher [4] has
given a simple derivation of the large sieve inequalities. It turned out that
the method of [4] can be applied to $L^{p}(\mathbb{T})$-norm. More precisely,
in [3, p. 96] was proved that if $s\in T_{N}$ and $2\leq p<\infty$, then

(3)
$\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\Delta_{p}p^{1/2}\max\biggl{(}N,\,\frac{4\pi}{\delta}\biggr{)}\Bigl{(}\sum_{k=-N}^{N}|c_{k}|^{q}\Bigr{)}^{p/q},$

where $\Delta_{p}$ is an absolute constant and $1/p+1/q=1$. Next, in [6, p.
533] the inequality (3) was extended for all $0<p<\infty$ as follows. Let
$\Psi$ be a convex, nonnegative, and nondecreasing function in $[0,\infty)$.
Then for any $s\in T_{N}$,

(4)
$\sum_{j=1}^{r}\Psi(|s(x_{j})|^{p})\leq\Bigl{(}\frac{N}{\pi}+\frac{1}{\delta}\Bigr{)}\int_{\mathbb{T}}\Psi\Bigl{(}|s(x)|^{p}(p+1)e/2\Bigr{)}\,dx,$

whenever $-\pi<x_{1}<x_{2}<\dots<x_{r}\leq\pi$ and $\delta$ is defined by (1).
If $\Psi(t)=t$, $0\leq t<\infty$, then (4) implies

(5)
$\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\Bigl{(}\frac{N}{\pi}+\frac{1}{\delta}\Bigr{)}\frac{(p+1)e}{2}\|s\|^{p}_{L^{p}(\mathbb{T})}.$

In [5, p. 164] the estimate (4) was extended to the case of generalized
trigonometric polynomials. In particular, for usual trigonometric polynomials
, the inequality (5) was improved as follows:

(6)
$\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\Bigl{(}\frac{N+1}{2\pi}+\frac{1}{\delta}\Bigr{)}\frac{(p+1)e}{2}\|s\|^{p}_{L^{p}(\mathbb{T})}.$

Note that inequalities (2)-(6) are also called forward Marcinkiewicz-Zygmund
inequalities (see, e.g., [2]).

In this paper, we will develop an approach to inequalities of the type (3) and
(5)-(6) that not use the usual $L^{2}$-technique. Our approach is based on the
spectral theory of convolution operators on $T_{N}$.

The main result is given in the following theorem. Note that in the sequel,
$[x]$ denotes the integer part of a positive number $x$. Also we use the
notation $\Gamma(\cdot)$ for the standard gamma function.

###### THEOREM 1

. Let $\\{x_{j}\\}_{j=1}^{r}$ be a sequence in $\mathbb{T}$ that satisfies
(1). If $s\in T_{N}$ and $1\leq p<\infty$, then

(7)
$\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\frac{pN\sigma(\delta;N)}{2\sqrt{\pi}}\cdot\frac{\Gamma\Bigl{(}p/2\Bigr{)}}{\Gamma\Bigl{(}p/2+1/2\Bigr{)}}\|s\|^{p}_{L^{p}(\mathbb{T})}$

with

(10)
$\displaystyle\sigma(\delta;N)=\begin{array}[]{\\{}{rl}.\frac{\pi}{N\delta},\
\ \ \ if&\frac{\pi}{N\delta}\in\mathbb{Z},\\\
1+\biggl{[}\frac{\pi}{N\delta}\biggr{]},&\mbox{otherwise.}\end{array}$

If $p$ is a positive integer, then the quantity

$\frac{\Gamma\Bigl{(}p/2\Bigr{)}}{\Gamma\Bigl{(}p/2+1/2\Bigr{)}}$

can be calculated directly by using the relations between
$\Gamma\Bigl{(}p/2\Bigr{)}$ and $\Gamma\Bigl{(}p/2+1/2\Bigr{)}$.

###### COROLLARY 2

. Let $l$ be a positive integer. Then under the conditions of Theorem 1 it
follows that:

(i) if $p=2l$, then

(11)
$\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\frac{pN\sigma(\delta;N)}{2\sqrt{\pi}}\cdot\frac{\Gamma\Bigl{(}p/2\Bigr{)}}{\Gamma\Bigl{(}p/2+1/2\Bigr{)}}\|s\|^{p}_{L^{p}(\mathbb{T})}=\frac{pN\sigma(\delta;N)\cdot
2^{l-1}(l-1)!}{\pi\Bigl{(}1\cdot 3\cdot
5\cdot\cdots\cdot(2l-1)\Bigr{)}}\|s\|^{p}_{L^{p}(\mathbb{T})};$

(ii) if $p=2l+1$, then

(12)
$\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\frac{pN\sigma(\delta;N)}{2\sqrt{\pi}}\cdot\frac{\Gamma\Bigl{(}p/2\Bigr{)}}{\Gamma\Bigl{(}p/2+1/2\Bigr{)}}\|s\|^{p}_{L^{p}(\mathbb{T})}=\frac{pN\sigma(\delta;N)\cdot\Bigl{(}1\cdot
3\cdot 5\cdot\cdots\cdot(2l-1)\Bigr{)}}{2^{l+1}\cdot
l!}\|s\|^{p}_{L^{p}(\mathbb{T})}.$

###### COROLLARY 3

. Assume that $s\in T_{N}$, $\|s\|_{L^{p}(\mathbb{T})}=1$, $p\geq 1$, and
$\\{x_{j}\\}_{j=1}^{r}$ satisfies (1).

(i) If $\frac{\pi}{N\delta}\in\mathbb{Z}$, then

(13) $\sum_{j=1}^{r}|s(x_{j})|^{p}<\frac{p+1}{\delta}.$

(ii) If $\frac{\pi}{N\delta}\not\in\mathbb{Z}$, then

(14)
$\sum_{j=1}^{r}|s(x_{j})|^{p}<(p+1)\biggl{(}\frac{N}{\pi}+\frac{1}{\delta}\biggr{)}.$

##  2  Proofs

Let $M(\mathbb{T})$ be the Banach algebra of finite complex-valued regular
Borel measures on $\mathbb{T}$. The norm in $M(\mathbb{T})$ is given by the
total variation $\|\mu\|$ of $\mu\in M(\mathbb{T})$. Therefore, the usual
Banach space $L^{1}(\mathbb{T})$ can be identified with the closed ideal in
$M(\mathbb{T})$ of all measures which are absolutely continuous with respect
to the Lebesgue measure $dt$ on $\mathbb{T}$.

Given $\mu\in M(\mathbb{T})$ and $f\in L^{1}(\mathbb{T})$, we define the
Fourier transform of $\mu$ and $f$ by

$\widehat{\mu}(x)=\int_{\mathbb{T}}e^{-ixt}\,d\mu(t)\quad{\text{
and}}\quad\widehat{f}(x)=\int_{\mathbb{T}}e^{-ixt}f(t)\,dt,$

respectively. For each $u\in L^{r}(\mathbb{T})$, $1\leq r\leq\infty$, on
$T_{N}$ is well defined the convolution operator

$\displaystyle A_{u}(s)(x)=s\ast
u(x)=\int_{\mathbb{T}}s(x-t)u(t)\,dt=\int_{\mathbb{T}}\Bigl{(}\sum_{k=-N}^{N}c_{k}e^{ik(x-t)}\Bigr{)}u(t)\,dt$
(15) $\displaystyle=\sum_{k=-N}^{N}c_{k}\widehat{u}(k)e^{ikx},$

for each $x\in\mathbb{T}$, where $s\in T_{N}$. Note that in this definition
and also below we assume that $s$ is a periodic function on the real line with
the period equal to $2\pi$. Below, the notation $T^{p}_{N}$, $1\leq
p\leq\infty$, means that $T_{N}$ is equipped with the usual
$L^{p}(\mathbb{T})$ norm.

Suppose that $S_{1}$ and $S_{2}$ are two measurable subsets of $\mathbb{T}$,
$\mu_{1}$ and $\mu_{2}$ are two non-negative finite measures on $\mathbb{T}$,
and $F:\mathbb{T}^{2}\to\mathbb{R}$ is a measurable function. For $1\leq
p<\infty$, Minkowski’s integral inequality [9, p. 37] states that

$\displaystyle\biggl{[}\int_{S_{2}}\biggl{|}\int_{S_{1}}F(x,y)\,d\mu_{1}(x)\biggr{|}^{p}\,d\mu_{2}(y)\biggr{]}^{1/p}$
(16)
$\displaystyle\leq\int_{S_{1}}\biggl{(}\int_{S_{2}}|F(x,y)|^{p}\,d\mu_{2}(y)\biggr{)}^{1/p}\,d\mu_{1}(x).$

###### PROPOSITION 4

. Let $u\in L^{q}(\mathbb{T})$, $1\leq q\leq\infty$,
$\|u\|_{L^{q}(\mathbb{T})}\neq 0$. Assume that $u$ is continuous, non-negative
and even on $\mathbb{T}$. If

(17) ${\text{supp}}\ u\subset\Bigl{[}-\frac{\pi}{2N}\,,\
\frac{\pi}{2N}\Bigr{]},$

then there exists a trigonometric sum

(18) $p_{u}(x)=\sum_{m=-N+1}^{N}\tau_{m}e^{-i\pi mx/N}$

such that

(19) $(-1)^{m}\tau_{m}>0,$

$m=-N+1,\dots,N$, and

(20) $p_{u}(n)=\frac{1}{\widehat{u}(n)}$

for all $n=-N,\dots,N$.

PROOF. We start by examining in more details the Fourier transform of $u$.
Under the assumptions on $u$, we see that

(21)
$\widehat{u}(x)=\int_{\mathbb{T}}u(t)e^{-ixt}\,dt=2\int_{0}^{\pi/2N}u(t)\cos
xt\,dt>0$

for all $x\in[-N,N]$. From this it follows that

(22) $(\widehat{u})^{\prime}(x)=-2\int_{0}^{\pi/2N}tu(t)\sin xt\,dt<0$

and

(23) $(\widehat{u})^{\prime\prime}(x)=-2\int_{0}^{\pi/2N}t^{2}u(t)\cos
xt\,dt<0$

for all $x\in[0,N]$.

Let $v=1/\widehat{u}$. By (19), the function $v$ is well defined and positive
on $[-N,N]$. Moreover, we conclude from (19)-(21) that $v$ is an even function
and

(24) $v^{\prime}(x)>0\quad{\text{and}}\quad v^{\prime\prime}(x)>0$

for all $x\in[0,N]$. Therefore, $v$ is increasing and convex on $[0,N]$, in
particular $v$ is of bounded variation on $[0,N]$. In particular, this means
that $v$ is a function of bounded variation on $\mathbb{T}$. The following is
well known: If $f$ is an $2\pi$-periodic continuously differentiable even
function on $\mathbb{T}$ such that $f$ is of bounded variation on
$\mathbb{T}$, then the Fourier series of $f$ converges absolutely (see, eg.
[10, p. 241]). Thus,

(25)
$v(x)=\sum_{k\in\mathbb{Z}}a_{k}e^{ikx\pi/N}=\sum_{k\in\mathbb{Z}}a_{k}\cos\Bigl{(}\frac{\pi}{N}kx\Bigr{)}$

with

(26) $\sum_{k\in\mathbb{Z}}|a_{k}|<\infty,$

where

(27)
$a_{k}=\int_{-N}^{N}v(t)e^{-ikt\pi/N}\,dt=2\int_{0}^{N}v(t)\cos\Bigl{(}\frac{\pi}{N}kt\Bigr{)}\,dt.$

We claim that

(28) $(-1)^{k}a_{k}>0$

for all $k\in\mathbb{Z}$. Combining (19) with (25), we see that $a_{0}>0$. Let
$k\geq 1$. Then using integration by parts, we conclude from (15) that

(29) $a_{k}=-\frac{2N}{\pi
k}\int_{0}^{N}v^{\prime}(t)\sin\Bigl{(}\frac{\pi}{N}kt\Bigr{)}\,dt=-\frac{2N}{\pi
k}\ \sum_{j=0}^{k-1}I_{j},$

where

(30) $I_{j}=\int_{E_{j}}v^{\prime}(t)\sin\Bigl{(}\frac{\pi}{N}kt\Bigr{)}\,dt$

and $E_{j}=[Nj/k,N(j+1)/k]$. Note that the length of $E_{j}$ is exactly half
length of period for $\sin\Bigl{(}\pi kx/N\Bigr{)}$. Combining this with (22),
we see that

(31) $(-1)^{j}I_{j}>0\quad{\text{and}}\quad|I_{j}|<|I_{j+1}|$

for all $j=0,\dots,k-1$. From this, it is easily seen that

$(-1)^{k-1}\sum_{j=0}^{k-1}I_{j}>0.$

In light of (27) this proving the claim (26).

For $m=-N+1,\dots,N$, let $\tau_{m}$ be defined by

(32) $\tau_{m}=\sum_{j\in\mathbb{Z}}a_{m+2jN}.$

From (24), we see that the series in (30) converges absolutely. Next, from
(26) it follows that, for each $m$, all terms of the sequence
$\\{a_{m+2jN}\\}_{j\in\mathbb{Z}}$ have the same sign. In particular, (26)
shows that $(-1)^{m}a_{m+2jN}>0$. Therefore, the trigonometric sum (16) is
well defined and satisfies (17).

Finally, for any $n\in\\{-N,\dots,N\\}$, combining (23) with (30), we get

$\displaystyle\frac{1}{\widehat{u}(n)}=v(n)=\sum_{k\in\mathbb{Z}}a_{k}e^{ik\pi
n/N}=\sum_{m=-N+1}^{N}\biggl{(}\sum_{j\in\mathbb{Z}}a_{m+2jN}e^{i(m+2jN)\pi
n/N}\Biggr{)}$
$\displaystyle=\sum_{m=-N+1}^{N}\biggl{(}\sum_{j\in\mathbb{Z}}a_{m+2jN}e^{im\pi
n/N}\biggr{)}=\sum_{m=-N+1}^{N}\biggl{(}\sum_{j\in\mathbb{Z}}a_{m+2jN}\biggr{)}e^{im\pi
n/N}\biggr{)}$
$\displaystyle=\sum_{m=-N+1}^{N}\tau_{m}e^{imn\pi/N}=p_{u}(-n).$

As $u$ and $\hat{u}$ are even functions we have $p_{u}(-n)=p(n)$ for all $n$.
Proposition 4 is proved.

We will denote by $\delta_{a}$ the usual Dirac measure supported on
$a\in\mathbb{T}$.

###### PROPOSITION 5

. Let positive numbers $p$ and $q$ satisfy $1/p+1/q=1$. Under the conditions
of Proposition 4 on $u\in L^{q}(\mathbb{T})$ it follows that the operator (13)
possesses on $T_{N}^{p}$ a bounded inverse of the type

(33) $A^{-1}_{u}(s)(x)=\int_{\mathbb{T}}s(x-t)\,d\mu(t),$

where

(34) $\mu=\sum_{m=-N+1}^{N}\tau_{m}\delta_{\pi m/N}$

and $\\{\tau_{m}\\}_{-N+1}^{N}$ are defined by (30), using (23) for
$v=1/\hat{u}$. Furthermore,

(35)
$\|A^{-1}_{u}\|_{T_{N}^{p}}=\|\mu\|_{M(\mathbb{T})}=\sum_{m=-N+1}^{N}|\tau_{m}|=\frac{1}{\widehat{u}(N)}.$

PROOF. Let $s(x)=\sum_{k=-N}^{N}c_{k}e^{ikx}\in T_{N}$. From (16) it follows
that

(36)
$s\ast\mu(x)=\sum_{k=-N}^{N}c_{k}\biggl{(}\int_{\mathbb{T}}e^{ik(x-t)}\,d\mu(t)\biggr{)}=\sum_{k=-N}^{N}c_{k}\widehat{\mu}(k)e^{ikx},$

where

$\widehat{\mu}(k)=\int_{\mathbb{T}}e^{-ikt}\,d\mu(t)=\sum_{m=-N+1}^{N}\tau_{m}e^{-i\pi
km/N}=p_{u}(k),$

for all $k=-N,\dots,N$. Now, taking into account (18), we conclude from from
(13) that (31) defines the inverse of the operator $A_{u}$.

Now, according to (34), we see that the set $\\{\widehat{\mu}(k)=p_{u}(k):\
k=-N,\dots,N\\}$ coincides with the spectrum of $A^{-1}_{u}$. Therefore, if
$|A^{-1}_{u}|_{T_{N}^{p}}$ denotes the spectral radius of $A^{-1}_{u}$, then

$|A^{-1}_{u}|_{T_{N}^{p}}=\max\\{|\widehat{\mu}(k)|=|p_{u}(k)|:\
k=-N,\dots,N\\}.$

Let us recall that $|H|_{X}\leq\|H\|_{X}$, i.e. the spectral radius is not
greater than the operator norm, for any bounded linear operator $H$ on a
normed space $X$. Combining this with (18), we get

$\displaystyle\|A^{-1}_{u}\|_{T_{N}^{p}}\geq\max_{-N\leq k\leq
N}|\widehat{\mu}(k)|=\max_{-N\leq k\leq N}|p_{u}(k)|=\max_{-N\leq k\leq
N}\biggl{\\{}\frac{1}{|\widehat{u}(k)|}\biggr{\\}}$
$\displaystyle=\frac{1}{\min_{-N\leq k\leq
N}|\widehat{u}(k)|}\geq\frac{1}{\widehat{u}(N)}=p_{u}(N)=\sum_{M=-N+1}^{N}\tau_{m}e^{i\pi
mN/N}=$ (37)
$\displaystyle\sum_{M=-N+1}^{N}(-1)^{m}\tau_{m}=\sum_{M=-N+1}^{N}|\tau_{m}|=\|\mu\|.$

Other hands, from Minkowski’s inequality (14) it is easily to see that

$\displaystyle\|A^{-1}_{u}(s)\|_{T_{N}^{p}}=\biggl{(}\int_{\mathbb{T}}\biggl{|}\int_{\mathbb{T}}s(x-t)\,d\mu(t)\biggr{|}^{p}\,dx\biggr{)}^{1/p}$
$\displaystyle\leq\biggl{(}\int_{\mathbb{T}}\biggl{|}\int_{\mathbb{T}}|s(x-t)|\,d|\mu|(t)\biggr{|}^{p}\,dx\biggr{)}^{1/p}$
(38)
$\displaystyle\leq\int_{\mathbb{T}}\biggl{(}\int_{\mathbb{T}}|s(x-t)|^{p}\,dx\biggr{)}^{1/p}\,d|\mu|(t)=\|s\|_{T_{N}^{p}}\|\mu\|,$

where $|\mu|$ denote the variation of $\mu$. Thus, (35) with (36) show (33),
and the Proposition 5 is proved.

PROOF OF THEOREM 1. Assume that $u\in L^{q}(\mathbb{T})$,
$\|u\|_{L^{q}(\mathbb{T})}=1$ and $u$ satisfies the conditions of Proposition
4. If $s\in T_{N}$, then by Hölder’s inequality, we get

$\displaystyle|s(x)|^{p}=\biggl{|}\int_{\mathbb{T}}A^{-1}_{u}(s)(x-t)u(t)\,dt\biggr{|}^{p}=\biggl{|}\int_{-\pi/2N}^{\pi/2N}A_{u}^{-1}(s)(x-t)u(t)\,dt\biggr{|}^{p}$
$\displaystyle\leq\biggl{(}\int_{-\pi/2N}^{\pi/2N}\Bigr{|}A_{u}^{-1}(s)(x-t)\biggl{|}^{p}\,dt\biggr{)}\biggl{(}\int_{-\pi/2N}^{\pi/2N}|u(t)|^{q}\,dt\biggr{)}^{p/q}$
$\displaystyle=\int_{-\pi/2N}^{\pi/2N}\Bigr{|}A^{-1}_{u}(s)(x-t)\biggl{|}^{p}\,dt$

for each $x\in\mathbb{T}$. Therefore,

$\displaystyle\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\sum_{j=1}^{r}\int_{-\pi/2N}^{\pi/2N}\biggl{|}A_{u}^{-1}(s)(x_{j}-t)\biggr{|}^{p}\,dt$
(39)
$\displaystyle=\sum_{j=1}^{r}\int_{x_{j}-\pi/2N}^{x_{j}+\pi/2N}\biggl{|}A_{u}^{-1}(s)(y)\biggr{|}^{p}\,dy.$

We will denote by $E_{j}$ the set

$E_{j}=\Bigl{(}x_{j}-\frac{\pi}{2N},x_{j}+\frac{\pi}{2N}\Bigr{]},$

$j=1,\dots,r$. Using the fact that $A_{u}^{-1}(s)$ is a trigonometric
polynomial, i.e., a continuous and periodical function on $\mathbb{R}$, we
conclude that

(40)
$\int_{x_{j}-\pi/2N}^{x_{j}+\pi/2N}\Bigl{|}A_{u}^{-1}(s)(y)\Bigr{|}^{p}\,dy=\int_{E_{j}}\Bigl{|}A_{u}^{-1}(s)(y)\Bigr{|}^{p}\,dy.$

Let $x\in\mathbb{R}$ and assume that $x\in E_{j}$, for $j=i,i+1,\dots,i+k$
with some non-negative integer $k$. Then we claim that:

(i) if $\pi/N\delta\in\mathbb{Z}$, then

(41) $k+1\leq\frac{\pi}{N\delta};$

(ii) if $\pi/N\delta\not\in\mathbb{Z}$, then

(42) $k\leq\Bigl{[}\frac{\pi}{N\delta}\Bigr{]},$

where $[\cdot]$ is the usual integer part of a real number.

Indeed, since $x\in E_{i}\cap E_{i+k}$, it follows that
$x_{i}+\pi/2N>x_{i+k}-\pi/2N$. Then

(43) $x_{i+k}-x_{i}<\frac{\pi}{N}.$

Other hands, by (1) we get

(44) $x_{i+k}-x_{i}\geq k\delta.$

Combining (42) with (41), we see that

(45) $k<\frac{\pi}{N\delta}.$

Now note that $k$ is an integer. Hence if $\pi/N\delta\in\mathbb{Z}$, then
(43) implies (39). For $\pi/N\delta\not\in\mathbb{Z}$, we get (40), which
yields our claim.

Thus, each $x\in\mathbb{T}$ can belong to at most $\sigma_{p}$ intervals
$E_{j}$, $j=1,\dots,p$, where $\sigma_{p}$ was defined by (8).

Now combining (35), (36) with (37), (38), (31) and (8), we get

$\displaystyle\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\sum_{j=1}^{r}\int_{x_{j}-\pi/2N}^{x_{j}+\pi/2N}\biggl{|}A_{u}^{-1}(s)(y)\biggr{|}^{p}\,dy\leq\sigma(\delta;N)\int_{\mathbb{T}}\Bigl{|}A^{-1}_{u}(s)(y)\Bigr{|}^{p}\,dy=$
(46)
$\displaystyle=\sigma(\delta;N)\|A^{-1}_{u}(s)\|^{p}_{L^{p}(\mathbb{T})}\leq\sigma(\delta;N)\|A_{u}^{-1}\|^{p}_{T^{p}_{N}}\|s\|^{p}_{L^{p}(\mathbb{T})}=\frac{\sigma_{p}}{\bigl{(}\hat{u}(N)\bigr{)}^{p}}\|s\|^{p}_{L^{p}(\mathbb{T})}.$

Next, the estimate (44) can be improved to

(47)
$\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\frac{\sigma(\delta;N)}{\sup_{u}\Bigl{(}\hat{u}(N)\Bigr{)}^{p}}\
\|s\|^{p}_{L^{p}(\mathbb{T})},$

where the supremum extends over all admissible $u$ as described in the
statement of Proposition 4. We claim that for such an $u$ we have

(48) $\sup_{u}(\widehat{u}(N))^{p}=\|\cos
Nx\|^{p}_{L^{p}[-\pi/2N,\pi/2N]}=\frac{\|\cos
t\|^{p}_{L^{p}[-\pi/2,\pi/2]}}{N}.$

Indeed, Hölder’s inequality implies that

$\displaystyle|\widehat{u}(N)|^{p}=\Bigl{|}\int_{\mathbb{T}}u(x)\cos Nx\
dx\Bigr{|}^{p}=\Bigl{|}\int_{-\pi/2N}^{\pi/2N}u(x)\cos Nx\ dx\Bigr{|}^{p}$
$\displaystyle\leq\int_{-\pi/2N}^{\pi/2N}|\cos Nx|^{p}\
dx\cdot\|u\|_{L^{q}[-\pi/2N,\pi/2N]}^{p}=\int_{-\pi/2N}^{\pi/2N}|\cos
Nx|^{p}\,dx$ (49) $\displaystyle=\frac{1}{N}\int_{-\pi/2}^{\pi/2}|\cos
t|^{p}\,dt=\frac{\|\cos t\|^{p}_{L^{p}[-\pi/2,\pi/2]}}{N}.$

Moreover, since we used Hölder’s inequality, it follows that the estimate (47)
is exact and the equality is attained if

$u(x)=\theta\cos^{p-1}Nx\cdot\chi_{[-\pi/2N,\pi/2N]}(x),$

where $\chi_{[-\pi/2N,\pi/2N]}$ is the indicator function of the interval
$[-\pi/2N,\pi/2N]$ and $\theta\in\mathbb{C}$ is such that
$\|u\|_{L^{q}[-\pi/2N,\pi/2N]}=1$. Therefore, our claim (46) is proved.

Combining (45) with (46), we get

(50) $\sum_{j=1}^{r}|s(x_{j})|^{p}\leq\frac{\sigma(\delta;N)\cdot N}{\|\cos
t\|^{p}_{L^{p}[-\pi/2,\pi/2]}}\|s\|^{p}_{L^{p}(\mathbb{T})}.$

Next,

(51) $\|\cos
t\|^{p}_{L^{p}[-\pi/2,\pi/2]}=\int_{-\pi/2}^{\pi/2}\cos^{p}t\,dt=B\Bigl{(}\frac{1}{2};\frac{p+1}{2}\Bigr{)},$

(see, e.g., [7, p. 142]), where $B$ is Euler’s beta function defined by

$B(a;b)=2\int_{0}^{\pi/2}\sin^{2a-1}\theta\cos^{2b-1}\theta\,d\theta$

for $\Re a$, $\Re b>0$. Applying the following connection between the beta and
the usual gamma function ([7, p. 142]

$B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},$

we conclude from (49) that

(52) $\|\cos
t\|^{p}_{L^{p}[-\pi/2,\pi/2]}=\frac{\Gamma\Bigl{(}\frac{1}{2}\Bigr{)}\Gamma\Bigl{(}\frac{p}{2}+\frac{1}{2}\Bigr{)}}{\Gamma\Bigl{(}\frac{p}{2}+1\Bigr{)}}$

Since $\Gamma\Bigl{(}p/2+1\Bigr{)}=\Gamma\Bigl{(}p/2\Bigr{)}\cdot p/2$ and
$\Gamma\Bigl{(}1/2\Bigr{)}=\sqrt{\pi}$ (see, e.g., [7, p.p. 137-138]), it
follows from (49) and (50) that

$\|\cos
t\|^{p}_{L^{p}[-\pi/2,\pi/2]}=\frac{2\sqrt{\pi}\cdot\Gamma\Bigl{(}\frac{p}{2}+\frac{1}{2}\Bigr{)}}{\Gamma\Bigl{(}\frac{p}{2}\Bigr{)}}.$

Substituting this into (48), we obtain (7). Theorem 1 is proved.

POOF OF COROLLARY 2. It is known that for a nonnegative integer $n$,

$\Gamma(n+1)=n!\qquad{\text{and}}\qquad\Gamma\Bigl{(}n+\frac{1}{2}\Bigr{)}=\sqrt{\pi}\cdot\frac{1\cdot
3\cdot 5\cdot\cdots\cdot(2-1)}{2^{n}}$

(see [7, p. 139]). Using this, the representations (9) and (10) can be
verified by straightforward calculation.

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