# Angle and angular momentum - new twist for an old pair

Ladislav Mišta, Jr<EMAIL_ADDRESS>Department of Optics, Palacký
University, 17. listopadu 12, 779 00 Olomouc, Czech Republic  Hubert de Guise
Department of Physics, Lakehead University, Thunder Bay, Ontario P7B 5E1,
Canada  Jaroslav Řeháček  Department of Optics, Palacký University, 17.
listopadu 12, 779 00 Olomouc, Czech Republic  Zdeněk Hradil  Department of
Optics, Palacký University, 17. listopadu 12, 779 00 Olomouc, Czech Republic

###### Abstract

Reaching ultimate performance of quantum technologies requires the use of
detection at quantum limits and access to all resources of the underlying
physical system. We establish a full quantum analogy between the pair of
angular momentum and exponential angular variable, and the structure of
canonically conjugate position and momentum. This includes the notion of
optimal simultaneous measurement of the angular momentum and angular variable,
the identification of Einstein-Podolsky-Rosen-like variables and states, and
finally a phase-space representation of quantum states. Our construction is
based on close interconnection of the three concepts and may serve as a
template for the treatment of other observables. This theory also provides a
new testbed for implementation of quantum technologies combining discrete and
continuous quantum variables.

Introduction.–Quantum limitations establish challenging problems for
contemporary science, and rapid progress in metrology and communications - two
important pillars of our technological world - bring us closer to this fully
unexplored ultimate regime. Though quantum effects are fundamentally distinct
from our classical intuition, they are manifested in variables which have a
classical interpretation. Conservation laws and the concept of complementary
variables offer the opportunity to be safely guided through this unfamiliar
world of intertwined quantum effects. Thus we see quantum limits more as a
sophisticated network of the interconnected rules and subtle conditions rather
than strict and impenetrable barriers.

Canonical pairs of variables like energy and time, position and momentum, and
angular momentum and angle provide the textbook examples. For instance, the
Schrödinger equation connecting the Hamiltonian with time evolution is a
starting point of quantum mechanics, whereas detection of energy of
electromagnetic field at the level of single photons opened the era of quantum
optics. Though these concepts are well understood, time is not an operator but
a parameter controlling the interaction, so care must be employed in
understanding the energy-time uncertainty relation. The celebrated pair of
position and momentum is the most famous example of non-commuting variables
and the starting point of quantum information science. The Heisenberg
uncertainty principle, Einstein-Podolsky-Rosen (EPR) states [1] and their
detection, coherent states and phase space representation formulated by Roy
Glauber [2], the Arthurs-Kelly concept of approximate simultaneous detection
[3] (see also [4]), as well as teleportation with continuous variables [5],
are the important milestones on the long way towards harnessing quantum
effects.

The angular momentum and angular variable have been treated similarly to the
energy and time rather than full bodied quantum (quadrature-like) variables
forming the phase space for complete description. The purpose of this Letter
is to formulate full quantum description for this conjugated pair. We show the
prominent role of the minimum uncertainty states for angular momentum and
angular variable in four tasks: the formulation of saturable uncertainty
relations, the simultaneous detection of non-commuting variables, the
construction of EPR-like variables and states, and finally the phase-space
representation of quantum states.

Our work is motivated by possible applications to metrology but more generally
by overarching questions about optimal measurements limited by the uncertainty
relations. The group E(2), the natural algebraic structure for angle and
angular momentum, is an interesting testbed for the extension of techniques
developed in the context of Heisenberg algebra. We mention for completeness
some expressions valid for the general case of quasi-periodic representations
[6, 7] but leave the consequences of quasi-periodicity and its potential
applications (as discussed for instance in [8]) for later work. As there is an
extensive body of work related to optical angular momentum as a tool for
quantum information processing [9, 10, 11], the theory developed here provides
theoretical framework for a full quantum description based on the concept of
complementary variables as a possible new platform fully implemented on the
E(2) symmetry. Astonishing experimental progress with sources based on
structured light with imprinted optical angular momentum [12, 13] is a promise
for the realization of such protocols and may trigger new experimental
techniques oriented to state engineering and detection at quantum limits.

Universal uncertainty relations.–Non-commutativity is an essential
differentiating concept between quantum and classical physics. We analyze in
detail the concept for the paradigmatic pair of angular momentum
$L=-i\partial_{\phi}$ and unitary exponential operator $E=e^{-i\phi}$
satisfying the commutation rule of Euclidean algebra $\mathfrak{e}$(2):
$[E,L]=E$. Rephrased in terms of Hermitian operators as
${[}S_{\alpha},L{]}=iC_{\alpha}$, where
$C_{\alpha}=(e^{-i\alpha}E^{\dagger}+e^{i\alpha}E)/2$ and
$S_{\alpha}=(e^{-i\alpha}E^{\dagger}-e^{i\alpha}E)/2i$, the rule implies the
uncertainty relations

$\langle(\Delta L)^{2}\rangle\langle(\Delta
S_{\alpha})^{2}\rangle\geq\frac{1}{4}|\langle C_{\alpha}\rangle|^{2}.$ (1)

The corresponding minimum uncertainty states (MUS) (in the $L$-representation)
[6, 14]

$\displaystyle|n+\delta,\alpha\rangle=\frac{1}{\sqrt{I_{0}(2\kappa)}}\sum_{l\in\mathbb{Z}}e^{i(n-l)\alpha}I_{n-l}(\kappa)|l+\delta\rangle,$
(2)

with $L|l+\delta\rangle=(l+\delta)|l+\delta\rangle$, yield the von Mises
distribution for the angle $\phi$:
$|\langle\phi|n+\delta,\alpha\rangle|^{2}=\exp{[2\kappa\cos(\phi-\alpha)]}/2\pi
I_{0}(2\kappa).$ As a result the states $|n+\delta,\alpha\rangle$ will be
referred to as von Mises states.

Here $n+\delta$, where $n\in\mathbb{Z}$ and $\delta\in[0,1)$, is the angular
momentum mean, $\alpha$ is an angle, $\kappa\geq 0$ represents the spread of
angular variable, and $I_{n}(z)$ is the modified Bessel function [15] (see
Supplemental Material Sec. I for its definition and other properties). Note
that we allow for angular momenta with generally fractional eigenvalues
$l+\delta$, whence the angular momentum eigenstates
$\\{|l+\delta\rangle\\}_{l\in\mathbb{Z}}$ possess quasi-periodic wave
functions $\langle\phi|l+\delta\rangle=\exp{[i(l+\delta)\phi]}/\sqrt{2\pi}$
[6].

For fixed $\alpha,$ the von Mises states $|n+\delta,\beta\rangle$ with
$\beta\neq\alpha+k\pi$, $k\in\mathbb{Z}$, do not saturate the uncertainty
relations (1). However, by setting $\alpha=-\mbox{arg}\langle E\rangle$ and
$\Delta S=S_{\alpha=-\mbox{arg}\langle E\rangle}$, we get the parameter-free
uncertainty relations

$\displaystyle\langle(\Delta
L)^{2}\rangle\,\omega^{2}\geq\frac{1}{4},\quad\omega^{2}=\frac{\langle(\Delta
S)^{2}\rangle}{|\langle E\rangle|^{2}},$ (3)

which is saturated by all von Mises states. Importantly, the measure of the
angular uncertainty $\omega^{2}$ is complementary to angular momentum in the
sense that

$\langle(\Delta
L)^{2}\rangle=\frac{\kappa}{2}\frac{I_{1}(2\kappa)}{I_{0}(2\kappa)},\quad\omega^{2}=\frac{1}{2\kappa}\frac{I_{0}(2\kappa)}{I_{1}(2\kappa)},$
(4)

where $\langle E^{l}\rangle=\exp(-il\alpha)I_{l}(2\kappa)/I_{0}(2\kappa)$
derived in the Supplemental Material Sec. II has been used. Saturable
uncertainty relations (3) for the complementary observables of angular
momentum and angular variable represent the first important result of this
Letter.

The spread parameter $\kappa$ has similar meaning as “squeezing” but here for
the angular momentum and the angular variable. Since the phase space of the
pair angle and angular momentum has cylindrical topology [6], one can
represent von Mises states by ellipses on the cylinder (see Fig. 1) similarly
to the representation of squeezed states of a harmonic oscillator by ellipses
in the plane.

Figure 1: Phase-space-representation of von Mises states $|n,\alpha\rangle$,
$n\in\mathbb{Z}$, Eq. (2). The pase space consists of parallel equidistant
rings (black rings), which are orthogonal to $z$-axis and their centres
possess the $z$-th coordinate $n$. The von Mises state $|n,\alpha\rangle$ is
represented by a noise ellipsis (red ellipsis) centered around a point on the
circle with $z$-th coordinate $n$ and polar angle $\alpha$ (positive angle
between blue line segment and positive $x$-axis). The shape of the ellipsis
depends on the value of the spread parameter $\kappa$, which is chosen to grow
from the bottom to the top. Accordingly, the uncertainties $\langle(\Delta
L)^{2}\rangle$ ($\omega^{2}$), Eq. (4), grow (decrease) from the bottom to the
top. The red ring represents von Mises state with $n=-2$ and $\kappa=0$, which
is an angular momentum eigenstate, so the other phase-space rings are images
of the respective angular momentum eigenstates. The red vertical line
represents the von Mises state with $n=2$ in the limit of
$\kappa\rightarrow\infty$. The red circle represents the von Mises state with
$n=0$ and symmetrical uncertainties $\langle(\Delta
L)^{2}\rangle=\omega^{2}=1/2$ for $\kappa\doteq 1.292$.

Moreover, MUS of Eq. (2) form an over-complete basis resolving the identity as
[6]

$\sum_{n\in\mathbb{Z}}\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}|n+\delta,\alpha\rangle\langle
n+\delta,\alpha|=\openone,$ (5)

and can be used as a generalised measurement for the discrete spectrum
$n+\delta$ of angular momentum and the continuous values $\alpha$ of the
angle. In the following we set to the choice $\delta=0.$

Optimal simultaneous measurement.–The deep analogy with $x$ and $p$ is obvious
from the operator formalism behind the measurement on a signal ($s$) and
ancilla ($a$) fields. Let us define the total sum angular momentum operator
and the exponential angular difference operator,

$\mathcal{L}=L_{s}+L_{a},\quad\mathcal{E}=E_{s}E_{a}^{\dagger}.$ (6)

Since $[\mathcal{L},\mathcal{E}]=0,$ the operator $\mathcal{L}$ and any
function of $\mathcal{E}$ and $\mathcal{E}^{{\dagger}}$ can be measured
simultaneously and may serve as meter variables, in analogy with the pair of
the EPR operators. We observe interestingly that if one assumes the unitary
operator $E$ is the exponential of some “hermitian angle” operator, one would
seemingly recover the same structure as EPR pair for quadrature operators.
However, such a conclusion cannot be justified here due to the issues of
periodicity.

We now move to finding optimal simultaneous measurement of the non-commuting
canonical pair $L_{s}$ and $S_{s}$. We implement the measurement via joint
measurement of the commuting bipartite observables $\mathcal{L}$ and
$\mathcal{S}=(\mathcal{E}^{\dagger}-\mathcal{E})/2i$. We seek the measurement
minimizing the uncertainty product
$\langle(\Delta\mathcal{L})^{2}\rangle\langle(\Delta\mathcal{S})^{2}\rangle$
with $\Delta\mathcal{S}=\mathcal{S}_{\beta=\mbox{arg}\langle
E_{a}\rangle-\mbox{arg}\langle E_{s}\rangle}$, where
$\mathcal{S}_{\beta}=(e^{-i\beta}\mathcal{E}^{\dagger}-e^{i\beta}\mathcal{E})/2i$,
with respect to the product state $|\varphi\rangle_{s}|\chi\rangle_{a}$. A
straightforward derivation with “unbiased” conditions $\langle
L_{a}\rangle=0,\arg\langle E_{a}\rangle=0$ and $\arg\langle
E_{a}^{2}\rangle=0$ is given in the Supplemental Material Sec. III and yields
the inequality

$\displaystyle\langle(\Delta\mathcal{L})^{2}\rangle\langle(\Delta\mathcal{S})^{2}\rangle\geq\frac{1}{4}\left(|\langle
E_{a}\rangle|+|\langle E_{s}\rangle|\sqrt{|\langle
E_{a}^{2}\rangle|}\right)^{2},$ (7)

which is the second main result of this Letter. The right-hand side of the
inequality represents the achievable lower bound for the simultaneous
measurement. Indeed, the inequality is saturated by the MUS for both the
system and ancilla fields satisfying the cross-condition $\langle(\Delta
L_{s})^{2}\rangle\langle(\Delta S_{a})^{2}\rangle=|\langle
E_{a}^{2}\rangle|\langle(\Delta L_{a})^{2}\rangle\langle(\Delta
S_{s})^{2}\rangle$. Consequently, the lower bound is saturated by von Mises
states $|\varphi\rangle_{s}=|n,\alpha,\kappa_{s}\rangle_{s}$ and
$|\chi\rangle_{a}=|0,0,\kappa_{a}\rangle_{a}$, with different spread
parameters $\kappa_{s}$ and $\kappa_{a}$ connected by the condition

$\displaystyle\kappa_{s}=\sqrt{|\langle
E_{a}^{2}\rangle|}\kappa_{a}=\sqrt{\frac{I_{2}(2\kappa_{a})}{I_{0}(2\kappa_{a})}}\kappa_{a}.$
(8)

In Fig. 2 we plot the optimally measurable uncertainty product (7) in
comparison with the uncertainty relations (3) which give the constant lower
bound of $1/4$. Note that the bound for (7) is approximately 4 times larger as
expected on the basis of the Arthurs-Kelly uncertainty relations [3], but only
in the regime where the measurement resolves the angular variable well. This
result is compared with the variance product mean
$\langle(\Delta\mathcal{L})^{2}(\Delta\mathcal{S})^{2}\rangle$ derived based
on the She-Heffner formalism [16] in the Supplemental Material Sec. V. The
analysis of the latter moment normalised with respect to $|\langle
E_{s}\rangle|^{2}|\langle E_{a}\rangle|^{2}$ is surprising: it is even below
the minimum value of uncorrelated product due to the anti-correlations (see
dashed blue line in Fig. 2). In the words, quantum mechanics allows to specify
the states (and the measurement), where each canonically conjugated variable
reaches its minimum in the uncertainty product, but the correlated product is
even below that. This indicates stronger correlations linked to the 4th order
moments. Note that such an effect, though mild in our system, is not possible
in the case of $x$ and $p$ operators.

The joint measurement is realized by a projection onto orthogonal common
eigenvectors of operators ${\cal L}$ and ${\cal E}$ corresponding to
eigenvalues $N,\Phi$ respectively. The common eigenstates are given as

$|N,\Phi\rangle_{sa}=\frac{1}{\sqrt{2\pi}}\sum_{l\in\mathbb{Z}}e^{-il\Phi}|l+N\rangle_{s}|-l\rangle_{a}$
(9)

resembling the EPR-states for the position and momentum operators [1]: when
the ancilla of the state is projected onto the von Mises ancilla state
$|0,0\rangle_{a}$ with the spread parameter $\kappa$, the signal collapses
into the von Mises system state $|N,\Phi\rangle_{s}$ with the same $\kappa$ as
the ancilla. Below we further develop the analogy with EPR states by showing
that the projective measurement onto the EPR-like states (9) plays the role
analogous to Bell measurement for position and momentum [5]. Generalization of
the states of Eq. (9) to signal and ancilla with generally different
fractional angular momenta is discussed in the Supplemental Material Sec. IV.
The full analogy between the structure of EPR pair and states for quadrature
operators and angular momentum and angular variable represents the third main
result of this Letter.

Figure 2: Uncertainties and uncertainty products for angular momentum and
angular variable for optimal states and measurements versus the signal-state
spread parameter $\kappa_{s}$. Uncertainties
$\langle(\Delta\mathcal{L})^{2}\rangle$ (green crosses) and
$\Omega^{2}=\langle(\Delta\mathcal{S})^{2}\rangle/|\langle
E_{s}\rangle|^{2}|\langle E_{a}\rangle|^{2}$ (black stars), and uncertainty
product $\langle(\Delta\mathcal{L})^{2}\rangle\Omega^{2}$ (solid red line) for
optimal simultaneous measurement with von Mises signal and ancilla states with
different spread parameters satisfying the optimal matching condition (8)
whose inverse is depicted in the inset. The same uncertainty product for
suboptimal simultaneous measurement with von Mises signal and ancilla states
with the same spread parameters $\kappa_{s}=\kappa_{a}$ (dotted magenta line).
The product mean
$\langle(\Delta\mathcal{L})^{2}(\Delta\mathcal{S})^{2}\rangle/|\langle
E_{s}\rangle|^{2}|\langle E_{a}\rangle|^{2}$ for von Mises signal and ancilla
states satisfying optimal matching condition (8) (dashed blue line). The
uncertainty product for optimal simultaneous measurement approaches
asymptotically its lower bound of $1$, which is four times larger than the
lower bound of 1/4 for uncertainty relations (3). The product mean always lies
below the uncertainty product for optimal measurement and it may even lie
below $1$. Equality $\langle(\Delta\mathcal{L})^{2}\rangle=\Omega^{2}\doteq
1.099$ is achieved for $\kappa_{s}\doteq 1.146$ and $\kappa_{a}\doteq 1.632$.

Phase-space representation.–Existing attempts to construct a phase-space
representation of angular momentum and angular variable focused exclusively on
the Wigner function [17] using group-theoretical methods [18, 19] or employing
analogies with the harmonic oscillator [20, 21]. Building on the latter ideas
and our previous results, we develop a complete hierarchy of phase-space
distributions exhibiting behaviours and connections very much like the
quasiprobability distributions of the standard harmonic oscillator. Here, we
only sketch the derivations, whereas the details can be found in the
Supplemental Material Sec. VI.

Our approach is based on identities linking the Fourier transformation of the
projectors onto the EPR-like states (9) and von Mises states (2) with the
ordering of the operators $E$ and $L$:

$\displaystyle 2\pi\left(\mathcal{F}|n,\alpha\rangle_{sa}\langle
n,\alpha|\right)(l,\phi)$ $\displaystyle=$ $\displaystyle
D_{s}(l,\phi)D_{a}(-l,\phi),$ (10)
$\displaystyle\left(\mathcal{F}|n,\alpha\rangle_{s}\langle
n,\alpha|\right)(l,\phi)$ $\displaystyle=$ $\displaystyle
o(l,\phi)D_{s}(l,\phi),$ (11)

where

$\displaystyle\left(\mathcal{F}A\right)(l,\phi)=\sum_{n\in\mathbb{Z}}\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}e^{i(l\alpha-\phi
n)}A(n,\alpha),$

is the Fourier transformation,

$\displaystyle D(l,\phi)=e^{-il\frac{\phi}{2}}E^{-l}e^{-iL\phi}$ (12)

is the displacement operator [21], and

$\displaystyle o(l,\phi)=e^{il\frac{\phi}{2}}\langle
l,\phi|0,0\rangle=\frac{I_{l}\left[2\kappa\cos\left(\frac{\phi}{2}\right)\right]}{I_{0}(2\kappa)}.$
(13)

The relation (10) follows immediately from the orthogonal expansion of the
operator $\mathcal{E}^{-l}e^{-i\mathcal{L}\phi}$ in terms of the states (9),
whereas the relation (11) is obtained by averaging Eq. (10) over the ancilla
von Mises state $|0,0\rangle_{a}$ with spread parameter $\kappa$. Based on the
equality (11) we can now construct the phase-space distributions for angular
momentum and angular variable, in a manner analogous to the quasiprobability
$Q$-function [22], Wigner function [17] and $P$-function [2, 23] of the
harmonic oscillator. In particular, the averaging of the formula (11) over the
rescaled density operator $\rho/(2\pi)$ yields immediately the relationship
between the characteristic function $C_{Q}(l,\phi)=(\mathcal{F}Q)(l,\phi)$ of
the $Q$-function,

$\displaystyle Q(n,\alpha)=\frac{\langle
n,\alpha|\rho|n,\alpha\rangle}{2\pi},$ (14)

and the Wigner characteristic function defined as
$C_{W}(l,\phi)=\mbox{Tr}\left[\rho D(l,\phi)\right]/2\pi$,

$\displaystyle C_{Q}(l,\phi)=o(l,\phi)C_{W}(l,\phi).$ (15)

The analogies with the phase-space distributions of the harmonic oscillator
can be taken further by defining the diagonal representation of a density
matrix $\rho$ as $P$-distribution, analogous to the Glauber-Sudarshan
quasiprobability distribution [23, 2]. Recall first that the displacement
operator (12) satisfies the following completeness property [21]:

$\displaystyle\mbox{Tr}\left[D^{{\dagger}}(l,\phi)D(l^{\prime},\phi^{\prime})\right]=2\pi\delta_{ll^{\prime}}\delta_{2\pi}(\phi-\phi^{\prime}),$
(16)

where $\delta_{2\pi}(\phi)$ is the $2\pi$-periodic delta function. Thus, one
can express any density matrix as

$\displaystyle\rho=\sum_{l\in\mathbb{Z}}\int_{-\pi}^{\pi}d\phi
C_{W}(l,\phi)D^{{\dagger}}(l,\phi).$ (17)

Insertion of $[o(l,\phi)]^{-1}o(l,\phi)=1$ into the integrand and application
of the unitarity of the Fourier transformation brings us straightforwardly to
the $P$-representation of any density matrix:

$\displaystyle\rho=\sum_{n\in\mathbb{Z}}\int_{-\pi}^{\pi}d\alpha
P(n,\alpha)|n,\alpha\rangle\langle n,\alpha|,$ (18)

where we introduced the analogy of the $P$-function as the Fourier
transformation $P(n,\alpha)=(\mathcal{F}C_{P})(n,\alpha)$ of the corresponding
characteristic function $C_{P}(l,\phi)$ defined by

$\displaystyle C_{W}(l,\phi)$ $\displaystyle=$ $\displaystyle
o(l,\phi)C_{P}(l,\phi).$ (19)

From Eqs. (15) and (19) it is apparent that the “Bessel” overlap (13) plays
for the pair of angular momentum and angular variable exactly the same role of
a universal smoothing factor as the Gaussian overlap
$\langle\alpha|0\rangle=\mbox{exp}(-|\alpha|^{2}/2)$ of the vacuum state
$|0\rangle$ and the coherent state $|\alpha\rangle$ of a harmonic oscillator.
The relationship between the respective phase-space distributions is given by
the convolution with the kernel comprised by the Fourier transformation of the
overlap (13). This phase-space structure and associated quasi-probability
distributions related to operator ordering constitute the final major result
of our Letter.

Quantum communication with von Mises states.–There were several experimental
attempts to use angular momentum and angle in a manner analogous to quadrature
operators for the purpose of quantum information processing [24, 25]. However
the formulation was burdened by periodicity of angular variable or missing an
analogue of Bell variables. The simultaneous measurement of $\mathcal{L}$ and
$\mathcal{S}$ with optimal ancillary state provides full analogy with the
quadrature heterodyne detection. This allows to translate protocols based on
optical quadratures and heterodyne detection into the realm of the $L$ and $S$
variables. For instance, the coherent state cryptography protocol with
heterodyne detection [26], which does not require switching of measurement
bases, becomes the analogous no-switching protocol with von Mises states.
Another application is obtained if we feed the investigated measurement with
other ancilla states, e.g., comprised by one part of the entangled state (9)
with $N=\Phi=0$. The generalized measurement then plays the role of the Bell
measurement for $L$ and $S$, which can be used for quantum teleportation [27,
5] of von Mises states. A generalization of such a protocol allowing
teleportation of von Mises states between systems with different fractional
angular momenta is provided in the Supplemental Material Sec. IV. Realization
of the proposed protocol would extend teleportation of finite superpositions
of angular momentum eigenstates [28] to the genuine “continuous-variable”
regime when states spanning entire unbounded state space are teleported.

Optical beams.–It is a challenging task to implement von Mises states as
optical beams by advanced techniques adopting twisted photons similar to [29,
25] \- either as non-diffracting Bessel or Laguerre-Gauss modes. Such states
would truly play the role of squeezed-like states carrying information about
both complementary observables of angular momentum and angular variable. New
fascinating progress in compact generation of optical angular momentum states
[13] together with optimal usage of information distributed into continuous
and discrete variables represent a step towards new communication schemes on
robust platform of optical beams.

Phase and intensity as conjugated variables.–Although the quantum phase
problem has a long history with many pitfalls [30], the canonical commutation
relation for $\mathfrak{e}$(2) can be modified to the case of phase and
intensity of the signal field. Considerations inspired by the analysis of the
phase of complex amplitudes allow to formulate the following two-mode
representation: $L=a^{\dagger}_{s}a_{s}-a^{\dagger}_{a}a_{a}$ and
$E=\sqrt{(a_{s}+a_{a}^{\dagger})/(a^{\dagger}_{s}+a_{a})}$. The phase of the
signal field enters through the phase of the complex amplitude
$Y=a_{s}+a_{a}^{\dagger},\quad[Y,Y^{\dagger}]=0.$ However, $L$ and $E$ are
represented here by non-commuting operators and simultaneous detection
requires strategies discussed above.

Conclusion.–We developed a full quantum description of the canonical pair of
angular momentum and angular variable obeying commutation rules associated
with the group E(2). A central role is played by the von Mises minimum
uncertainty states, allowing the performance of optimal measurement as well as
the provision of a phase-space representation of states. Since the optimality
is linked to saturable uncertainty relations, our theory has important
metrological consequences and may trigger new experimental techniques oriented
to state engineering and detection at quantum limits, fully employing the E(2)
symmetry.

###### Acknowledgements.

We thank Jan Bílek for help with graphics. J.R. and Z.H. acknowledge the
support from the project ApresSF supported by the MEYS, Czech Republic, under
the QuantERA programme, which has received funding from the European Union
Horizon 2020 research and innovation programme and
H2020-FETOPEN-2018-2019-2020-01 StormyTune. The work of H.dG. is supported by
NSERC of Canada.

Supplemental Material

##  I Modified Bessel function

Here we review useful formulas to help with some explicit calculations
involving Bessel functions. The modified Bessel function of integer order $n$,
is defined by the integral formula [15]

$I_{n}(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}d\phi e^{z\cos\phi+in\phi}.$ (20)

From the definition one can see easily that $I_{n}(z)$ is real for real $z$,
and satisfies

$I_{n}(z)=I_{-n}(z),\quad I_{n}(-z)=(-1)^{n}I_{n}(z),\quad
I_{n}(0)=\delta_{n0}.$ (21)

In addition, the modified Bessel functions fulfil the recurrence relations
[15]

$I_{n-1}(z)-I_{n+1}(z)=\frac{2n}{z}I_{n}(z),$ (22)

and

$I_{n-1}(z)+I_{n+1}(z)=2\frac{d}{dz}I_{n}(z).$ (23)

Our calculations with modified Bessel functions are greatly simplified by the
addition theorem [15]

$\displaystyle\sum_{m\in\mathbb{Z}}(-1)^{m}I_{r+m}(Z)I_{m}(z)e^{im\phi}=e^{ir\psi}I_{r}(\omega),$
(24)

where $r\in\mathbb{Z}$ and

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\sqrt{Z^{2}+z^{2}-2Zz\cos\phi},$ $\displaystyle Z-z\cos\phi$
$\displaystyle=$ $\displaystyle\omega\cos\psi,\quad z\sin\phi=\omega\sin\psi.$
(25)

In particular, the addition formula yields

$\displaystyle\sum_{m\in\mathbb{Z}}I_{m}(\kappa)I_{m+r}(\kappa)e^{im\phi}=e^{-ir\frac{\phi}{2}}I_{r}\left[2\kappa\cos\left(\frac{\phi}{2}\right)\right],$

with the special case

$\displaystyle\sum_{m\in\mathbb{Z}}I_{m}^{2}(\kappa)=I_{0}(2\kappa).$ (27)

The modified Bessel functions can also be obtained from the following
generating function [31]:

$\displaystyle\sum_{m\in\mathbb{Z}}I_{m}(z)e^{im\phi}=e^{z\cos\phi}.$ (28)

##  II Properties of von Mises states

In this section we summarise some useful properties of the von Mises states,
Eq. (2) of the main text:

$\displaystyle|n+\delta,\alpha\rangle=\frac{1}{\sqrt{I_{0}(2\kappa)}}\sum_{l\in\mathbb{Z}}e^{i(n-l)\alpha}I_{n-l}(\kappa)|l+\delta\rangle,$
(29)

where $\delta\in[0,1)$ and $\kappa\geq 0$.

Recall first that von Mises states (29) are defined as the states saturating
the uncertainty relations (1) of the main text,

$\langle(\Delta L)^{2}\rangle\langle(\Delta
S_{\alpha})^{2}\rangle\geq\frac{1}{4}|\langle C_{\alpha}\rangle|^{2}.$ (30)

In the $\phi$-representation von Mises states read

$\displaystyle\psi_{n+\delta,\alpha}(\phi)=\frac{1}{\sqrt{2\pi
I_{0}(2\kappa)}}e^{i(n+\delta)\phi+\kappa\cos(\phi-\alpha)},$ (31)

where the generating function (28) has been used. The states can be seen as a
special type of states introduced previously in [6] given in
$\phi$-representation by

$\displaystyle\tilde{\psi}_{n+\delta,\alpha}^{\sigma}(\phi)=\frac{1}{\sqrt{2\pi
I_{0}(2s)}}e^{i[(n+\delta)(\phi-\alpha)+\sigma\sin(\phi-\alpha)]},$

where $\sigma=\gamma-is$. The states of (II) can be shown to saturate the
uncertainty relations

$\langle(\Delta L)^{2}\rangle\langle(\Delta
C_{\alpha})^{2}\rangle\geq\frac{1}{4}\left(|\langle
S_{\alpha}\rangle|^{2}+|\langle\\{\Delta L,\Delta
C_{\alpha}\\}\rangle|^{2}\right)$ (33)

and their relationship to our states (31) is given by

$\tilde{\psi}_{n+\delta,\alpha-\frac{\pi}{2}}^{-i\kappa}(\phi)=e^{-i(n+\delta)(\alpha-\frac{\pi}{2})}\psi_{n+\delta,\alpha}(\phi).$
(34)

In what follows, it is advantageous to use the states (31) as they represent
the “standard form” of von Mises states in the $\phi$-representation with
$\gamma=0$ guaranteeing vanishing of the anticommutator mean:
$\langle\\{\Delta L,\Delta S_{\alpha}\\}\rangle=0$. This form is simpler for
calculations yet it captures all essential features of minimum uncertainty
states (MUS) for angular momentum and angular variable.

We start with the overlap $\langle
n^{\prime}+\delta,\alpha^{\prime}|n+\delta,\alpha\rangle$ of two von Mises
states with the same fractional parts $\delta$. By inserting the resolution of
identity $\int_{-\pi}^{\pi}d\phi|\phi\rangle\langle\phi|=\openone$ into the
overlap we obtain

$\displaystyle\langle
n^{\prime}+\delta,\alpha^{\prime}|n+\delta,\alpha\rangle$ $\displaystyle=$
$\displaystyle\int_{-\pi}^{\pi}d\phi\psi^{\ast}_{n^{\prime}+\delta,\alpha^{\prime}}(\phi)\psi_{n+\delta,\alpha}(\phi)\stackrel{{\scriptstyle
1}}{{=}}\frac{1}{I_{0}(2\kappa)}\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}e^{i(n-n^{\prime})\phi+2\kappa\cos\left(\frac{\alpha-\alpha^{\prime}}{2}\right)\cos\left[\phi-\left(\frac{\alpha+\alpha^{\prime}}{2}\right)\right]}$
(35) $\displaystyle\stackrel{{\scriptstyle 2}}{{=}}$ $\displaystyle
e^{i(n-n^{\prime})\left(\frac{\alpha+\alpha^{\prime}}{2}\right)}\frac{I_{n-n^{\prime}}\left[2\kappa\cos\left(\frac{\alpha-\alpha^{\prime}}{2}\right)\right]}{I_{0}(2\kappa)},$

where to get equality 1, Eq. (31) and the identity
$\cos(\phi-\alpha)+\cos(\phi-\alpha^{\prime})=2\cos\left[\phi-\left(\frac{\alpha+\alpha^{\prime}}{2}\right)\right]\cos\left(\frac{\alpha-\alpha^{\prime}}{2}\right)$
were used, whereas in equality $2$ we used the definition (20). Alternatively,
the overlap formula can be derived using the definition (29) and the addition
theorem (I) as

$\displaystyle\langle
n^{\prime}+\delta,\alpha^{\prime}|n+\delta,\alpha\rangle$ $\displaystyle=$
$\displaystyle\frac{e^{i(n\alpha-n^{\prime}\alpha^{\prime})}}{I_{0}(2\kappa)}\sum_{l\in\mathbb{Z}}e^{il(\alpha^{\prime}-\alpha)}I_{n-l}(\kappa)I_{n^{\prime}-l}(\kappa)=e^{i(n-n^{\prime})\left(\frac{\alpha+\alpha^{\prime}}{2}\right)}\frac{I_{n-n^{\prime}}\left[2\kappa\cos\left(\frac{\alpha-\alpha^{\prime}}{2}\right)\right]}{I_{0}(2\kappa)}.$
(36)

Interestingly, since $I_{n}(0)=\delta_{n0}$ according to the last of equations
(21), von Mises states with $\alpha^{\prime}=\alpha+(2k+1)\pi$,
$k\in\mathbb{Z}$, and $n\neq n^{\prime}$ are orthogonal. Thus, contrary to the
usual intuition, the over-complete von Mises-state basis contains not only
nonorthogonal, but also orthogonal states. A more generic overlap formula for
states (II) with generally different fractional parts can be found in [6].

The overlap formula (35) together with the addition theorem (I) allow us to
calculate arbitrary moments of von Mises states. To show this, let us note
first how the operators $\mbox{exp}(-iL\phi)$ and $E^{-l}$ act on von Mises
states (29),

$\displaystyle e^{-iL\phi}|n+\delta,\alpha\rangle$ $\displaystyle=$
$\displaystyle e^{-i(n+\delta)\phi}|n+\delta,\alpha+\phi\rangle,$
$\displaystyle E^{-l}|n+\delta,\alpha\rangle$ $\displaystyle=$
$\displaystyle|n+l+\delta,\alpha\rangle,$ (37)

where in derivation of the second equality the relation
$E^{{\dagger}}|n+\delta\rangle=|n+1+\delta\rangle$ has been used. Let us now
adopt a conventional definition of the moment generating function of a quantum
state $\rho$ as a mean $G(l,\phi)=\mbox{Tr}[\rho\tilde{D}(l,\phi)]$ of the
operator

$\displaystyle\tilde{D}(l,\phi)=E^{-l}e^{-iL\phi}.$ (38)

Making use of equations (II) and the overlap formula (35) one can show easily,
that the moment generating function for the von Mises state
$|n+\delta,\alpha\rangle$ is given by

$\displaystyle
G(l,\phi)=e^{il\alpha}e^{-i\left(n+\delta-\frac{l}{2}\right)\phi}\frac{I_{l}\left[2\kappa\cos\left(\frac{\phi}{2}\right)\right]}{I_{0}(2\kappa)}.$
(39)

From here one can then get straightforwardly all moments as derivatives

$\displaystyle\left\langle
E^{-l}L^{N}\right\rangle=i^{N}\left.\frac{d^{N}}{d\phi^{N}}G(l,\phi)\right|_{\phi=0}.$
(40)

For $N=0$ we can combine equations (39) and (40) to get

$\displaystyle\langle
E^{-l}\rangle=\left.G(l,\phi)\right|_{\phi=0}=e^{il\alpha}\frac{I_{l}(2\kappa)}{I_{0}(2\kappa)}.$
(41)

Moving to $N>0$, let us now express the $N$-th derivative on the right-hand
side of equation (40) as $i^{N-1}(d^{N-1}/d\phi^{N-1})i(d/d\phi)G(l,\phi)$,
calculate the first derivative $i(d/d\phi)G(l,\phi)$ with the help of
generating function (39) and use the recurrence relation (23) to express the
resulting formula for the first derivative in terms of $G(l,\phi)$ and $G(l\pm
1,\phi)$. This yields the $N$-th derivative of the generating function as a
linear combination of $(N-1)$-st derivatives of the generating functions
$G(l,\phi)$ and $G(l\pm 1,\phi)$, which in turn leads, when combined with the
formula (40), to the following recurrence relation for the von Mises states:

$\displaystyle\left\langle E^{-l}L^{N}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{\kappa}{4}\left\\{e^{i\alpha}\left\langle
E^{-(l-1)}\left[L^{N-1}-\left(L-\openone\right)^{N-1}\right]\right\rangle-e^{-i\alpha}\left\langle
E^{-(l+1)}\left[L^{N-1}-\left(L+\openone\right)^{N-1}\right]\right\rangle\right\\}$
(42) $\displaystyle+\left(n+\delta-\frac{l}{2}\right)\left\langle
E^{-l}L^{N-1}\right\rangle.$

Hence, we can rederive moments of the angular momentum [6]

$\displaystyle\langle L\rangle=n+\delta,\quad\langle
L^{2}\rangle=(n+\delta)^{2}+\frac{\kappa}{2}\frac{I_{1}(2\kappa)}{I_{0}(2\kappa)},$
(43)

and

$\displaystyle\langle(\Delta
L)^{2}\rangle=\frac{\kappa}{2}\frac{I_{1}(2\kappa)}{I_{0}(2\kappa)},$ (44)

or derive new moments, e.g.,

$\displaystyle\left\langle(E)^{\pm 2}\Delta L\right\rangle$ $\displaystyle=$
$\displaystyle\pm e^{\mp i2\alpha}\frac{I_{2}(2\kappa)}{I_{0}(2\kappa)}$ (45)

and

$\displaystyle\left\langle(E)^{\pm 2}(\Delta L)^{2}\right\rangle=\frac{e^{\mp
i2\alpha}}{2I_{0}(2\kappa)}\left[I_{2}(2\kappa)+\kappa I_{1}(2\kappa)\right],$
(46)

where $(E)^{\pm 2}$ stands for the $(\pm 2)$-nd power of $E$. Later in this
Supplemental material we use the latter joint moments to calculate the joint
moment appearing in an alternative approach to simultaneous detection of
incompatible observables put forward by She and Heffner [16].

Before doing this, let us briefly comment on another interesting property of
von Mises states, which stems from the recurrence relation (42). Namely, as
$\langle L\rangle=n+\delta$ for von Mises states, the joint moment
$\left\langle E^{-l}L^{N}\right\rangle$ can be expressed via the mean $\langle
L\rangle$ and joint moments involving at most $(N-1)$-st power of the angular
momentum operator. Repeated application of the recurrence relation (42) on the
moments on right-hand side thus allows us to express any joint moment
$\left\langle E^{-l}L^{N}\right\rangle$ only in terms of powers of the mean
value $\langle L\rangle$ and the moments of powers of the operator $E$. This
can be viewed as an analogy of a similar property of Gaussian quantum states
[32]. These states are fully determined by the first-order and second-order
moments of the quadrature operators and thus any higher-order moment can be
expressed only in terms of the first two moments.

##  III Simultaneous detection of angular momentum and angular variable

In this section we derive a fundamental lower bound for product of variances
of the outcomes of simultaneous measurements of the non-commuting observables
$L_{s}$ and $S_{s}$. This can be done most easily using a joint measurement of
commuting observables $\mathcal{L}=L_{s}+L_{a}$ and
$\mathcal{S}=(\mathcal{E}^{\dagger}-\mathcal{E})/2i$ of the signal $s$ and
ancilla $a$, where $\mathcal{E}=E_{s}E_{a}^{{\dagger}}$. We seek the
measurement which would reach the minimum of the uncertainty product
$\langle(\Delta\mathcal{L})^{2}\rangle\langle(\Delta\mathcal{S})^{2}\rangle$
over the product state $|\varphi\rangle_{s}|\chi\rangle_{a}$ with
$\Delta\mathcal{S}=\mathcal{S}_{\beta=\mbox{arg}\langle
E_{a}\rangle-\mbox{arg}\langle E_{s}\rangle}$ and
$\mathcal{S}_{\beta}=(e^{-i\beta}\mathcal{E}^{\dagger}-e^{i\beta}\mathcal{E})/2i$,
acting on both the ancilla and system spaces. We require the measurement to
preserve the signal angular momentum mean, i.e.,
$\langle\mathcal{L}\rangle=\langle{L}_{s}\rangle$, as well as the angular
dependence of relevant moments of angular variable,
$\langle\mathcal{S}^{l}\rangle$, $l=1,2$, on the signal state. This restricts
the ancilla state as

$\langle L_{a}\rangle=0,\quad\arg\langle E_{a}\rangle=0,\quad\arg\langle
E_{a}^{2}\rangle=0.$ (47)

Making use of the latter two conditions one can cast the uncertainties of
angular variables in the form:

$\displaystyle\langle(\Delta\mathcal{S})^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-e_{s}e_{a}\cos\psi_{s}\right),\quad\langle(\Delta
S_{a})^{2}\rangle=\frac{1}{2}(1-e_{a}),$ $\displaystyle\langle(\Delta
S_{s})^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-e_{s}\cos\psi_{s}\right),$ (48)

where $e_{s,a}=|\langle E_{s,a}^{2}\rangle|$ and $\psi_{s}=2\arg\langle
E_{s}\rangle-\arg\langle E_{s}^{2}\rangle$. Consequently, the measurable
uncertainties are simply given as

$\displaystyle\langle(\Delta\mathcal{L})^{2}\rangle$ $\displaystyle=$
$\displaystyle\langle(\Delta L_{s})^{2}\rangle+\langle(\Delta
L_{a})^{2}\rangle,$ $\displaystyle\langle(\Delta\mathcal{S})^{2}\rangle$
$\displaystyle=$ $\displaystyle\langle(\Delta
S_{a})^{2}\rangle+e_{a}\langle(\Delta S_{s})^{2}\rangle.$ (49)

Hence, the uncertainty product is lower bounded as

$\displaystyle\langle(\Delta\mathcal{L})^{2}\rangle\langle(\Delta\mathcal{S})^{2}\rangle$
$\displaystyle=$ $\displaystyle\left[\langle(\Delta
L_{s})^{2}\rangle+\langle(\Delta L_{a})^{2}\rangle\right]\left[\langle(\Delta
S_{a})^{2}\rangle+e_{a}\langle(\Delta S_{s})^{2}\rangle\right]$ (50)
$\displaystyle=$ $\displaystyle\langle(\Delta L_{a})^{2}\rangle\langle(\Delta
S_{a})^{2}\rangle+e_{a}\langle(\Delta L_{s})^{2}\rangle\langle(\Delta
S_{s})^{2}\rangle+\langle(\Delta L_{s})^{2}\rangle\langle(\Delta
S_{a})^{2}\rangle+e_{a}\langle(\Delta L_{a})^{2}\rangle\langle(\Delta
S_{s})^{2}\rangle$ $\displaystyle\stackrel{{\scriptstyle 1}}{{\geq}}$
$\displaystyle\left[\sqrt{\langle(\Delta L_{a})^{2}\rangle\langle(\Delta
S_{a})^{2}\rangle}+\sqrt{e_{a}\langle(\Delta L_{s})^{2}\rangle\langle(\Delta
S_{s})^{2}\rangle}\right]^{2}$ $\displaystyle\stackrel{{\scriptstyle
2}}{{\geq}}$ $\displaystyle\frac{1}{4}\left(|\langle E_{a}\rangle|+|\langle
E_{s}\rangle|\sqrt{|\langle E_{a}^{2}\rangle|}\right)^{2}.$

Here, inequality $1$ follows from

$\displaystyle\left[\sqrt{\langle(\Delta L_{s})^{2}\rangle\langle(\Delta
S_{a})^{2}\rangle}-\sqrt{e_{a}\langle(\Delta L_{a})^{2}\rangle\langle(\Delta
S_{s})^{2}\rangle}\right]^{2}\geq 0$

and it is saturated if

$\displaystyle\langle(\Delta L_{s})^{2}\rangle\langle(\Delta
S_{a})^{2}\rangle=e_{a}\langle(\Delta L_{a})^{2}\rangle\langle(\Delta
S_{s})^{2}\rangle.$ (52)

The inequality $2$ is a consequence of the uncertainty relations

$\displaystyle\langle(\Delta L_{s,a})^{2}\rangle\langle(\Delta
S_{s,a})^{2}\rangle\geq\frac{1}{4}|\langle E_{s,a}\rangle|^{2},$ (53)

and it is saturated by the von Mises MUS of both the signal and the ancilla.
For the condition of Eq. (52) to hold, we will see that the parameters
$\kappa_{a}$ and $\kappa_{s}$ for these states must be related. Recall that
for a general von Mises state $|n+\delta,\alpha\rangle$ one has $\langle
L\rangle=n+\delta$ and $\langle
E^{l}\rangle=\mbox{exp}(-il\alpha)I_{l}(2\kappa)/I_{0}(2\kappa)$, Eqs. (43)
and (41), and the unbiasedness conditions (47) imply the optimal ancilla state
to be the von Mises “vacuum” state $|0,0\rangle_{a}$. If the condition
$\langle L_{a}\rangle=0$ is relaxed, the optimal ancilla state reads
$|\delta_{a},0\rangle_{a}$, where $\delta_{a}\in[0,1)$. Similarly, the optimal
signal state is also a von Mises state $|n+\delta_{s},\alpha\rangle_{s}$. What
is more, substituting the variances for signal and ancilla von Mises states,

$\langle(\Delta
L_{j})^{2}\rangle=\frac{\kappa_{j}}{2}\frac{I_{1}(2\kappa_{j})}{I_{0}(2\kappa_{j})},\quad\langle(\Delta
S_{j})^{2}\rangle=\frac{1}{2\kappa_{j}}\frac{I_{1}(2\kappa_{j})}{I_{0}(2\kappa_{j})},$
(54)

$j=s,a$, into the condition of Eq. (52) one finds that the signal and ancilla
spread parameters $\kappa_{s}$ and $\kappa_{a}$ of optimal states must fulfill
the non-trivial condition

$\displaystyle\kappa_{s}=\sqrt{\frac{I_{2}(2\kappa_{a})}{I_{0}(2\kappa_{a})}}\kappa_{a}.$
(55)

Thus in accordance with our intuition, it is optimal to carry out a von Mises
measurement on von Mises states, but contrary to our intuition, the spread
parameter of the ancilla $\kappa_{a}$ and of the measured state $\kappa_{s}$
differ.

##  IV Quantum teleportation of von Mises states

This section deals with the unconditional teleportation of von Mises states.
Let us consider two quantum systems $A$ and $B$ with angular momenta $L_{A}$
and $L_{B}$, and angular variables $E_{A}$ and $E_{B}$, respectively. The
simultaneous measurement of the total orbital angular momentum
$\mathcal{L}=L_{A}+L_{B}$ and of the sine of the angular difference
$\mathcal{S}=(\mathcal{E}^{\dagger}-\mathcal{E})/2i=(E_{A}^{{\dagger}}E_{B}-E_{A}E_{B}^{{\dagger}})/2i$
plays in the optimal simultaneous measurement of $L_{A}$ and $S_{A}$ the same
role as the EPR operators $x_{A}-x_{B}$ and $p_{A}+p_{B}$ in the optimal
simultaneous measurement of $x_{A}$ and $p_{A}$. Since the latter measurement
is nothing but the Bell measurement for continuous-variable systems [5], one
expects that the former measurement will realize the Bell measurement for
orbital angular momentum and angular variable. In the following we confirm
this by showing that the measurement can be used for perfect quantum
teleportation [27] of unknown von Mises states.

Assume the two systems $A$ and $B$ under consideration carry generally
different angular momenta characterised by fractional parts $\delta_{A}$ and
$\delta_{B}$, respectively. Consider further the vectors

$|N+\Delta_{AB},\Phi\rangle_{AB}=\frac{1}{\sqrt{2\pi}}\sum_{l\in\mathbb{Z}}e^{-il\Phi}|l+\delta_{A}+N-I_{AB}\rangle_{A}|-l+\delta_{B}\rangle_{B},$
(56)

where $I_{jk}=[\delta_{j}+\delta_{k}]\in\\{0,1\\}$ and
$\Delta_{jk}=(\delta_{j}+\delta_{k})\,\mbox{mod}\,1$, $\Delta_{jk}\in[0,1)$,
$j,k=A,B$, are the integer part and the fractional part of
$\delta_{j}+\delta_{k}$, respectively. The normalisation factor
$1/\sqrt{2\pi}$ ensures that the states are normalized as

$\langle
M+\Delta_{AB},\Psi|N+\Delta_{AB},\Phi\rangle=\delta_{MN}\delta_{2\pi}(\Psi-\Phi),$
(57)

where

$\delta_{2\pi}(\phi)=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}e^{in\phi}=\sum_{n\in\mathbb{Z}}\delta(\phi-2n\pi)$
(58)

is the $2\pi$-periodic delta function (or Dirac comb). From relations
$E|n+\delta\rangle=|n-1+\delta\rangle$ and
$E^{{\dagger}}|n+\delta\rangle=|n+1+\delta\rangle$ it further follows
straightforwardly, that

$\displaystyle\mathcal{L}|N+\Delta_{AB},\Phi\rangle_{AB}$ $\displaystyle=$
$\displaystyle(N+\Delta_{AB})|N+\Delta_{AB},\Phi\rangle_{AB},$
$\displaystyle\mathcal{E}|N+\Delta_{AB},\Phi\rangle_{AB}$ $\displaystyle=$
$\displaystyle e^{-i\Phi}|N+\Delta_{AB},\Phi\rangle_{AB},$
$\displaystyle\mathcal{E}^{{\dagger}}|N+\Delta_{AB},\Phi\rangle_{AB}$
$\displaystyle=$ $\displaystyle e^{i\Phi}|N+\Delta_{AB},\Phi\rangle_{AB},$
(59)

where $\mathcal{E}=E_{A}E_{B}^{{\dagger}}$, and thus the vectors of Eq. (56)
are common eigenvectors of $\mathcal{L}$ and $\mathcal{S}$ corresponding to
eigenvalues $(N+\Delta_{AB})$ and $\sin\Phi$, respectively.

Adopting the line of argument of Ref. [33] we can now design the following
teleportation protocol. The goal of the protocol is to transmit faithfully an
unknown von Mises state $|n+\delta_{\rm in},\alpha\rangle_{\rm in}$ of an
input system “in” characterized by the fractional part of angular momentum
$\delta_{\rm in}$, from a sender Alice to a receiver Bob. For this purpose,
the participants can use the shared “EPR-like” state of Eq. (56)

$\displaystyle|\Delta_{AB},0\rangle_{AB}=\frac{1}{\sqrt{2\pi}}\sum_{l\in\mathbb{Z}}|l+\delta_{A}-I_{AB}\rangle_{A}|-l+\delta_{B}\rangle_{B}$

corresponding to eigenvalue $\Delta_{AB}$ of $\mathcal{L}$ and zero eigenvalue
of $\mathcal{S}$. First, Alice performs measurement of EPR-like states (56) on
subsystem “in” and her part $A$ of the shared state. Provided that the
outcomes of her measurement are $(M,\Psi)$, the global state $|n+\delta_{\rm
in},\alpha\rangle_{\rm in}|\Delta_{AB},0\rangle_{AB}$ collapses to the
(unnormalized) state

${}_{{\rm in}A}\langle M+\Delta_{{\rm in}A},\Psi|n+\delta_{\rm
in},\alpha\rangle_{\rm in}|\Delta_{AB},0\rangle_{AB}$ $\displaystyle=$
$\displaystyle\frac{e^{i(I_{AB}-\delta_{B})\Psi}}{2\pi}e^{-i(M+I_{AB}-I_{{\rm
in}A})\Psi}E_{B}^{M+I_{AB}-I_{{\rm
in}A}}e^{iL_{B}\Psi}|n+\delta_{B},\alpha\rangle_{B}$ (61) $\displaystyle=$
$\displaystyle\frac{e^{i(I_{AB}+I_{{\rm
in}A}-M-2\delta_{B})\frac{\Psi}{2}}}{2\pi}D_{B}^{-1}(M+I_{AB}-I_{{\rm
in}A},\Psi)|n+\delta_{B},\alpha\rangle_{B},$

where

$\displaystyle D(l,\phi)=e^{-il\frac{\phi}{2}}E^{-l}e^{-iL\phi}$ (62)

is the displacement operator [21], and where in the second equality we used
the relation

$E^{l}e^{iL\phi}=e^{il\phi}e^{iL\phi}E^{l}=e^{il\frac{\phi}{2}}D^{-1}(l,\phi).$
(63)

Alice subsequently sends the outcomes of her measurement to Bob via classical
channel and he applies to his part of the shared state the correcting
operation $D_{B}(M+I_{AB}-I_{{\rm in}A},\Psi)$. Up to an irrelevant phase
factor and generally different fractional part $\delta_{B}$ from $\delta_{\rm
in}$, Bob recreates a perfect replica $|n+\delta_{B},\alpha\rangle_{B}$ of the
original von Mises state on his system and thus he completes the
teleportation.

The result above strengthens the attractiveness of a laboratory implementation
of the von Mises measurement. First, the measurement would allow teleportation
of von Mises states thereby extending teleportation of finite superpositions
of angular momentum eigenstates [28] to the “continuous-variable” regime in
which infinite superpositions of angular momentum eigenstates, which span
entire infinite-dimensional Hilbert state space, are teleported. In addition,
the presented protocol allows, at least in principle, to teleport quantum
states between systems with generally different fractional angular momenta. It
can be expected that the utility of von Mises measurement will also further
carry over to all other translations of quantum information protocols to
angular momentum - angle, which utilize Bell measurement, such as entanglement
swapping [34] or quantum cryptography without measurement switching [26].

Note finally, that here we demonstrated perfect teleportation of von Mises
states using the non-normalizable EPR-like state of Eq. (56). Analysis of the
realistic protocol with physical approximation of the state (IV), such as, for
instance, the entangled state
$\sum_{l\in\mathbb{Z}}c_{l,-l}|l\rangle_{A}|-l\rangle_{B}$ generated in the
process of spontaneous parametric down-conversion [35], is outside the
immediate scope of the present work.

##  V She-Heffner approach to simultaneous measurement

This section contains analysis of the simultaneous detection of the angular
momentum and angular variable based on the statistical perspective introduced
in the seminal paper of She and Heffner [16]. Following their argumentation
simultaneous detection can be cast as a two-stage process - state preparation
specified by the moments and repeated detection conditioned by the same
constraints as in the state preparation step.

The EPR-like states of Eq. (56) allow us to bridge the Arthurs-Kelly and She-
Heffner approaches. Note first that the states (56) satisfy the completeness
condition

$\displaystyle\sum_{N\in\mathbb{Z}}\int_{-\pi}^{\pi}d\Phi|N+\Delta_{sa},\Phi\rangle_{sa}\langle
N+\Delta_{sa},\Phi|=\openone_{sa},$

where we have done the following identification $s\equiv A$ and $a\equiv B$.
With the help of the resolution of identity and the eigenvalue equations (IV)
we can express the product of squares of operators $\Delta\mathcal{L}$ and
$\Delta\mathcal{S}$ as

$\displaystyle(\Delta\mathcal{L})^{2}(\Delta\mathcal{S})^{2}=\sum_{N\in\mathbb{Z}}\int_{-\pi}^{\pi}d\Phi(N+\Delta_{sa}-\langle\mathcal{L}\rangle)^{2}\sin^{2}\left(\Phi-\beta\right)|N+\Delta_{sa},\Phi\rangle_{sa}\langle
N+\Delta_{sa},\Phi|,$ (65)

where once again $\beta=\mbox{arg}\langle E_{a}\rangle-\mbox{arg}\langle
E_{s}\rangle$. Let us now calculate the partial average of the latter operator
over the optimal ancilla state $|\delta_{a},0\rangle_{a}$. Taking into account
that for this ancilla $\langle L_{a}\rangle=\delta_{a}$, $\mbox{arg}\langle
E_{a}\rangle=0$ and
${}_{a}\langle\delta_{a},0|N+\Delta_{sa},\Phi\rangle_{sa}=|N-I_{sa}+\delta_{s},\Phi\rangle_{s}/\sqrt{2\pi}$
we get after some algebra the following signal operator, diagonal in the von
Mises states $|N+\delta_{s},\Phi\rangle_{s}$:

$\displaystyle\langle(\Delta\mathcal{L})^{2}(\Delta\mathcal{S})^{2}\rangle_{a}=\sum_{N\in\mathbb{Z}}\int_{-\pi}^{\pi}\frac{d\Phi}{2\pi}(N+\delta_{s}-\langle{L}_{s}\rangle)^{2}\sin^{2}\left(\Phi+\mbox{arg}\langle
E_{s}\rangle\right)|N+\delta_{s},\Phi\rangle_{s}\langle N+\delta_{s},\Phi|,$
(66)

where $\langle
X_{sa}\rangle_{a}=\\!\,_{a}\langle\delta_{a},0|X_{sa}|\delta_{a},0\rangle_{a}$.
Further, by averaging the latter operator over the signal state $\rho_{s}$, we
get the analogue of the She-Heffner integral [16] for the angular momentum and
angular variable:

$\displaystyle\langle(\Delta\mathcal{L})^{2}(\Delta\mathcal{S})^{2}\rangle=\sum_{N\in\mathbb{Z}}\int_{-\pi}^{\pi}\frac{d\Phi}{2\pi}(N+\delta_{s}-\langle{L}_{s}\rangle)^{2}\sin^{2}\left(\Phi+\mbox{arg}\langle
E_{s}\rangle\right)_{s}\\!\langle
N+\delta_{s},\Phi|\rho_{s}|N+\delta_{s},\Phi\rangle_{s}.$ (67)

Making use of the expressions for the moments given in Eq. (41) and Eqs.
(44)-(46), we can finally calculate the She-Heffner moment for von Mises
states with spread parameters $\kappa_{s}$ and $\kappa_{a}$ in the form

$\langle(\Delta\mathcal{L})^{2}(\Delta\mathcal{S})^{2}\rangle=\frac{1}{4I_{0}(2\kappa_{s})I_{0}(2\kappa_{a})}\left[\left(\frac{\kappa_{s}}{\kappa_{a}}+\frac{\kappa_{a}}{\kappa_{s}}\right)I_{1}(2\kappa_{s})I_{1}(2\kappa_{a})+2I_{2}(2\kappa_{s})I_{2}(2\kappa_{a})\right].$
(68)

In Fig. 2 of the main text we plot the properly normalized moment
$\langle(\Delta\mathcal{L})^{2}(\Delta\mathcal{S})^{2}\rangle/|\langle
E_{s}\rangle|^{2}|\langle E_{a}\rangle|^{2}$ versus the spread parameter
$\kappa_{s}$ and $\kappa_{a}$ satisfying condition (55). The figure reveals
that the correlated uncertainties represented by the latter moment lie below
the uncorrelated ones (50) for the same von Mises states:

$\displaystyle\langle(\Delta\mathcal{L})^{2}\rangle\langle(\Delta\mathcal{S})^{2}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{4}\left(|\langle
E_{a}\rangle|+|\langle E_{s}\rangle|\sqrt{|\langle
E_{a}^{2}\rangle|}\right)^{2}=\frac{1}{4}\left[\frac{I_{1}(2\kappa_{a})}{I_{0}(2\kappa_{a})}+\sqrt{\frac{I_{2}(2\kappa_{a})}{I_{0}(2\kappa_{a})}}\frac{I_{1}(2\kappa_{s})}{I_{0}(2\kappa_{s})}\right]^{2}.$
(69)

##  VI Phase-space representation

In this section we show that von Mises states allow the development of a
phase-space representation for angular momentum and angular variable, which
closely resembles the phase-space representation for quadrature operators
based on standard coherent states. For the sake of simplicity, we restrict our
attention to integer angular momentum, the generalization to the fractional
angular momenta being deferred for further research. The key mathematical tool
used for the development of the phase-space methods is the Fourier
transformation [20]

$\displaystyle\left(\mathcal{F}A\right)(l,\phi)=\sum_{n\in\mathbb{Z}}\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}e^{i(l\alpha-\phi
n)}A(n,\alpha)$ (70)

of an operator (or function) $A(n,\alpha)$. Making use of the filtration
property of the $2\pi$-periodic delta function (58) on the interval of the
length $2\pi$, one can show easily that the Fourier transformation (70)
fulfils the following analogue of the Parseval formula:

$\displaystyle\sum_{l\in\mathbb{Z}}\int_{-\pi}^{\pi}d\phi(\mathcal{F}A)(l,\phi)(\mathcal{F}B)^{{\dagger}}(l,\phi)=\sum_{n\in\mathbb{Z}}\int_{-\pi}^{\pi}d\alpha
A(n,\alpha)B^{{\dagger}}(n,\alpha),$ (71)

where the symbol † stands for the Hermitian conjugate. Analogously, one can
show that the Fourier transformation of a product is a convolution of the
Fourier transformations of the factors,

$\displaystyle\left[\mathcal{F}(AB)\right](n,\alpha)=\sum_{l\in\mathbb{Z}}\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}(\mathcal{F}A)(n-l,\alpha-\phi)(\mathcal{F}B)(l,\phi)=\left(\mathcal{F}A\right)\ast\left(\mathcal{F}B\right)(n,\alpha).$
(72)

Finally, for $2\pi$-periodic $A(n,\alpha)$ the Fourier transformation (70) is
also its own inverse.

The phase-space representation relies on the identity linking the ordering of
the operators $\mathcal{L}$ and $\mathcal{E}$ with the Fourier transformation
of the projectors onto common eigenstates of the operators, Eq. (56) with
$\Delta_{sa}=0$:

$\displaystyle 2\pi\left(\mathcal{F}|n,\alpha\rangle_{sa}\langle
n,\alpha|\right)(l,\phi)$ $\displaystyle=$
$\displaystyle\mathcal{E}^{-l}e^{-i\mathcal{L}\phi}=D_{s}(l,\phi)D_{a}(-l,\phi),$
(73)

where $D_{j}(l,\phi)$ is the displacement operator of the subsystem $j=s,a$.
The latter relation follows directly from the application of the operator
$\mathcal{E}^{-l}e^{-i\mathcal{L}\phi}$ to the resolution of identity for
states (56), Eq. (V) with $\Delta_{sa}=0$. Further, by averaging both sides of
the equation (73) over the von Mises vacuum state $|0,0\rangle_{a}$ of the
ancillary system $a$ with spread parameter $\kappa$, we obtain

$\displaystyle\left(\mathcal{F}|n,\alpha\rangle_{s}\langle
n,\alpha|\right)(l,\phi)$ $\displaystyle=$ $\displaystyle
o(l,\phi)D_{s}(l,\phi),$ (74)

where

$\displaystyle o(l,\phi)=e^{il\frac{\phi}{2}}\langle
l,\phi|0,0\rangle=\frac{I_{l}\left[2\kappa\cos\left(\frac{\phi}{2}\right)\right]}{I_{0}(2\kappa)}.$
(75)

Here, to get the left-hand side we used ${}_{a}\langle
0,0|n,\alpha\rangle_{sa}=|n,\alpha\rangle_{s}/\sqrt{2\pi}$, and to calculate
the mean ${}_{a}\langle 0,0|D_{a}(-l,\phi)|0,0\rangle_{a}$ on the right-hand
side we used the Eq. (39). The Fourier transformation of the projector onto
von Mises state (74) plays a central role in our approach to development of
the phase-space methods for angular momentum and angular variable. An
interesting feature of the formula (74) is the $c$-number function
$o(l,\phi)$, Eq. (75), in front of the displacement operator $D_{s}(l,\phi)$.
Below we show, among other things, that for angular momentum and angular
variable the “overlap” (75) plays exactly the same role as plays overlap
$\langle\alpha|0\rangle=\mbox{exp}(-|\alpha|^{2}/2)$ of the vacuum state
$|0\rangle$ and the coherent state $|\alpha\rangle$ of a harmonic oscillator.

The relation (74) allows us to arrive in an elegant way to analogies of the
(Husimi) $Q$-function [22], Wigner function [17] and Glauber-Sudarshan
$P$-function [23, 2] of the standard harmonic oscillator. Namely, let us
average the relation (with the index $s$ dropped for simplicity) over the
rescaled density operator $\rho/(2\pi)$, i.e.,

$\displaystyle\mbox{Tr}\left[\frac{\rho}{2\pi}\left(\mathcal{F}|n,\alpha\rangle\langle
n,\alpha|\right)(l,\phi)\right]=\left[\mathcal{F}\frac{\langle
n,\alpha|\rho|n,\alpha\rangle}{2\pi}\right](l,\phi)=o(l,\phi)\frac{1}{2\pi}\mbox{Tr}\left[\rho
D(l,\phi)\right].$ (76)

In analogy with the phase-space distributions of a harmonic oscillator we now
introduce the $Q$-function of a density matrix $\rho$ by a prescription

$\displaystyle Q(n,\alpha)=\frac{\langle
n,\alpha|\rho|n,\alpha\rangle}{2\pi},$ (77)

which is normalized as $\sum_{n\in\mathbb{Z}}\int_{-\pi}^{\pi}d\alpha
Q(n,\alpha)=1$. Likewise, we define the Wigner characteristic function as the
average of the displacement operator,

$\displaystyle C_{W}(l,\phi)=\frac{1}{2\pi}\mbox{Tr}\left[\rho
D(l,\phi)\right].$ (78)

As the Fourier transformation of the $Q$–function is just its characteristic
function, $(\mathcal{F}Q)(l,\phi)=C_{Q}(l,\phi)$, we get from the formula (76)
the relationship

$\displaystyle C_{Q}(l,\phi)$ $\displaystyle=$ $\displaystyle
o(l,\phi)C_{W}(l,\phi).$ (79)

Surprisingly, the analogy with the quadrature phase-space can be developed
even further. Recall first that the displacement operator (62) exhibits the
following completeness property [21]:

$\displaystyle\mbox{Tr}\left[D^{{\dagger}}(l,\phi)D(l^{\prime},\phi^{\prime})\right]=2\pi\delta_{ll^{\prime}}\delta_{2\pi}(\phi-\phi^{\prime}).$
(80)

The property (80) enables us to decompose any density matrix $\rho$ as

$\displaystyle\rho=\sum_{l\in\mathbb{Z}}\int_{-\pi}^{\pi}d\phi
C_{W}(l,\phi)D^{{\dagger}}(l,\phi).$ (81)

Consider now the Hermitian conjugate of the equality (74) (with the index $s$
again dropped)

$\displaystyle\left(\mathcal{F}|n,\alpha\rangle\langle
n,\alpha|\right)^{{\dagger}}(l,\phi)$ $\displaystyle=$ $\displaystyle
o(l,\phi)D^{{\dagger}}(l,\phi).$ (82)

By multiplying both sides with $C_{P}(l,\phi)=[o(l,\phi)]^{-1}C_{W}(l,\phi)$
and performing summation over $l$ and integration over $\phi$, we get

$\displaystyle\sum_{l\in\mathbb{Z}}\int_{-\pi}^{\pi}d\phi
C_{P}(l,\phi)\left(\mathcal{F}|n,\alpha\rangle\langle
n,\alpha|\right)^{{\dagger}}(l,\phi)$ $\displaystyle=$
$\displaystyle\sum_{l\in\mathbb{Z}}\int_{-\pi}^{\pi}d\phi
C_{W}(l,\phi)D^{{\dagger}}(l,\phi)=\rho,$ (83)

where the rightmost equality is a consequence of Eq. (81). If we now apply to
the left-hand side the formula (71), we obtain

$\displaystyle\rho=\sum_{n\in\mathbb{Z}}\int_{-\pi}^{\pi}d\alpha
P(n,\alpha)|n,\alpha\rangle\langle n,\alpha|,$ (84)

where we defined the $P$-function by the formula

$\displaystyle P(n,\alpha)=(\mathcal{F}C_{P})(n,\alpha).$ (85)

Equation (84) reveals that any density matrix can be expressed in diagonal
form in von Mises states. This is a direct analogy of the celebrated Glauber-
Sudarshan representation [23, 2] for the harmonic oscillator.

Summarizing the results, the characteristic functions of different phase-space
distributions are related as

$\displaystyle C_{Q}(l,\phi)$ $\displaystyle=$ $\displaystyle
o(l,\phi)C_{W}(l,\phi)=o^{2}(l,\phi)C_{P}(l,\phi).$ (86)

The overlap $o(l,\phi)$, Eq. (75), plays for the pair of angular momentum and
angular variable the same role of a universal “smoothing” factor as plays the
overlap $\langle\alpha|0\rangle=\mbox{exp}(-|\alpha|^{2}/2)$ for the
canonically conjugate quadrature operators, where [36]

$\displaystyle
C_{Q}(\alpha)=e^{-\frac{|\alpha|^{2}}{2}}C_{W}(\alpha)=e^{-|\alpha|^{2}}C_{P}(\alpha).$
(87)

Application of the Fourier transformation to equation (86) and utilization of
the formula (72) yield finally the following relationship between the adjacent
phase-space distributions:

$\displaystyle Q(n,\alpha)$ $\displaystyle=$
$\displaystyle\left[(\mathcal{F}o)\ast W\right](n,\alpha),$ $\displaystyle
W(n,\alpha)$ $\displaystyle=$ $\displaystyle\left[(\mathcal{F}o)\ast
P\right](n,\alpha).$ (88)

We see that the Fourier transformation of the overlap (75) plays the role of a
kernel of the convolution relating different phase-space distributions. As the
$P$-function of the von Mises state $|n,\alpha\rangle$ takes the form

$\displaystyle
P^{|n,\alpha\rangle}(m,\beta)=\frac{1}{2\pi}\delta_{nm}\delta_{2\pi}(\alpha-\beta),$
(89)

one finds from the second equality of (VI) the kernel to be

$\displaystyle(\mathcal{F}o)(n,\alpha)=2\pi W^{|0,0\rangle}(n,\alpha),$ (90)

where $W^{|0,0\rangle}(n,\alpha)$ is the Wigner function of the von Mises
state $|0,0\rangle$. The Wigner function is given by a sum of two terms both
involving third Jacobi theta function [21] and we can combine it with the
formulas (VI) and (90) to calculate phase-space distributions for other basic
states of the investigated system. This programme as well as further
development of the phase-space methods introduced here is beyond the scope of
the present manuscript and will be addressed elsewhere.

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# Improving Contrastive Learning of Sentence Embeddings with Focal-InfoNCE

Pengyue Hou
University of Alberta
<EMAIL_ADDRESS>
Xingyu Li
University of Alberta
<EMAIL_ADDRESS>

###### Abstract

The recent success of SimCSE has greatly advanced state-of-the-art sentence
representations. However, the original formulation of SimCSE does not fully
exploit the potential of hard negative samples in contrastive learning. This
study introduces an unsupervised contrastive learning framework that combines
SimCSE with hard negative mining, aiming to enhance the quality of sentence
embeddings. The proposed focal-InfoNCE function introduces self-paced
modulation terms in the contrastive objective, downweighting the loss
associated with easy negatives and encouraging the model focusing on hard
negatives. Experimentation on various STS benchmarks shows that our method
improves sentence embeddings in terms of Spearman’s correlation and
representation alignment and uniformity. Our code is available at:
https://github.com/puerrrr/Focal-InfoNCE.

##  1 Introduction

Unsupervised learning of sentence embeddings has been extensively explored in
natural language processing (NLP) (Cer et al., 2018; Giorgi et al., 2020; Yan
et al., 2021), aiming to generate meaningful representations of sentences
without the need for labeled data. Among various approaches, SimCSE (Gao et
al., 2021) achieves state-of-the-art performance in learning high-quality
sentence embeddings through contrastive learning. Due to its simplicity and
effectiveness, various efforts have been made to improve the contrastive
learning of sentence embeddings from different aspects, including alleviating
false negative pairs (Wu et al., 2021a; Zhou et al., 2022) and incorporating
more informative data augmentations (Wu et al., 2021b; Chuang et al., 2022).

Leveraging hard-negative samples in contrastive learning is of significance
(Schroff et al., 2015; Oh Song et al., 2016; Robinson et al., 2020).
Nevertheless, unsupervised contrastive learning approaches often face
challenges in hard sample mining. Specifically, the original training paradigm
of unsupervised-SimCSE proposes to use contradiction sentences as "negatives".
But such implementation only guarantees that the contradiction sentences are
"true negatives" but not necessarily hard. With a large number of easy
negative samples, the contribution of hard negatives is thus prone to being
overwhelmed,

To address this issue, we propose a novel loss function, namely Focal-InfoNCE,
in the paradigm of unsupervised SimCES for sentence embedding. Inspired by the
focal loss (Lin et al., 2017), the proposed Focal-InfoNCE loss assigns higher
weights to the harder negative samples in model training and reduces the
influence of easy negatives accordingly. By doing so, focal Info-NCE
encourages the model to focus more on challenging pairs, forcing it to learn
more discriminative sentence representations. In addition, to adapt the
dropout strategy for positive pair construction in SimCSE, we further
incorporate a positive modulation term in the contrastive objective, which
reweights the positive pairs in model optimization. We conduct extensive
experiments on various STS benchmark datasets (Agirre et al., 2012, 2013,
2014, 2015, 2016; Cer et al., 2017; Marelli et al., 2014) to evaluate the
effectiveness of Focal Info-NCE. Our results demonstrate that Focal Info-NCE
significantly improves the quality of sentence embeddings and outperforms
unsupervised-SimCSE by an average of 1.64%, 0.82%, 1.51%, and 0.75% Spearan’s
correlation on BERT-base, BERT-large, RoBERTa-base, and RoBERTa-large,
respectively.

##  2 Related Work

###  2.1 Unsupervised SimCSE

SimCSE (Gao et al., 2021) provides an unsupervised contrastive learning
solution to SOTA performance in sentence embedding. Following previous work
(Chen et al., 2020), it optimizes a pre-trained model with the cross-entropy
objective using in-batch negatives. Formally, given a mini-batch of $N$
sentences, $\\{x_{i}\\}^{N}_{i=1}$, let $h_{i}$ be the sentence representation
of $x_{i}$ with the pre-trained language model such as BERT (Devlin et al.,
2018) or RoBERTa (Liu et al., 2019). SimCSE’s training objective, InfoNCE, can
be formulated as

$l_{i}=-log\frac{e^{sim(h_{i},h_{i}^{+})/\tau}}{\sum^{N}_{j=1}e^{sim(h_{i},h_{j})/\tau}},$
(1)

where $\tau$ is a temperature hyperparameter and $sim(h_{i},h_{j})$ represents
the cosine similarity between sentence pairs $(x_{i},x_{j})$. Note,
$h_{i}^{+}$ is the representation of an augmented version of $x_{i}$, which
constitutes the positive pair of $x_{i}$. For notation simplification, we will
use $s_{p}^{i}$ and $s_{n}^{i,j}$ to represent the similarities between
positive pairs and negative pairs in this paper.

Unsupervised SimCSE uses model’s built-in dropout as the minimal "data
augmentation" and passes the same sentence to the same encoder twice to obtain
two sentence embeddings as positive pairs. Any two sentences within a mini-
batch form a negative pair. It should be noted that in contrastive learning,
model optimizaiton with hard negative samples helps learn better
representations. But SimCSE doesn’t distinguish hard negatives and easy ones.
We show in this work that incorporating hard negative sample mining in SimCSE
boosts the quality of sentence embedding.

###  2.2 Sample Re-weighting in Machine Learning

Re-weighting is a simple yet effective strategy for addressing biases in
machine learning. It down-weights the loss from majority classes and obtains a
balanced learning solution for minority groups. Re-weighting is also a common
technique for hard example mining in deep metric learning Schroff et al.
(2015) and contrastive learning Chen et al. (2020); Khosla et al. (2020).
Recently, self-paced re-weighting is explored in various tasks, such as object
detection(Lin et al., 2017), person re-identification(Sun et al., 2020), and
adversarial training Hou et al. (2023). It re-weights the loss of each sample
adaptively according to model’s optimization status and encourages a model to
focus on learning hard cases. To the best of our knowledge, this study
constitutes the first attempt to incorporate self-paced re-weighting strategy
in unsupervised sentence embedding.

##  3 Focal-InfoNCE for Sentence Embedding

This study follows the unsupervised SimCSE framework for sentence embedding.
Instead of taking the InfoNCE loss in Eq. (1), we introduce a self-paced
reweighting objective function, Focal-InfoNCE, to up-weight hard negative
samples in contrastive learning. Specifically, for each sentence $x_{i}$,
Focal-infoNCE is formulated as

$l_{i}=-log\frac{e^{(s_{p}^{i})^{2}/\tau}}{\sum^{N}_{j\not=i}e^{s_{n}^{i,j}(s_{n}^{i,j}+m)/\tau}+e^{(s_{p}^{i})^{2}/\tau}},$
(2)

where $m$ is a hardness-aware hyperparameter that offers flexibility in
adjusting the re-weighting strategy. Within a mini-batch of $N$ sentences, the
final loss function, $L=\sum_{i=1}^{N}l_{i}$, can be derived as

$L=-log\frac{e^{\sum_{i=1}^{N}(s^{i}_{p})^{2}/\tau}}{\Pi_{i=1}^{N}[\sum^{N}_{j\not=i}e^{s_{n}^{i,j}(s_{n}^{i,j}+m)/\tau}+e^{(s_{p}^{i})^{2}/\tau}]}$
(3)

Analysis of Focal-InfoNCE: Compare with InfoNCE in Eq. (1), Focal-InfoNCE
introduces self-paced modulation terms on $s_{p}$ and $s_{n}$, proportional to
the similarity quantification. Let’s first focus on the modulation term,
$s_{n}^{i,j}+m$, on negative pairs. Prior arts have shown that pre-trained
language models usually suffer from anisotropy in sentence embedding Wang and
Isola (2020). Finetuning the pretrained models with contrastive learning on
negative samples, especially hard negative samples, improves uniformity of
representations, mitigating the anisotropy issue. In SimCSE, $s_{n}^{i,j}$
quantifies the similarity between negatives $x_{i}$ and $x_{j}$. If
$s_{n}^{i,j}$ is large, $x_{i}$ and $x_{j}$ are hard negatives for current
model. Improving the model with such hard negative pairs encourage
representation’s uniformity. To this end, we propose to upweight the
corresponding term $s_{n}^{i,j}/\tau$ by a modulation factor $s_{n}^{i,j}+m$.
The partial derivative of Focal-InforNCE with respect to $s_{n}^{i,j}$ is

$\frac{\partial L}{\partial
s_{n}^{i,j}}=\sum^{N}_{j\not=i}\frac{2}{\tau}\frac{e^{s_{n}^{i,j}(s_{n}^{i,j}+m)/\tau}}{Z_{i}}(s_{n}^{i,j}+m),$
(4)

where
$Z_{i}=\sum^{N}_{j\not=i}e^{s_{n}^{i,j}(s_{n}^{i,j}+m)/\tau}+e^{(s_{p}^{i})^{2}/\tau}$.
According to Eq. (4), comparing to easy negatives, hard negative samples that
associates with higher similarity score $s_{n}^{i,j}$ contribute more to the
loss function. This implies that a model optimized with the proposed Focal-
InfoNCE focuses more on hard-negative samples. Our experiments also show that
Focal-InfoNCE improves uniformity in sentence embeddings.

To uncover the insight of the modulation term $s_{p}^{i}$ on positive cases,
let’s revisit SimCSE. In SimCSE, the positive pair is formed by dropout with
random masking. Thus a low similarity score $s_{p}$ indicates semantic
information loss introduced by dropout. Since such a low similarity is not
attributed to model’s representation capability, we should mitigate its effect
on model optimization. Hence, Focal-inforNCE assigns a small weight to the
dissimilary positive pair. The partial derivative with respect to
$s_{n}^{i,j}$ is

$\frac{\partial L}{\partial
s_{p}^{i}}=\frac{2}{\tau}(\frac{e^{(s_{p}^{i})^{2}/\tau}}{Z_{i}}-1)s_{p}^{i},$
(5)

which suggests that positive pairs with lower similarity scores in SimCSE
contributes less to model optimization. We show in the experiments that Focal-
InfoNCE improves the allignment of sentense embeddings as well.

Due to the modulation terms on both positive and negative samples, Focal-
InfoNCE reduces the chances of the model getting stuck in sub-optimal
solutions dominated by easy pairs. We show in our experiment that the proposed
Focal-InfoNCE can easily fit into most contrastive training frameworks for
sentence embeddings.

##  4 Experiments

We evaluate Focal-InfoNCE on 7 semantic similarity tasks: STS 12-16 (Agirre et
al., 2012, 2013, 2014, 2015, 2016), STS Benchmark (Cer et al., 2017) and SICK-
Relatedness (Marelli et al., 2014). Spearman’s correlation between predicted
and ground truth scores is used as the numerical performance metric. Our
implementation closely follows unsupervised-SimCSE (Gao et al., 2021).
Briefly, starting with pre-trained models $\mbox{BERT}_{base}$,
$\mbox{BERT}_{large}$ (Devlin et al., 2018) and $\mbox{RoBERTa}_{base}$, and
$\mbox{RoBERTa}_{large}$ (Liu et al., 2019), we take the $[cls]$ embeddings as
the sentence representations and fine-tune the models with $10^{6}$ randomly
sampled English Wikipedia sentences. The hyperparameter $\tau$ for the four
models are {0.7, 0.7, 0.5, 0.5} respectively and we use $m=0.3$ in this
experiment. To make a fair comparison, we adopted the same batch size and
learning rate as unsupervised-SimCSE, shown in Table 1.

| BERT | RoBERTa
---|---|---
| base | large | base | large
batch size | 64 | 64 | 512 | 512
learning rate | 3e-5 | 1e-5 | 1e-5 | 3e-5
Table 1: Experimental setting for our main results.

Model STS12 STS13 STS14 STS15 STS16 STS-B SICK-R Avg. GloVe embeddings (avg.)♣
55.14 70.66 59.73 68.25 63.66 58.02 53.76 61.32 $\mbox{BERT}_{base}$(first-
last avg.)* 39.70 59.38 49.67 66.03 66.19 53.87 62.06 56.70
$\mbox{BERT}_{base}$-flow* 58.40 67.10 60.85 75.16 71.22 68.66 64.47 66.55
$\mbox{BERT}_{base}$-whitening* 57.83 66.90 60.90 75.08 71.31 68.24 63.73
66.28 $\mbox{SimCSE-BERT}_{base}$ 67.33 82.45 72.30 80.76 79.04 76.38 71.58
75.68 \+ Focal-InfoNCE $\textbf{68.50}_{\pm 0.08}$ $\textbf{83.70}_{\pm 0.50}$
$\textbf{79.00}_{\pm 0.15}$ $\textbf{82.71}_{\pm 0.23}$ $\textbf{79.43}_{\pm
0.40}$ $\textbf{78.85}_{\pm 0.37}$ $\textbf{72.99}_{\pm 0.10}$
$\textbf{77.33}_{\pm 0.08}$ $\mbox{SimCSE-BERT}_{large}$ 70.31 84.73 75.52
83.06 78.90 78.20 74.56 77.90 \+ Focal-InfoNCE $\textbf{71.86}_{\pm 1.55}$
$\textbf{84.50}_{\pm 0.20}$ $\textbf{75.81}_{\pm 0.49}$ $\textbf{84.56}_{\pm
0.62}$ $\textbf{79.13}_{\pm 0.97}$ $\textbf{80.97}_{\pm 1.00}$ $72.74_{\pm
0.45}$ $\textbf{78.51}_{\pm 0.28}$ $\mbox{RoBERTa}_{base}$(first-last avg.)*
40.88 58.74 49.07 65.63 61.48 58.55 61.63 56.57
$\mbox{RoBERTa}_{base}$-whitening* 46.99 63.24 57.23 71.36 68.99 61.36 62.91
61.73 DeCLUTR-$\mbox{RoBERTa}_{base}$* 52.41 75.19 65.52 77.12 78.63 72.41
68.62 69.99 $\mbox{SimCSE-RoBERTa}_{base}$ 67.54 81.39 72.82 81.61 80.29 80.00
68.86 76.07 \+ Focal-InfoNCE $\textbf{70.84}_{\pm 0.59}$ $\textbf{82.52}_{\pm
0.48}$ $\textbf{73.99}_{\pm 0.85}$ $\textbf{83.02}_{\pm 0.99}$
$\textbf{82.26}_{\pm 0.45}$ $\textbf{81.16}_{\pm 0.50}$ $\textbf{69.43}_{\pm
0.77}$ $\textbf{77.60}_{\pm 0.26}$ $\mbox{SimCSE-RoBERTa}_{large}$ 71.54 83.11
75.04 84.20 80.54 81.52 69.99 77.99 \+ Focal-InfoNCE $71.45_{\pm 1.10}$
$\textbf{83.24}_{\pm 0.88}$ $\textbf{74.48}_{\pm 1.07}$ $\textbf{85.10}_{\pm
0.35}$ $\textbf{82.54}_{\pm 1.06}$ $\textbf{82.24}_{\pm 0.67}$
$\textbf{72.56}_{\pm 0.96}$ $\textbf{78.80}_{\pm 0.04}$

Table 2: Sentence embedding performance on STS tasks with Spearman’s
correlation. ♣: results from Reimers and Gurevych (2019); *: results from Gao
et al. (2021).

###  4.1 Comparison to Prior Arts

Table 2 shows the performance of the different models with and without the
Focal-InfoNCE loss. In general, we observe improvements in Spearman’s
correlation scores when incorporating the proposed Focal-InfoNCE. For example,
with $\mbox{SimCSE-BERT}_{base}$, the average score increases from 75.68 to
77.32 when using Focal-InfoNCE.

###  4.2 Alignment and Uniformity

Alignment and uniformity are two key properties to measuring the quality of
contrastive representations Gao et al. (2021). By specifically focusing on
challenging negative samples, focal-InfoNCE encourages the model to pay closer
attention to negative instances that are difficult to distinguish from
positive pairs. In Table. 3, we incorporate the proposed focal-InfoNCE into
different contrastive learning for sentence embeddings and show improvements
in both alignment and uniformity of the resulting representations.

Model | ALGN $\downarrow$ | UNIF $\downarrow$ | SpCorr $\uparrow$
---|---|---|---
SimCSE | 0.190 | -2.400 | 76.38
+Focal InfoNCE | 0.134 | -1.906 | 79.31
DiffCSE | 0.081 | -1.195 | 79.53
+Focal InfoNCE | 0.070 | -1.335 | 80.09
DCLR | 0.231 | -2.790 | 78.31
+Focal InfoNCE | 0.164 | -2.366 | 79.30
Table 3: Representation alignment (ALGN), uniformity (UNIF), and Spearman’s
correlation (SpCorr) with $\mbox{BERT}_{base}$ on the STS Benchmark (Cer et
al., 2017).

###  4.3 Ablation Studies on Hyperparameters

We conducted ablation studies to analyze two key factors in Focal-InfoNCE:
temperature $\tau$ and the hardness hyperparameter $m$.

Temperature $\tau$ is a hyper-parameter used in the InfoNCE loss function that
scales the logits before computing the probabilities. Wang and Liu (2021) show
that the temperature plays a key role in controlling the strength of penalties
on hard negative samples. In this ablation, we set the temperature as 0.03,
0.05, 0.07, and 0.1, explore the effect of different temperature values on the
model’s performance and report the results in Table 4.

| BERT | RoBERTa
---|---|---
$\tau$ | base | large | base | large
0.03 | 74.66 | 76.96 | 80.62 | 77.55
0.05 | 78.98 | 79.11 | 81.72 | 82.46
0.07 | 79.31 | 80.43 | 81.00 | 82.03
0.1 | 77.33 | 73.23 | 79.99 | 81.09
Table 4: Effects of temperature $\tau$ on STS Benchmark (Cer et al., 2017) with Spearman’s correlation. | $\mbox{BERT}_{base}$ | $\mbox{RoBERTa}_{base}$
---|---|---
$m$ | STS-B | SICK-R | STS-B | SICK-R
0.1 | 77.23 | 71.30 | 80.33 | 68.86
0.2 | 79.37 | 71.70 | 81.42 | 69.60
0.3 | 79.31 | 72.72 | 81.72 | 69.52
0.4 | 78.99 | 71.75 | 80.07 | 64.21
Table 5: Effects of hyperparameter $m$ on STS Benchmark (Cer et al., 2017)
with Spearman’s correlation. Figure 1: Plot of re-scaled negative pairs vs.
their original value with different choices of $m$. Orange: up-weighted
negative examples; green: down-weighted negative examples

The hardness hyper-parameter $m$ controls the rescaling of the negative
samples in the contrastive loss function. Figure. 1 visualizes the rescaling
effects of $m$. Specifically, our Focal-InfoNCE loss regards negative pairs
with cosine similarity larger than (1-$m$) as hard negative examples and vice
versa. The loss is then up-weighted or down-weighted proportionally. Table. 5
demonstrates that our method is not sensitive to $m$ and the optimal setting
can usually be found between 0.2 and 0.3. One possible explanation for this
could be the simplicity of the data augmentation employed by SimCSE, resulting
in a consistent positive similarity score of approximately 0.85 (Wu et al.,
2021a).

##  5 Discussion

###  5.1 Qualititave Analysis

We conducted a qualitative analysis as follows. We sampled a subset of hard
negative pairs from the STS-B training data and analyzed their
characteristics. We noticed that these hard negative pairs often involved
sentences with domain-specific terms, or sentences with a high degree of
syntactic similarity. For example, sentences like "This is a very unusual
request." and "I think it’s fine to ask this question." have a notable cosine
similarity of 0.64 by the pre-trained BERT base model, though their underlying
tones and attitudes are completely different. The proposed Focal InfoNCE
effectively guides the model’s attention by putting more penalties toward
these cases and encourages the model to learn a better sentence embedding to
separate them.

###  5.2 Statistical Significance Analysis

To validate the statistical significance of performance increase, we conducted
three more sets of experiments, each with different random seed, and reported
the mean and standard deviation (std) in Table 2. Based on the results, we
calculated paired-t tests with a standard significance level of 0.05. All
p-values over the various STS tasks with different base models are smaller
than 0.05, which indicate that our focal-InfoNCE improves the performance
significantly in the statistical sense.

###  5.3 Compatibility with Existing Methods

As reported in Table 3, focal-InfoNCE improves DiffCSE in terms of
representation alignment, uniformity, and Spearman’s correlation. In contrast,
our attempts to integrate focal-InfoNCE with PromptBERTJiang et al. (2022) did
not yield any improvements over the baseline models. We hypothesize that this
outcome could be attributed to the different treatments of noise in positive
pairs. In SimCSE and DiffCSEChuang et al. (2022), positive pairs are generated
with the same template but different dropout noise. Our focal-InfoNCE
downweights the false positive pairs introduced by the random masking
mechanism. But in PromptBERT where different positive templates are used for
contrastive learning, the template biases have been incorporated in the
contrastive loss. Consequently, downweighting false positive pairs with focal-
InfoNCE might be unnecessary on PromptBERT.

##  6 Conclusions

This paper introduced a novel unsupervised contrastive learning objective
function, Focal-InfoNCE, to enhance the quality of sentence embeddings. By
combining SimCSE with self-paced hard negative re-weighting, model
optimization was benefited from hard negatives. Extensive experiments shows
the effectiveness of the proposed method on various STS benchmarks.

##  7 Limitations

The effectiveness of the Focal-InfoNCE depends on the quality of pre-trained
models. When a pre-trained model leads to bad representations, the similarity
scores may mislead model finetuing. In addition, the positive re-weighting
strategy in this study is quite simple. We believe that more sophisticated
mechanisms to address semantic information loss in positive pairs would
further improve the performance.

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