# The operational reality of quantum nonlocality

R. Srikanth  Theoretical Sciences Division, Poornaprajna Institute of
Scientific Research (PPISR),
Bidalur post, Devanahalli, Bengaluru 562164, India

###### Abstract

Does the remote measurement-disturbance of the quantum state of a system $B$
by measurement on system $A$ entangled with $B$, constitute a real disturbance
– i.e., an objective alteration– of $B$ in an operational sense? Employing
information theoretic criteria motivated by operational considerations alone,
we argue that the disturbance in question is real for a subset of steerable
correlations. This result highlights the distinction between quantum no-
signaling and relativistic signal-locality. It furthermore suggests a natural
reason why a convex operational theory should be non-signaling: namely, to
ensure the consistency between the properties of reduced systems and those of
single systems.

##  I Introduction

A basic phenomenon underlying important quantum information processing tasks
such as remote state preparation Pati (2000), quantum teleportation Hu  _et
al._ (2020), device-independent (DI) certification of randomness Pironio  _et
al._ (2010) and DI cryptography Vazirani and Vidick (2019), is the remote
“collapse” or reduction of the quantum state of an entangled system $B$ by
means of a local measurement on its partner system $A$. This phenomenon, which
famously figures in the Einstein, Podolsky and Rosen (EPR) paradox Einstein
_et al._ (1935), lies at the heart of quantum nonlocality Brunner  _et al._
(2014). As it happens, the basic nature of this remote measurement-disturbance
of system $B$’s state– in particular, the issue of whether the disturbance
constitutes an objective change of $B$ in an operational sense, or merely a
subjective update to the observer’s knowledge of $B$– has remained moot in
quantum mechanics (QM).

This state of affairs is part of the broader question whether the quantum
state real, and is tied to the fact that the interpretation of QM is still not
universally agreed upon Leifer (2014); Cabello (2017). Certain interpretations
of QM are consistent with the idea that the quantum state is real Cushing  _et
al._ (2013); Beltrametti and Bugajski (1995); Penrose (1996); Pusey  _et al._
(2012); Bassi  _et al._ (2013); Patel and Kumar (2017); Ghirardi and Bassi
(2020), while others that are inspired by the Copenhagen interpretation of QM
Faye (2002) are consistent with the idea that the quantum state only
represents our knowledge about measurement outcomes or underlying ontic
variables Fuchs  _et al._ (2014); Peres (2006); Brukner and Zeilinger (2003);
Spekkens (2007); Harrigan and Spekkens (2010). Correspondingly, the
wavefunction collapse takes on an ontic sense, or not, in these
interpretations.

Bell’s theorem Brunner  _et al._ (2014) itself can’t help here. Its
operational significance is not an indication of the reality of this remote
disturbance, but rather a complementarity between signaling and
unpredictability Cavalcanti and Wiseman (2012); Hall (2010); Kar  _et al._
(2011); Aravinda and Srikanth (2015, 2016). This leads, via the assumption of
no-signaling, to the possibility of device-independent randomness generation
Herrero-Collantes and Garcia-Escartin (2017) and key distribution Vazirani and
Vidick (2019), but doesn’t give any guidance on addressing the foundational
issue raised above.

These considerations prompt the question of whether a relevant concept of
reality can be indicated that is independent of the physical interpretation or
mathematical representation of QM. Here, we wish to show that this can be
achieved based only on basic operational considerations about measurement-
induced disturbance. Here it will be convenient to use the framework of
generalized probability theories (GPTs) Hardy (2001); Barrett (2007); Janotta
and Hinrichsen (2014); Chiribella and Scandolo (2015), which will make it
clear that the concept of reality developed here is purely of operational
origin and independent of the Hilbert space formalism. Our point of departure
is the intuitive idea that if a measurement-disturbance can be the basis for
communication, then it constitutes an objective fact and is thus real.
Requiring only operational criteria and not extraneous paraphernalia such as
hidden variables or parallel worlds, this approach promises to provide an
intrinsic interpretation of QM.

The remaining article is arranged as follows. Starting with a brief note on
the concept of “operational”, we define the concept of operational reality of
local measurement disturbance in Section II. This is then extended in Section
III to the case of remote measurement-disturbance, which is shown in Section
IV to be related to Einstein-Podolsky-Rosen (EPR) steering. Building on this,
a device-independent approach to indicate remote measurement-disturbance is
discussed. Finally, we present our conclusions and related discussions in
Section V. Here, we argue that the case for the operationally real nonlocality
highlights the tension between quantum no-signaling and relativistic signal
locality.

##  II Measurement disturbance and uncertainty in single systems

The operational formulation of a physical theory (such as quantum mechanics)
consists in an abstract characterization of the theory as a GPT aka convex
operational theory, i.e., in terms of rules governing preparation procedures,
probabilities for measurement outcomes and reversible operations, while
avoiding concepts that cannot be accessed directly, such as the Hilbert space,
complex global phase, etc. Janotta and Hinrichsen (2014); Barnum and Wilce
(2011). In other words, the operational formulation corresponds to the basic
“syntax” of a theory, devoid of the “semantics” pertaining to its physical
interpretation or mathematical representation.

An operational feature of a theory is one that can be defined as an element of
its GPT formulation. For example, a state $\varphi$ is operationally
understood as an equivalence class of preparation procedures. Given a
bipartite state $\varphi_{AB}$, the marginal state of system $A$, denoted
$\varphi_{A}^{\\#}$, is the GPT analogue of the reduced density operator
obtained via partial tracing over $B$. An entangled state in a GPT corresponds
to a valid composite state that cannot be expressed as a convex combination of
product states.

In practice, we would require a detailed characterization of the subsystems in
order to indicate a given joint state in a GPT is entangled. A device-
independent formulation represents a further level of abstraction where even
such a characterization is not required, as the input-output statistics can
self-test, typically based on the violation of Bell-type inequalities Clauser
_et al._ (1969); Brunner  _et al._ (2014).

A basic feature of QM, and indeed of any nonclassical GPT, is that the act of
measurement can disturb– i.e., randomly alter– the measured state of the given
system, even in the case of a pure state. This phenomenon is ultimately due to
the non-simpliciality of the relevant state space of the theory Aravinda  _et
al._ (2017).

######  Definition 1.

Suppose measurement $x$ is performed on a quantum or GPT system $A$ prepared
in state $\varphi_{A}$, producing outcome $a$, with probability
$\mathfrak{p}(a|x,\varphi)$. Let the normalized post-measurement state be
denoted $\varphi_{A}^{a|x}$. The state change

$\varphi_{A}\longrightarrow\varphi_{A}^{a|x},$ (1)

is called the measurement(-induced) disturbance of the initial state
$\varphi_{A}$. $\blacklozenge$

Here the measurement used, including the state update rule, may be considered
as the GPT analogue of the Lüders instrument in the context of quantum
measurement Busch  _et al._ (1996). As it happens, no experiment to date has
succeeded in observing such disturbance of a quantum state, called a collapse
or state reduction in this context. The quantum measurement problem is
concerned with the question of whether state reduction happens objectively.
Revisiting this issue, we ask: Is the measurement disturbance represented in
Eq. (1) real? That is, does the measurement objectively alter system $A$, or
is measurement disturbance only epistemic, i.e., one that only updates the
observer’s knowledge about a pre-existing property of $A$, without objectively
altering $A$?

Given a GPT, consider a communication protocol $\mathfrak{P}_{1}$ between
Alice and Bob, wherein Alice transmits the non-classical system $A$ either in
the state $\varphi_{A}$ or in the post-selected state $\varphi_{A}^{a|x}$.
(She measures $x$ and discards states $\varphi_{A}^{a^{\prime}|x}$ for which
$a^{\prime}\neq a$). Importantly, she doesn’t send any supplementary classical
information to Bob. By performing on $A$ a measurement $x^{\prime}$
incompatible with $x$ Busch  _et al._ (2013), Bob tries to determine which out
of $\varphi_{A}$ and $\varphi_{A}^{a|x}$ Alice transmitted. If he succeeds
with a probability $p$ better than a random guess’s (i.e., $p>\frac{1}{2})$,
then it is quite natural to infer that Alice’s measurement objectively
disturbed system $A$. Letting $\mathfrak{S}_{A\rightarrow A}$ denote the
amount of information (in bits) that Alice can communicate on average to Bob
using protocol $\mathfrak{P}_{1}$, we have the following intuitive,
operational concept of the reality of Alice’s measurement disturbance,
applicable to any GPT:

######  Definition 2.

Given state $\varphi_{A}$ and measurement $x$, if it is the case in protocol
$\mathfrak{P}_{1}$ that for some outcome $a$

$\mathfrak{S}_{A\rightarrow A}>0,$ (2)

then the measurement disturbance Eq. (1) constitutes an operationally real
disturbance of $A$. $\hfill\blacklozenge$

Note that the concept of reality in Definition 2 avoids any reference to a
“hidden variable” (HV) ontology. In a practical implementation of protocol
$\mathfrak{P}_{1}$, it is not necessary to post-select on a specific outcome
$a$. Let
$\mathcal{M}_{x}(\varphi_{A})\equiv\sum_{a}\mathfrak{p}(a|x)\varphi_{A}^{a|x}$
represent the non-selective state obtained by ignoring the outcome. Alice can
use $\varphi_{A}$ and $\mathcal{M}_{x}(\varphi_{A})$ as symbols for
communicating. If this works, then obviously
$\|\varphi_{A}-\mathcal{M}_{x}(\varphi_{A})\|>0$, and by convexity,
$\exists_{a}\|\varphi_{A}-\varphi^{a|x}_{A}\|>0$, from which Eq. (2) follows,
and reality is inferred via Definition 2.

As an illustration in QM, suppose Alice chooses to perform or not to perform
measurement $\sigma_{X}$ on qubit $A$ initialized in the state
$\varphi_{A}=\cos^{2}(\theta/2)\ket{0}\bra{0}+\sin^{2}(\theta/2)\ket{1}\bra{1}$,
with $0\leq\theta\leq\frac{\pi}{2}$ and $\theta\neq\pi/4$. She communicates
with Bob by sending either the state $\varphi_{A}$ or
$\mathcal{M}_{\sigma_{X}}(\varphi_{A}):=\frac{\mathbb{I}}{2}$. Since these two
states are probabilistically distinguishable, it follows that
$\mathfrak{S}_{A\rightarrow A}>0$. Thus, we infer that Alice’s measurement
really disturbs the system in the operational sense.

If $A$ is not an isolated single system, but entangled with another system
$B$, then the above criterion may not be applicable. For example, suppose
Alice and Bob share the bipartite state
$\ket{\Psi(\theta)}_{AB}=\cos(\theta)\ket{00}_{AB}+\sin(\theta)\ket{11}_{AB},~{}~{}~{}\theta\in[0,\frac{\pi}{2}].$
Alice measures $A$ in the basis $\sigma_{Z}$, and sends $A$ to Bob. As
$\mathfrak{S}_{A\rightarrow A}=0$ here, thus Definition 2 is unable to
indicate the reality of $A$’s measurement disturbance.

In this situation, an entanglement-assisted version of protocol
$\mathfrak{P}_{1}$ can be used, which we call $\mathfrak{P}_{2}$. Here, Alice
and Bob share the entangled state $\varphi_{AB}$, with $A$ being initially
with Alice and $B$ with Bob. As in $\mathfrak{P}_{1}$, Alice sends $A$ to Bob
after measuring $x_{0}$ on it. Bob performs a joint measurement on $A$ and $B$
that is incompatible Busch and Lahti (1984); Busch  _et al._ (2013) with
$x_{0}$ (see Sec. A). If he can (probabilistically) determine whether or not
Alice measured $x_{0}$, he infers that her measurement really disturbed $A$ in
an operational sense. We denote by $\mathfrak{S}_{A\rightarrow AB}$ the amount
of information (in bits) about Alice’s measurement choice that can be
communicated to Bob in this way. In place of Definition 2, we have the
following operational criterion (applicable to any sufficiently rich GPT):

######  Definition 3.

Given bipartite state $\varphi_{AB}$ and local measurement $x$ on $A$ in
protocol $\mathfrak{P}_{2}$, if it is the case that

$\mathfrak{S}_{A\rightarrow AB}>0,$ (3)

then the measurement disturbance Eq. (1) constitutes an operationally real
disturbance of $A$. $\hfill\blacklozenge$

In the context of the above example with state $\ket{\Psi(\theta)}_{AB}$, the
post-measurement bipartite state with Bob is distinguishable from the initial
bipartite state, i.e.,
$\rho_{AB}^{(Z)}\equiv\cos^{2}(\theta)\ket{00}_{AB}\bra{00}+\sin^{2}(\theta)\ket{11}_{AB}\bra{11}\neq\ket{\Psi(\theta)}_{AB}\bra{\Psi(\theta)}$,
implying the satisfaction of Eq. (3). Accordingly, Bob infers the operational
reality of the measurement disturbance of $A$ per Definition 3. Analogous
situations of disturbance can be pointed out for example in the case of
Spekkens’ toy theory Spekkens (2007) and the box world Oppenheim and Wehner
(2010).

##  III Remote measurement-disturbance in a bipartite system

Suppose Alice and Bob share the quantum state $\ket{\Psi(\theta)}$. Alice
measures $\sigma_{X}$ on qubit $A$ obtaining outcome
$\ket{\pm}\equiv\frac{1}{\sqrt{2}}(\ket{0}\pm\ket{1})_{A}$. Correspondingly,
Bob’s particle $B$ collapses to the state
$\ket{\theta^{\pm}}\equiv\cos(\theta)\ket{0}_{B}\pm\sin(\theta)\ket{1}_{A}$.
Does this remote collapse constitute a real disturbance of system $B$?

More generally, in a GPT context, suppose Alice and Bob share an entangled
state $\varphi_{AB}=\sum_{\lambda}p_{\lambda}\varphi_{AB}^{\lambda}$,
$p(\lambda)$ being a probability distribution, where Bob’s marginal (or,
reduced) state is denoted $\varphi_{B}^{\\#}$. Alice’s measurement of $x$ on
$A$ conditioned on her obtaining outcome $a$ leaves Bob’s system in the
unnormalized (indicated by a tilde) state:

$\tilde{\varphi}^{a|x}_{B}=\sum_{\lambda}p_{\lambda}\mathfrak{p}(a|x,\lambda)\varphi_{B}^{a|x,\lambda}.$
(4)

Denote its normalized version by
$\varphi_{B}^{a|x}\equiv\mathcal{N}\tilde{\varphi}^{a|x}_{B}$ where
$\mathcal{N}\equiv[\sum_{\lambda}p(\lambda)\mathfrak{p}(a|x,\lambda)]^{-1}$.
Does the state change

$\varphi_{B}^{\\#}\longrightarrow\varphi_{B}^{a|x},$ (5)

which is the remote analogue of Eq. (1), constitute a real disturbance of $B$
in an operational sense? That is, can one advance a purely operational
argument in support of the claim that Alice’s remote measurement-disturbance
of system $B$ constitutes an objective change of $B$?

An appeal here to the direct analogue of Definition 2 is obviously ruled out
by virtue of no-signaling, which entails that $\mathfrak{S}_{A\rightarrow
B}=0$. Furthermore, in an instance where Eq. (3) holds true, the signal would
be attributed to the (local) disturbance of $A$ rather than to the (remote)
disturbance of $B$. Thus, a simple adaptation of Definition 3 is also ruled
out. We now present an indirect, operational criterion to indicate the reality
of a remote disturbance.

We will require a specific feature of our ontological formalism, which is that
it should satisfy a reasonable consistency principle in assigning reality to
the measurement disturbances of multiple particles in a given measurement
situation. In particular, given a pure joint state $\varphi_{AB}$, if the
disturbances to $A$ and $B$ are identical (to each other) when either particle
is measured, then the operational reality status of the two disturbances must
also be identical. This notion of consistency is natural since holding the
disturbance of one of the particles to be real, but not that of the other
particle that is identically disturbed in the same measurement situation,
would make this operationally inspired ontological system somewhat incoherent.
Formally:

######  Feature 1.

Suppose $\varphi_{AB}$ is a pure state such that the disturbances of $A$ and
$B$ are identical (to each other) under measurement of $x$ on $A$. That is,
the marginal states of $A$ and $B$ in $\varphi_{AB}$ are identical:

$\varphi_{A}^{\\#}=\varphi_{B}^{\\#},$ (6)

and furthermore, their respective post-measurement states are also identical:

$\varphi_{A}^{a|x}=\varphi_{B}^{a|x}.$ (7)

Then, consistency requires that any attribution of operational reality to the
disturbances of $A$ and $B$ should be identical, i.e., either both
disturbances are deemed operationally real, or both are deemed not.
$\hfill\blacklozenge$

Here we note that in QM, any pure bipartite state has the symmetric property
of Eq. (6). Therefore, to fulfill the conditions of Feature 1, it suffices to
choose a suitable measurement satisfying the symmetry property Eq. (7).
Feature 1 is a means for (indirectly) addressing the question of the reality
of $B$’s remote measurement-disturbance Eq. (5), given the constraint of no-
signaling. This argument may be formalized as follows:

######  Theorem 1.

Given pure state $\varphi_{AB}$, and Alice’s measurement $x_{0}$ on system
$A$, if the (local) measurement-disturbance of $A$ is: (a) operationally real
per the criterion of Definition 3, and: (b) is identical to the (remote)
measurement-disturbance of system $B$, then the latter disturbance is
operationally real.

Intuitively, the idea here is that if the correlations between $A$ and $B$ are
local, then Alice’s measurement should operationally disturb only $A$, but not
$B$. In particular, non-trivial disturbances of $A$ and $B$ couldn’t be
identical. But Theorem 1 gives conditions under which $A$ is operationally
disturbed and yet this identicality holds, thereby negating the locality
supposition. The proof below of Theorem 1 is essentially a formal elaboration
of this idea.

###### Proof.

Suppose that locality holds on the operational level, meaning that Alice’s
measurement on $A$ doesn’t disturb the state of system $B$ in an operational
sense. More precisely, there exists a pre-existing ensemble of states on Bob’s
side $\chi\equiv\\{q(\mu),\phi_{B}^{\mu}\\}$, such that the post-measurement
state of $B$ can be expressed as
$\tilde{\varphi}^{a|x}_{B}=\sum_{\mu}q(\mu)\mathfrak{q}(a|x,\mu)\phi_{B}^{\mu},$
where $q(\mu)$ and $\mathfrak{q}(a|x,\mu)$ are probability distributions. It
follows in view of the symmetry condition (b), specifically requirement Eq.
(7), that the joint state of the composite system $AB$ conditioned on Alice’s
measurement, is given by

$\tilde{\varphi}^{a|x}_{AB}=\sum_{\mu}q(\mu)\mathfrak{q}(a|x,\mu)\varphi_{A}^{\mu}\varphi_{B}^{\mu},$

such that $\varphi_{A}^{\mu}=\varphi_{B}^{\mu}$. This implies that, from Bob’s
perspective the non-selective joint state is given by

$\sum_{a}\tilde{\varphi}^{a|x}_{AB}=\sum_{\mu}q(\mu)\varphi_{A}^{\mu}\varphi_{B}^{\mu},$

which is independent of $x$, entailing that Alice’s choice $x$ could not be
deduced by Bob via his joint measurement of $AB$, contradicting condition (a).
Therefore, given the satisfaction of condition (a), it follows that particle
$B$ is necessarily disturbed from afar by Alice’s measurement in the sense
that no such local ensemble $\chi$ exists for Bob. In this case, the general
expression Eq. (4) should hold, where the specific outcome state
$\tilde{\varphi}^{a|x}_{AB}$ depends on $x$.

To show that this remote disturbance of $B$ is operationally real, we note
that by condition (b), the disturbance at $B$ is identical to that at $A$. By
Feature 1, the same reality status should be assigned to both these
disturbances. Therefore, the operational reality of $B$’s disturbance follows,
given that of $A$’s disturbance by virtue of fulfillment of condition (a). ∎

A state $\varphi_{AB}$ that admits such a remote measurement-disturbance in
the above sense may conveniently be called nonlocal in an operationally real
(OR) sense. The state $\ket{\Psi(\theta)}$ is evidently OR nonlocal according
to Theorem 1, with measurement $x_{0}\equiv\sigma_{Z}$.

In the standard Bell test and EPR steering scenarios, the particles $A$ and
$B$ are measured in geographically separated stations. By contrast, in the
present scenario, Alice sends $A$ to Bob such that he can subsequently perform
a joint measurement on the composite system $AB$. Specifically, condition (a)
above corresponds to the violation of a type of no-signaling-in-time (NSIT) by
particle $A$. In the context of temporal correlations, NSIT is a statistical
characterization of non-invasive measurability Kofler and Brukner (2013);
Clemente and Kofler (2016). Given that both particles are available at the
same place for the second measurement in the above scenario, it would be
desirable to elucidate the operational sense in which Theorem 1 establishes
the nonlocality of the correlations between $A$ and $B$.

To this end, we may consider an alternative, equivalent protocol that makes
this nonlocality explicit. We now make the important if somewhat
straightforward observation that conditions (a) and (b) can be guaranteed
through local operations and classical communication (LOCC), without requiring
Alice to transmit particle $A$ to Bob for his joint measurement. We note that
the initial state $\varphi_{AB}$ must be entangled in order for Alice to be
able to re-prepare $B$’s state and thereby disturb it. Alice’s measurement
disentangles $\varphi_{AB}$ into a product state
$\hat{\varphi}_{A}^{a|x}\otimes\tilde{\varphi}_{B}^{a|x}$, where
$\hat{\varphi}_{A}^{a|x}$ denotes the normalized outcome obtained by Alice on
$A$, and $\tilde{\varphi}_{B}^{a|x}$ is given by Eq. (4). Both the initial
state $\varphi_{AB}$ and the final product state can be ascertained by LOCC in
a GPT with the local tomographic property. Thus, the symmetry condition (b)
can be directly checked. Moreover, noting that
$\sum_{a}\hat{\varphi}_{A}^{a|x}\otimes\tilde{\varphi}_{B}^{a|x}\neq\varphi_{AB}$,
it follows that condition (a) can also be checked by LOCC.

The above considerations show that the conditions of Theorem 1, and thus the
reality of the remote disturbance, can be verified using only LOCC and without
any quantum or nonclassical communication from Alice to Bob, thereby bringing
the scenario of the OR nonlocality closer to that of Bell nonlocality or EPR
steering. As shown below, this allows OR nonlocality to be deduced by a
steering-like statistical inequality. Later, this will be shown to correspond
to a one-sided device-independent (DI) characterization of OR nonlocality.

Mixed states present a kind of “operational preparation-contextuality”,
somewhat reminiscent of HV-ontological preparation contextuality Spekkens
(2005). This is related to the fact that a nonclassical GPT, being
characterized in general by a non-simplicial state space, admits multiple
pure-state decompositions of a given mixed state $\varphi_{AB}$, whereas the
concept of reality will arguably depend on the actual decomposition that is
the case. Accordingly, with regard to indicating OR nonlocality, we have the
following criterion.

######  Definition 4.

Given state $\varphi_{AB}$ prepared by a known probabilistic procedure of
mixing an OR nonlocal component and a separable component, with probabilities
$p$ and $1-p$, respectively, the remote measurement-disturbance of $B$ in
state $\varphi_{AB}$ is OR nonlocal with probability $p$.
$\hfill\blacklozenge$

For example, suppose Charlie prepares state $\ket{\Psi(\theta)}$ with
probability $f$ and the four computational basis states
$\ket{00},\ket{01},\ket{10}$ and $\ket{11}$ with equal probability $(1-f)/4$.
He sends the first particle to Alice and the second to Bob. Accordingly, Alice
and Bob share the Werner-like state $W(f,\theta)\equiv
f\outerproduct{\Psi(\theta)}{\Psi(\theta)}+(1-f)\frac{\mathbb{I}_{4}}{4},$
with mixing parameter $f\in[0,1]$. Then Charlie can assert that the remote
measurement-disturbance of Bob’s particle is operationally real with
probability $f$, with the choice $x_{0}\equiv\sigma_{Z}$.

As well, the state $W(f,\theta)$ can be prepared by Charlie mixing the state
$\ket{\Psi(\theta+\frac{\pi}{2})}$ with probability $\frac{1-f}{2}$ and
$\ket{\Psi(\theta)}$ with probability $\frac{1+f}{2}$. Since both
$\ket{\Psi(\theta)}$ and $\ket{\Psi(\theta+\frac{\pi}{2})}$ are nonlocal in an
operationally real sense, Charlie asserts that given the state $W(f,\theta)$
under this preparation, the remote measurement-disturbance of Bob’s particle
is operationally real with probability 1. Setting $f\equiv 0$ here, we find
that even a maximally mixed state can be potentially OR nonlocal to the
maximum extent. This surprising observation essentially has to do with the
idea that the underlying reality of disturbance should be unaffected by the
observer’s state of knowledge.

If the preparation information of a given mixed state $\varphi_{AB}$ is
unavailable, then its remote measurement-disturbance under measurement $x$ is
said to be probabilistically OR nonlocal if $\varphi_{AB}$ contains a non-
vanishing OR nonlocal component under any pure-state decomposition. Here, a
direct application of conditions (a) and (b) of Theorem 1 can be misleading.
For example, the separable state
$\frac{1}{2}(\ket{\theta^{+}}\bra{\theta^{+}}\otimes\ket{0}\bra{0}+\ket{\theta^{-}}\bra{\theta^{-}}\otimes\ket{1}\bra{1})$,
when $x\equiv\sigma_{Z}$, satisfies (a) and also, with high probability,
condition (b), by choosing sufficiently small $\theta$. Yet, it is evident
that there can be no remote measurement-disturbance of $B$ for any separable
state.

Therefore, to obtain a lower bound on OR nonlocality a different approach is
required for mixed states. Here a key observation, noted above in the proof of
Theorem 1, is that Alice steers Bob’s state. Given a steerable state $\rho$,
the required lower bound is associated with the pure-state decomposition that
gives the lowest fraction of OR nonlocality. In the above example of the state
$W(f,\theta)$, the first decomposition provides the minimal decomposition.
Thus if $f$ is sufficiently large to guarantee steerability, then the mixture
is OR nonlocal with probability at least $f$.

To witness steering in an arbitrary GPT, we use the uncertainty principle.
This refers to the feature whereby two or more observables cannot
simultaneously assume exact values Oppenheim and Wehner (2010). Let
$\mathfrak{p}(b|y)$ represent the probability of outcome $b$ upon measurement
of $y$ on a given operational state $\varphi$. Define
$\mathcal{P}(y)\equiv\max_{b}\mathfrak{p}(b|y)$. Given measurements
$y_{0},y_{1}$ and $y_{2}$, an uncertainty relation exists if for any state
$\varphi$

$\mathcal{P}(y_{0})+\mathcal{P}(y_{1})+\mathcal{P}(y_{2})\leq\upsilon$ (8)

such that $\upsilon<3$. For classical theory, $\upsilon=3$ for any triple of
sharp measurements. Here it is assumed that the measurements $y_{j}$ aren’t
“trivial”, such as one that produces a fixed outcome for any measured state
$\varphi$. In this case, of course there is trivially no uncertainty.

Consider protocol $\mathfrak{P}_{3}$, where Alice and Bob share the state
$\varphi_{AB}$. Alice performs measurement $x_{j}$ on the particle $A$ and
predicts the outcome for Bob, who performs the corresponding measurement
$y_{j}$ on $B$. We consider the conditional version of Eq. (8), namely:

$\mathcal{P}(y_{0}|x_{0})+\mathcal{P}(y_{1}|x_{1})+\mathcal{P}(y_{2}|x_{2})~{}\leq~{}\upsilon,$
(9)

where $\mathcal{P}(y_{j}|x_{j})$ represents Bob’s certainty in $y_{j}$
measured on $B$, conditioned on Alice’s measurement of $x_{j}$ on $A$ and
knowing the outcome. The violation of Eq. (9) certifies that Alice’s
measurement genuinely re-prepares– and thereby disturbs– the state of $B$. As
such, Eq. (9) represents an EPR (spatial) steering inequality. A necessary
condition here is that the measurement pairs $x_{0},x_{1}$ and $x_{3}$ must be
pairwise incompatible for its violation (Appendix, Part II).

Given a pure state $\varphi_{AB}$ in a GPT and a pair of measurements
$(x_{0},y_{0})$, if measuring $x_{0}$ on $A$ and $y_{0}$ on $B$ produce
identical results, i.e., Eq. (7) holds, then the pair $(x_{0},y_{0})$ is said
to be commensurate for $\varphi_{AB}$. For a mixed state prepared by combining
such $\varphi_{AB}$ (with probability $p$) and other states, the pair
$(x_{0},y_{0})$ is said to be commensurate with probability (at least) $p$.
For example, the measurements $x_{0}=y_{0}:=\sigma_{Z}$ is commensurate for
the state $\ket{\Psi(\theta)}$. Consider a mixed states such as $W(f,\theta)$,
that are known to be noisy versions of pure states that have identical
marginal states (i.e., satisfy Eq. (6)) and either admit a pair of
commensurate measurements or can be brought to that form by application of
suitable local reversible operations (i.e., satisfy Eq. (7)). Promised a mixed
state with this commensurate measurement property, the following result shows
that the degree of violation of Eq. (9) can be used to lower-bound the degree
of OR nonlocality.

######  Theorem 2.

Given the violation of the inequality Eq. (9) with observed correlation
$\upsilon^{\ast}>\upsilon$, the remote measurement-disturbance of $B$ under
measurement $x_{0}$ on $A$ is operationally real with probability at least
$\frac{\upsilon^{\ast}-3/2}{\upsilon_{\max}-3/2}$, where $\upsilon_{\max}$ is
the largest violation ($\leq 3$) allowed in the given GPT.

###### Proof.

In Eq. (4), if the remote repreparation of the state of $B$ can be explained
by a pre-existing, hidden state ensemble of $B$-states
$\\{q(\mu),\phi_{B}^{\mu}\\}$, then conditioning on $x$ won’t provide
information to beat the uncertainty relation Eq. (8) for system $B$.
Therefore, the EPR steering inequality Eq. (9) will hold. It follows that
under its violation, the measurement disturbance of $A$ disturbs the system
$B$.

Let the minimal pure fraction leading to the violation in all possible pure
state decompositions of the given state $\varphi_{AB}$ be $f$. Then $f\geq
f_{\min}$, where
$f_{\min}\upsilon_{\max}+(1-f_{\min})\frac{3}{2}=\upsilon^{\ast}$, and
$\upsilon_{\max}$ and $\frac{3}{2}$ are, respectively, the theory-dependent
maximum and the algebraic minimum value attainable by $\upsilon$ in Eq. (9).
Solving the equation, we find
$f_{\min}=\frac{\upsilon^{\ast}-3/2}{\upsilon_{\max}-3/2}$. By assumption, the
mixed state is a noisy version of a pure state with identical marginal states
and the commensurate measurement associated with the pair $(x_{0},y_{0})$.
Therefore, with probability at least $f_{\min}$, the conditions of Theorem 1
must be satisfied. ∎

To clarify Theorem 2, we note that the violation of the inequality Eq. (9), as
discussed, entails the remote preparation of $B$’s state (Eq. (4)) and hence
the disentanglement of the initial state $\varphi_{AB}$. This in turn implies
the verification of the reality condition (a) in the equivalent LOCC scenario.
However, quantifying how closely condition (b) is supported by the
experimental conditional probabilities $\mathcal{P}(y_{j}|x_{k})$ would
require certain theory-dependent assumptions. Furthermore, experimentally
estimating the violation of inequality Eq. (9) in terms the theory-specific
quantity $\upsilon_{\max}$ presupposes that the states of $B$ are well
characterized. On the other hand, the probability $\mathfrak{p}(a|x,\lambda)$
in Eq. (4) can be arbitrary. This situation corresponds to one-sided device-
independence, which is of practical importance in cryptography Branciard  _et
al._ (2012). In Section IV, we show how the above result can be strengthened
to a fully device-independent characterization of OR nonlocality.

As a specific realization of Eq. (9), we consider
$\mathcal{P}(y_{0}|x_{0})+\mathcal{P}(y_{1}|x_{1},a_{1})+\mathcal{P}(y_{2}|x_{1},\overline{a}_{1})\leq\upsilon$,
where Alice has only two measurement choices: $x_{0}$ and $x_{1}$, whilst Bob
has three. If Alice measures $x_{0}$, then Bob measures $y_{0}$. If her
measurement is $x_{1}$, then corresponding to her outcome $a_{1}$ (resp.,
$\overline{a}_{1}$), Bob measures $y_{1}$ (resp., $y_{2})$. For an application
to the quantum context, we set $x_{0}=y_{0}:=\sigma_{Z}$, $x_{1}:=\sigma_{X}$,
$y_{1}:=\sin(2\theta)\sigma_{X}+\cos(2\theta)\sigma_{Z}$ and
$y_{2}:=\sin(2\theta)\sigma_{X}-\cos(2\theta)\sigma_{Z}$. For these settings
of Bob, the single system uncertainty bound is given by
$\upsilon=\frac{5}{2}$.

The state $\ket{\Psi(\theta)}$ under the above settings entails a violation of
Eq. (9) up to its algebraic maximum of 3 for any $\theta$ in the above range.
Moreover, the margin of maximal violation over the local bound $\upsilon$ is
$\frac{3}{\upsilon}=\frac{6}{5}$, for the optimal choice $\theta=\pi/6$ here.
This can be shown to be larger than the optimal margin of
$\frac{2}{\upsilon}=\frac{2\sqrt{2}}{\sqrt{2}+1}\approx 1.17$ in the case of
the analogous two-term steering inequality (Appendix). Thus, inequality Eq.
(9) is suitable for the non-maximally entangled state $\ket{\Psi(\theta)}$.

##  IV Relation to Bell nonlocality

Bell-nonlocality is a stronger condition than steering. With measurement
settings as above, the bound $\upsilon:=\frac{5}{2}$ in Eq. (9). If
$f>\frac{2}{3}$, Werner-like states $W(f,\frac{\pi}{6})$ violate the
inequality. Using the two-qubit nonlocality criterion Horodecki  _et al._
(1995), we find that the state $W(f,\frac{\pi}{6})$ is Bell nonlocal for
$f>\frac{2}{\sqrt{7}}$. Thus, in the range
$f\in[\frac{2}{3},\frac{2}{\sqrt{7}}]$, the state $W(f,\frac{\pi}{6})$ is Bell
local but nonlocal in an operationally real way. It is important to stress
that the existence of such Bell-local states doesn’t undermine the operational
reality of the remote measurement-disturbance precisely because the hidden
variables of the Bell-local theory are not part of the operational theory of
our interest. Indeed, the basic premise here that we decide the reality of a
disturbance by operational considerations alone, without recourse to hidden-
variable ontology. Another point is that given correlation $P(a,b|x,y)$ of OR-
nonlocal but Bell-local correlations in a GPT, the classical dimension
$|\lambda|$ of shared randomness in a local model will be larger than the
dimension $\mathfrak{d}$ of the correlated systems of the GPT, i.e., such
correlations will be superlocal Donohue and Wolfe (2015); Jebaratnam  _et al._
(2017). It is reasonable to assume that the GPT encompasses classical theory
and so any local correlation requiring shared randomness
$|\lambda|\leq\mathfrak{d}$ can be produced by local measurements on separable
states of the GPT. Thus, OR nonlocality entails superlocality, i.e.,
$|\lambda|>\mathfrak{d}$.

Finally, let us point out that the above characterization of OR nonlocality in
the LOCC scenario naturally leads to a device-independent (DI)
characterization thereof. The idea is to obtain a sufficient condition for OR
nonlocality without reference to theory-dependent parameters such as
$\upsilon_{\max}$. We suppose that in protocol $\mathfrak{P}_{3}$, Alice and
Bob perform their measurements simultaneously on the pre-shared state
$\varphi_{AB}$. Importantly, there is no classical communication from her to
Bob or vice versa until after their measurements. Further, we suppose that
from a subset of the resulting conditional probabilities $P(a,b|x,y)$, Alice
and Bob construct a Bell-type inequality, e.g.,

$P_{11}^{=}+P_{12}^{=}+P_{21}^{=}+P_{22}^{\neq}\leq 3,$ (10)

where $P_{jk}^{=}\equiv P(a_{j}=b_{k}|x_{j},y_{k})$ and $P_{jk}^{\neq}\equiv
P(a_{j}\neq b_{k}|x_{j},y_{k})$. In an arbitrary GPT, under maximal violation
of this inequality, the quantity in the l.h.s can go up to the algebraic
maximum of 4, while in QM it is $2+\sqrt{2}\approx 3.414$.

By design, if Alice’s conditional probability has the local-realist form
$P(b|a,x,y)=\sum_{\lambda}p_{\lambda}\mathfrak{p}(a|x,\lambda)\mathfrak{p}_{B}(b|y,\lambda)$,
i.e., Bob’s measurement merely reveals a pre-existing value of $y$ on $B$,
then the satisfaction of the inequality Eq. (10) follows. Accordingly, the
violation of the inequality implies the absence of such a value and can
thereby serve as the basis to certify the entanglement in state $\varphi_{AB}$
in a DI manner, i.e., irrespective of the details of the operational theory
governing the subsystems $A$ and $B$. Alice and Bob may perform a follow-up
measurement on their respective particles to confirm the absence of
entanglement in the post-measured particles, which trivially requires no
theory-specific assumptions. By thus verifying the disentangling action of
their measurements, they obtain a DI checking of the reality condition (a) of
Theorem 1.

As regards condition (b), it turns out that in the context of device
independence, it can be relaxed, by not requiring the disturbances of
particles $A$ and $B$ to be identical. As the violation of Eq. (10) precludes
the possibility of a pre-existing value of $y$ on $B$, Bob’s conditional state
here is in a sense created by Alice’s act of measurement. (To improve the
semantics, we let Bob’s measurement to happen slightly later in their common
reference frame, but in such a way that their measurements are spacelike
separated.) Ontological consistency of the formalism then requires that the
remote measurement-disturbance of $B$ be OR, given that the measurement-
disturbance of $A$ is OR under a violation of inequality Eq. (10). For
otherwise, we would have the incoherent situation that $A$’s measurement
disturbance is real, and yet not so the remote state-preparation that it is
certified (by the Bell inequality) to have produced. This observation provides
a natural extension to the consistency requirement of Feature 1 in the DI
scenario.

We thus have the following strengthening of Theorem 2: Given the violation of
the inequality Eq. (10), the remote measurement-disturbance of $B$ under
measurement $x$ on $A$ is operationally real with a non-zero probability.

##  V Discussion and conclusions

The ontological question of whether measurement-induced disturbance is real in
QM or other nonclassical operational theories is addressed in the present work
by employing– ironically– only operational considerations. As such, it can be
construed as providing a different response to the EPR paradox Einstein  _et
al._ (1935); Wiseman (2013), than both Bohr’s Bohr (1935) and Bell’s Bell
(1964) responses.

EPR asserted essentially that quantum spatial steering entailed a “spooky
action-at-a-distance” (Bell, 2004, Article 16), which they hoped could be
banished in a more complete version of QM. Employing a different criterion of
reality than EPR, the present work argued that the action-at-a-distance is an
unavoidable feature of operational QM itself, and not just of QM’s ontological
completion (as follows from Bell’s theorem).

The operationally reality of remote measurement-disturbance sheds new light on
the tension Gisin (2009) between quantum nonlocality and special relativity.
In particular, it highlights that quantum no-signaling is distinct from
relativistic signal locality. The former is a consequence of the tensor
product structure of the state space, with no association to light speed,
whereas the latter is essentially a prohibition on superluminal transmission
of information arising from the Lorentz invariance of the light cone.
Evidently, these two no-go conditions belong to two distinct frameworks,
respectively. In point of fact, quantum nonlocality is non-signaling even in
non-relativistic QM.

In an instance of OR nonlocality, as far as Alice can say, when she measures
$x$ and obtains outcome $a$, particle $B$ is instantaneously left in the state
$\tilde{\varphi}_{B}^{a|x}$, given by Eq. (4). The remote measurement-
disturbance of $B$, given by Eq. (5), thus represents a spacelike influence
linking the event of $A$’s measurement and the event of re-preparation of
$B$’s state. Per Theorem 2, this influence is nevertheless real in an
operational sense, and thus arguably imposes an intrinsic time-ordering on the
two events. This observation underscores a further aspect of the distinction
between quantum no-signaling and signal locality in special relativity.
Whereas the latter is conceivably a not unanticipated speed-limit on
information propagation, by contrast the former seems surprising in light of
OR nonlocality, and prompts the question– especially relevant in the context
of reconstructing QM from operational or information theoretic principles
Clifton  _et al._ (2003); Chiribella  _et al._ (2011); Dakic and Brukner
(2011)– of why an OR nonlocal theory, such as QM, is non-signaling. Indeed, we
understand the complementaristic role played here by the randomness of
measurement outcomes in suppressing the signaling (cf. Aravinda and Srikanth
(2015); Kar  _et al._ (2011); Hall (2010)), but this only answers the “how”,
rather than the “why” aspect of the question.

Here we offer a simple and brief answer: that no-signaling is a consequence of
the natural requirement of consistency between the properties of single
systems and those of reduced systems. Suppose no-signaling could be violated
in a GPT. Then, the inequality Eq. (9) could be violated without Alice’s
classical communication of the outcome of her measurement $x_{j}$. But if so,
then it would essentially mean that violation of the local uncertainty
principle Eq. (8) at Bob’s end, contradicting this property in the context of
single systems.

###### Acknowledgements.

The author thanks S. Aravinda for discussions. He acknowledges the support of
Dept. of Science and Technology (DST), India, Grants No. MTR/2019/001516 and
also the Interdisciplinary Cyber Physical Systems (ICPS) programme of the DST,
India, Grant No. DST/ICPS/QuST/Theme-1/2019/14.

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##  Appendix A Measurement incompatibility and disturbance

Suppose measuring $x_{0}$ doesn’t disturb the joint state $\varphi_{AB}$,
i.e., $\mathcal{M}_{x_{0}\otimes u}(\varphi_{AB})=\varphi_{AB}$ (where $u$
represents an identity operation on $B$). Let $x_{J}$ be the joint measurement
on $A$ and $B$ that is used to check for disturbance of $A$. This would entail
that $x_{0}\otimes u$ and $x_{J}$ are compatible, because we can construct
their joint measurement, given by
$J(a_{0},a_{J}|x_{0},x_{J},\varphi_{AB})\equiv\mathfrak{p}(a_{0}|x_{0},\varphi_{AB})\mathfrak{p}(a_{J}|x_{J},\varphi_{AB})$
for this state, simply by first measuring $x_{0}$ and then $x_{J}$, and noting
the respective outcome probabilities. Thus, the measurement of $x_{0}$ must
(globally) disturb $\varphi_{AB}$. This disturbance can in principle be
detected by joint measurement $x_{J}$ on systems $A$ and $B$, whereby
$\mathfrak{S}_{A\rightarrow AB}>0$.

##  Appendix B Incompatibility and steering

The violaton of Eq. (9) implies that the measurements $x_{0},x_{1}$ and
$x_{2}$ are pairwise incompatible. To show this, for simplicity consider the
two-term variant of the above inequality:

$\mathcal{P}(y_{0}|x_{0})+\mathcal{P}(y_{1}|x_{1})~{}\leq~{}\upsilon_{2},$
(11)

where $\upsilon_{2}$ is the local uncertainty bound for two measurements.
Suppose $x_{0}$ and $x_{1}$ are jointly measurable Busch and Lahti (1984);
Busch  _et al._ (2013). In place of Eq. (4), we would have
$\tilde{\varphi}^{a_{0},a_{1}|x_{0},x_{1}}_{B}\equiv\sum_{\lambda}p(\lambda)\mathfrak{p}(a_{0},a_{1}|x_{0},x_{1},\lambda)\varphi_{B}^{a_{0},a_{1}|x_{0},x_{1},\lambda}$,
where the conditional probability $\mathfrak{p}(a_{j}|x_{j},\lambda)$ for
either measurement should be derivable as the marginal statistics of a “master
measurement”:
$\mathfrak{p}(a_{j}|x_{j},\lambda)=\sum_{a_{\overline{j}}}\mathfrak{p}(a_{0},a_{1}|x_{0},x_{1},\lambda)$
where $\overline{j}\equiv j+1{\rm~{}mod~{}}2$. But this means that the states
$\varphi_{B}^{a_{0},a_{1}|x_{0},x_{1},\lambda}$ constitute hidden states to
reproduce the result of the two measurements. Specifically, to implement the
measurement of $x_{0}$, one marginalizes over $a_{1}$:
$\tilde{\varphi}^{a_{0}|x_{0}}_{B}\equiv\sum_{a_{1}}\tilde{\varphi}^{a_{0},a_{1}|x_{0},x_{1}}_{B}$,
and vice versa.


# Optically- and thermally-driven huge lattice orbital and spin angular
momenta from spinning fullerenes

G. P. Zhang∗ Department of Physics, Indiana State University, Terre Haute,
Indiana 47809, USA  Y. H. Bai  Office of Information Technology, Indiana State
University, Terre Haute, Indiana 47809, USA  Thomas F. George  Departments of
Chemistry & Biochemistry and Physics & Astronomy, University of Missouri-St.
Louis, St. Louis, MO 63121, USA

###### Abstract

Lattice vibration in solids may carry angular momentum. But unlike the
intrinsic spin of electrons, the lattice vibration is rarely rotational. To
induce angular momentum, one needs to find a material that can accommodate a
twisted normal mode, two orthogonal modes or excitation of magnons. If
excitation is too strong, one may exceed the Lindemann limit, so the material
melts. Therefore these methods are not ideal. Here, we theoretically propose a
new route to phonon angular momentum in a molecular crystal $\rm C_{60}$. We
find that a single laser pulse is able to inject a significant amount of
angular momentum to $\rm C_{60}$, and the momentum transfer is helicity-
dependent. Changing from right-circularly polarized light to left-circularly
polarized light switches the direction of phonon angular momentum. On the
ultrafast time scale, the orbital angular momentum change closely resembles
the displacive excitation of coherent phonons, with a cosine-function
dependence on time, different from the spin counterpart. Atomic displacements,
even under strong laser excitation, remain far below the Lindemann criterion.
Under thermal excitation, spinning $\rm C_{60}$ even at room temperature
generates a huge angular momentum close to several hundred $\hbar$. Our
finding opens the door to a large group of fullerenes, from $\rm C_{60}$, C70
to their endohedral derivatives, where angular momentum can be generated
through light or temperature. This paves the way to the phononic control
electronic spin and harvesting thermal energy through phonon angular momentum.

###### pacs:

42.65.Ky, 78.66.Tr

Nuclear vibration is ubiquitous and important to biological, chemical, and
physical processes, ranging from photosynthesis brixner2005 ,
photoisomerization gai1998  to spin manipulation garanin2015 ; fahnle2017 ;
zahn2020 . Microscopically, each material has its own vibrational normal
modes. However, due to the symmetry constraint coh2019 , normal modes rarely
have a rotational eigenvector, so the phonon angular momentum (PAM) is feeble.
The concept of phonon spin is not new vonsovskii1962 ; levine1962 , and
reemerged zhang2014 . McLellan mclellan1988  investigated PAM in a
rotationally invariant harmonic model. Interest in PAM is more recent
zhang2014 ; nakane2018 ; ruckriegel2020 ; park2020 ; streib2021 . It was shown
that one may use an electric field moseni2021  or temperature gradient
hamada2018  to control PAM. By coupling PAM to the electronic spin degrees of
freedom, one can use PAM to control spin devices stupakiewicz2021 . However,
the main obstacle is how to generate PAM in the first place. It was predicted
that one could introduce chiral phonons in monolayer molybdenum disulfide
zhang2015 , but the real calculation juraschek2019  showed that the effect is
tiny in transition metal dichalcogenides. The same thing is true for Pt3 and
Pt5 bistoni2021 , where the most of vibrational modes has nearly zero angular
momentum, with the maximum close to $0.094(\hbar/2)$. Experimentally, it was
demonstrated that one could construct a superimposed vibrational state which
consists of two orthogonal vibrational modes to generate an artificial
circular phonon mode nova2017 . One may also couple the phonon to magnons in a
magnetic material holanda2018 ; streib2018 ; an2020 ; ruckriegel2020a  to
induce spin angular momentum of phonons, but one must first have magnons. In
general, these methods are not easy.

Molecular crystals are different. The weak bonding between neighboring sites
render atom motifs rotating freely. Buckministerfullerene $\rm C_{60}$
kroto1985  is a prime example. At room temperature, $\rm C_{60}$ is spinning
rapidly johnson1992 ; dresselhaus ; bubenchikov2019  around its equilibrium
lattice positions. Spinning is temperature-dependent. As temperature cools
down below 261 K, the rotation slows down, where 13C NMR spectra of solid $\rm
C_{60}$ show a significant broadening at a chemical shift of 143 ppm tycko1991
; tycko1991a ; johnson1992 . Its rotation can also be manipulated through
substrates wang2004 ; bozhko2011 , the tip of scanning tunneling microscope
neel2008  and encapsulation inside carbon nanotubes zou2009 . $\rm C_{60}$
rotation also affects its energy spectrum lima2014  and vibrational spectra
saito1994 . $\rm C_{60}$ and C70 are not the only fullerenes that spin.
Endohedral fullerenes bethune1993 , with additional atoms encapsulated inside
fullerenes, offer additional kinds of rotational dynamics hernandez1997 , with
applications to organic photovoltaic devices ross2009  and quantum computing
twamley2003 . Recently, possible light-induced superconductivity was reported
in K3C60 mitrano2016 . It is conceivable that light can inject angular
momentum into K3C60. These readily available molecular crystals represent a
new frontier for phonon angular momentum investigation.

In this Letter, instead of relying on artificial rotational motion of atoms to
search for phonon angular momentum, we start with a material that rotates in
the first place at room temperature. Specifically, we employ $\rm C_{60}$ as
an example, and we show that a laser pulse can induce a significant orbital
and spin angular momentum change. PAM depends on the laser helicity.
Circularly polarized light injects more angular momentum to the system than
linearly polarized light. Changing from right- to left-circularly polarized
light switches angular momentum direction. The total angular momentum exactly
follows the energy change, thus reaffirming that it is physical. The orbital
and spin angular momenta depend on time differently. The orbital has a cosine
dependence, very similar to displacive excitation of coherent phonons
zeiger1992 . The lattice distortion of $\rm C_{60}$ is small even under strong
laser excitation, with the atom displacement far below the Lindemann criterion
lindemann1910 . The global rotation of $\rm C_{60}$ generates several hundred
of $\hbar$. Our finding here represents a different direction for phonon
angular momentum research, and is expected to motivate experimental and
theoretical investigations.

We define the total angular momentum of $\rm C_{60}$ as

${\bf J}=\sum_{i}{\bf j}_{i}(t)=\sum_{i}{\bf r}_{i}(t)\times{\bf p}_{i}(t),$
(1)

where ${\bf p}_{i}(t)$ is its momentum of atom $i$ at time $t$ and ${\bf
r}_{i}(t)$ is its position vector. The reason why we use J instead of L will
become clear below. Although we do not quantize the atomic vibration formally
here, we still use phonons below. To find the initial coordinates $\\{{\bf
r}_{i}(0)\\}$ of carbon atoms in $\rm C_{60}$, we use the spiral generation
method proposed by Fowler and Manolopoulos fowler  for the topological problem
of fullerenes. Here, in brief, we first construct the adjacency matrix, and
diagonalize it. The three lowest eigenvectors correspond to the $x$, $y$ and
$z$ coordinates of the carbon atoms, and then we properly rescale them
according to the radius of $\rm C_{60}$.

To optimize the geometry, we employ the Su-Schrieffer-Heeger (SSH) model
su1979  often used in conjugated polymers, where only $\pi$ electrons are
treated quantum mechanically, and the nuclear motions are described by the
classical potentials heeger1988  since the mass of carbon atoms is more than
three orders of magnitude larger than that of the electron. Recently, the SSH
model finds an important application in topological insulators asboth2016 .
The Hamiltonian reads prb03 ; prl04 ; prl05

$H_{0}=-\sum_{\langle
ij\rangle,\sigma}t_{ij}(c^{\dagger}_{i,\sigma}c_{j,\sigma}+h.c.)+\frac{K_{1}}{2}\sum_{\langle
i,j\rangle}(r_{ij}-d_{0})^{2}+\frac{K_{2}}{2}\sum_{i}\delta\theta_{i,p}^{2}+\frac{K_{3}}{2}\sum_{i}(\delta\theta_{i,h,1}^{2}+\delta\theta_{i,h,2}^{2}),$
(2)

where $c^{\dagger}_{i,\sigma}$ is the electron creation operator at site $i$
with spin $\sigma(=\uparrow\downarrow)$ prl03  and the summation $\langle
ij\rangle$ over $i(j)$ runs from 1 to 60 with $i\neq j$. The first term on the
right-hand side represents the electron hopping between nearest-neighbor atoms
at positions ${\bf r}_{i}$ and ${\bf r}_{j}$, $t_{ij}=t_{0}-\alpha(|{\bf
r}_{i}-{\bf r}_{j}|-d_{0})$, where $r_{ij}=|{\bf r}_{i}-{\bf r}_{j}|$, $t_{0}$
is the average hopping constant, and $\alpha$ is the electron-lattice coupling
constant. The last three terms are the lattice stretching, pentagon-hexagon
and hexagon-hexagon bending energies, respectively, (see further details in sm
). You et al. you1993  parametrized the Hamiltonian by fitting the energy gap,
bond lengths and 174 normal mode frequencies to their respective experimental
values. They found that $t_{0}=1.91$ eV, $\alpha=5.0$ eV/$\rm\AA$, $K_{1}=42$
eV/$\rm\AA^{2}$, $K_{2}=8$ eV/$\rm rad^{2}$, $K_{3}=7$ eV/$\rm rad^{2}$ and
$d_{0}=1.5532~{}\rm\AA$. These parameters are consistent with the literature
values adams1991 ; menon1991 ; feldman1992 ; faulhaber1993 ; friedman1993 ;
harigaya1994 ; wang1994 ; liu1994 . With You’s six parameters and with the
electrons initially occupying 30 lowest energy levels, we optimize the
structure. Figure 1(a) shows a two-dimensional structure of our optimized $\rm
C_{60}$, where the $z$ axis is out of the page. The $\rm C_{60}$ molecule has
the highest $I_{h}$ point symmetry. 60 carbon atoms form 90 bonds, 60 single
bonds around 12 pentagons, and 30 single bonds shared by 20 hexagons (see Fig.
1(a)). Our double bond length is 1.403 $\rm\AA$, and the single bond length is
1.443 $\rm\AA$, both of which indeed match the experimental values of $1.40\pm
0.015\rm\AA$ and $1.45\pm 0.015\rm\AA$ hedberg1991 ; yannoni1991 . Figure 1(b)
shows the energy spectrum with the degeneracy. The highest occupied molecular
orbital (HOMO) is a $H_{u}$ state, while the lowest unoccupied molecular
orbital (LUMO) is a $T_{1u}$ state. The first dipole-allowed transition is
between HOMO and LUMO+1, which is highlighted by the double arrow. All these
features are fully consistent with the prior study you1993 . The challenge is
to fit 174 normal mode frequencies menendez2000  with only six parameters. We
find that the calculated frequencies within our theory slightly deviate from
the experimental values menendez2000  within 3%. For instance, our breathing
mode $A_{g}(1)$ frequency is 481 $\rm cm^{-1}$, while the experimental one is
between 492 dresselhaus  and 496 menendez2000  $\rm cm^{-1}$. Our pentagonal
pinch mode $A_{g}(2)$’s frequency is 1477 $\rm cm^{-1}$, while the
experimental one is 1470 dresselhaus ; menendez2000  $\rm cm^{-1}$. Here the
error is 0.5 %. Because our Hamiltonian does not include high-order terms
wu1987 , these small differences are expected and do not affect our main
conclusion of the paper. For this reason, we do not adjust those parameters.

As seen above, there are varieties of ways to induce angular momentum change.
We perturb the system with an ultrafast laser pulse, whose interaction with
$\rm C_{60}$ is described by

$H_{I}=-e\sum_{i\sigma}{\bf E}(t)\cdot{\bf r}_{i}~{}n_{i\sigma}~{},$ (3)

where ${\bf E}(t)$ is the electric field of the laser and $n_{i\sigma}$ is the
electron number operator at site $i$. We choose a Gaussian pulse for ${\bf
E}(t)$: ${\bf E}(t)=A_{0}\exp(-t^{2}/\tau^{2})[\cos(\omega
t)\hat{x}+\epsilon\sin(\omega t)\hat{y}]/\sqrt{1+\epsilon^{2}}$, where
$A_{0}$, $\omega$, $\tau$, $t$, $\epsilon$ and $\tau$ are the field amplitude,
laser frequency, pulse duration or width, time, helicity and pulse duration,
respectively. $\hat{x}$ and $\hat{y}$ are the unit vectors along the $x$ and
$y$ axes, respectively. We ensure that $A_{0}$ is large enough, so the quantum
effect of light is small and the classical treatment of our laser field is
adequate. We treat the electrons quantum mechanically through the time-
dependent Liouville equation prl03 ; prl05 ,

$i\hbar\frac{\partial\langle\rho_{ij}^{\sigma}\rangle}{\partial
t}=\langle[H,\rho_{ij}^{\sigma}]\rangle,$ (4)

where $H=H_{0}+H_{I}$, and $\rho_{ij}^{\sigma}$ is the density matrix. The
lattice vibrations are described by the Newtonian equation. This classical
treatment of nuclear motion is reasonable if the de Broglie wavelength is
smaller than the system size, but if the de Broglie wavelength becomes
comparable to the system size, then our approach is less accurate. We solve
the coupled Liouville and Newtonian equations numerically.

We excite $\rm C_{60}$ with a 60-fs laser pulse with electric field amplitude
of 0.01 $\rm V/\AA$. The photon energy is 2.76 eV, which is tuned to be
resonant with the transition between HOMO and LUMO+1 (see Fig. 1b). Figure
2(a) displays the total energy (thin line), lattice kinetic energy (thick
line) and lattice potential energy (thick dashed line) as a function of time.
The curve around zero denotes the laser pulse. Upon laser excitation, energy
enters the electronic system first. The peak in the total energy is mainly due
to the electron energy. Through the electron-lattice interaction and laser
excitation, the lattice starts to vibrate. Most of the lattice energy is in
the potential energy, and the kinetic energy is very small. In the figure, we
multiply the kinetic energy by 100\. Carbon atoms move at a speed of
$10^{-3}-10^{-2}\rm\AA/fs$, or 100-1000 m/s, very typical for nuclear
vibrations. The oscillation is due to the energy exchange between the electron
and lattice subsystems (for details, see sm ). Since we want to check the
energy conservation as well as vibrational oscillations, we do not dampen our
system. We find that the oscillation has a period of 69.3 fs, which exactly
matches the frequency of the breathing mode $A_{g}(1)$ dexheimer1993 ; prb06 .
Such a coherent phonon excitation has been observed experimentally
dexheimer1993  and in other systems as well zeiger1992 . The reason why this
$\rm A_{g}(1)$ is excited strongly is due to our chosen laser parameters prl04
; sm . Our interest is in the lattice angular momentum. Figure 2(b) shows the
lattice angular momentum for three laser helicities ($\epsilon$): right
($\sigma^{+}$), left ($\sigma^{-}$) and linear ($\pi$) pulses, with the
polarization in the $xy$ plane (see Fig. 1(a)). There is a general trend as to
how the angular momentum is transferred to the lattice. Upon laser excitation,
the flow of angular momentum to the lattice occurs on the 100 fs time scale,
which corresponds to the total energy absorbed. Although each atom still
vibrates and exchanges its angular momentum with the rest of the atoms, the
total angular momentum must remain constant and is conserved in the absence of
an external field. This is rigorously reproduced in our calculation. The
angular momentum shows a strong dependence on $\epsilon$. For $\sigma^{-}$,
the lattice angular momentum is along the $+z$ axis, while for $\sigma^{+}$,
it is along the $-z$ axis. The $\pi$ pulse induces a much weaker angular
momentum (see the long-dashed line). This is expected because the angular
momentum of light is directly related to the helicity of light. When it
interacts with $\rm C_{60}$, the lattice angular momentum has the hallmark of
incident light. Figures 2(c), (d) and (e) show the detailed change in $J_{x}$,
$J_{y}$ and $J_{z}$ as a function of helicity $\epsilon$. When $\epsilon=0$,
the laser polarization is linear along the $x$ axis. One can see that only
$J_{y}$ differs from zero, because the atom moves mainly along the $x$ axis.
Going from $\epsilon=0$ to 1, our pulse changes from a linearly polarized
pulse to elliptically, and finally to a circularly polarized pulse. Both
$J_{x}$ and $J_{z}$ increase significantly as the atom also moves along the
$y$ axis. $J_{z}$ is negative as expected. When we change $\epsilon$ from 0 to
-1, we have a left elliptically to circularly polarized pulse, so $J_{z}$
flips its sign, fully consistent with our finding in Fig. 2(b).

What is unknown or less familiar is whether and how the lattice angular
momentum J can characterize a system. This is an important conceptual question
because angular momentum is rarely used to characterize the dynamics of a
system, except in atoms and some simple structures. Prior studies zhang2014 ;
nakane2018 ; ruckriegel2020 ; park2020 ; streib2021  are all based on this
assumption. We compare J against the absorbed energy $\Delta E$ into $\rm
C_{60}$. When the laser photon energy is off resonance, the energy change
follows the perturbative path and has a simple dependence on the laser field
amplitude $A_{0}$ shen , so the agreement between the angular momentum ${\bf
J}$ and $\Delta E$ could be coincidental. The harder question is what happens
if the laser is on resonance. Figure 3(a) shows the energy absorbed as a
function of laser field amplitude where the photon energy is tuned to the
dipole-allowed transition between the HOMO and LUMO+1 states (Fig. 1(b)). It
is clear that the dependence on $A_{0}$ is highly nonlinear. When $A_{0}$ is
far below 0.005 $\rm V/\AA$, the change is linear, but as far as it closes to
0.005 $\rm V/\AA$, it peaks, after which it starts to decrease. Because
resonant excitation activates multiple real electronic excitation, this opens
other channels and $\Delta E$ decreases. Once $A_{0}$ exceeds 0.01 $\rm
V/\AA$, $\Delta E$ increases again, which is consistent with our prior study
prb03 . Figure 3(b) is our angular momentum. One can see that its change with
$A_{0}$ matches the absorbed energy change, down to some minor details. This
proves that J is physical, and it represents the acquired angular momentum
from light.

It has been proposed that if one rewrites ${\bf r}(t)$ as a sum of the
equilibrium position ${\bf r}_{i}(0)$ and the displacement ${\bf u}_{i}(t)$,
${\bf r}_{i}(t)={\bf r}_{i}(0)+{\bf u}_{i}(t)$, the angular momentum for atom
$i$ can be separated into two terms:

${\bf j}_{i}(t)={\bf r}_{i}(0)\times{\bf p}_{i}(t)+{\bf u}_{i}(t)\times{\bf
p}_{i}(t)\equiv{\bf l}_{i}(t)+{\bf s}_{i}(t).$ (5)

${\bf l}_{i}(t)$ is called the orbital angular momentum of atom $i$, while
${\bf s}_{i}(t)$ is called the spin angular momentum mclellan1988 ; zhang2014
; garanin2015 ; ruckriegel2020  because ${\bf s}_{i}$ is independent of the
initial position of an atom. By separating orbital from spin angular momentum,
we can directly investigate the interplay between spin and orbital angular
momenta on an ultrafast time scale.

Figure 3(c) compares the spin angular momentum with the orbital counterpart.
We employ a 60-fs laser pulse with $\hbar\omega=2.76$ eV and $A_{0}=0.01\rm
V/\AA$. We notice that the angular moment mainly enters the orbital part. From
Eq. 5, we see that it is the momentum ${\bf p}_{i}$ that is mainly responsible
for this increase. $S_{z}$ is very small and magnified by 10 in the figure.
What is interesting is that the spin angular momentum does not follow the
laser pulse, very different from the orbital angular momentum. The reason for
this difference is straightforward. $S_{z}$ depends on both ${\bf u}_{i}(t)$
and ${\bf p}_{i}$. Although the pulse peaks at 0 fs, the atom cannot follow
instantaneously and needs time to respond. The behavior of $L_{z}$ is like a
cosine function, with its maximum at 0 fs, similar to displacive excitation of
coherent phonons (DECP) zeiger1992 . But $S_{z}$ is like a sine function. The
sum of $S_{z}$ and $L_{z}$ is $J_{z}$, which is constant after the laser field
is gone because the system does not have a way to exchange angular momentum.
This demonstrates that our definition of angular momentum through Eq. 5 is
physically sound.

It has been a big challenge to induce phonon spin angular momentum in real
materials nova2017 ; holanda2018 ; juraschek2019 . We can increase spin
angular momentum by tuning the laser photon energy to vibrations directly
stupakiewicz2021 . We choose $\hbar\omega=0.185$ eV, or 44.7 THz, to be
resonant with nuclear vibrations. The laser pulse duration is also 60 fs and
the field amplitude is increased to 0.15 $\rm V/\AA$. We caution that our
current treatment of nuclear vibration may not be adequate to compute the
infrared spectrum since we do not include the nuclear dipole moment and the
nuclear vibration is not treated quantum mechanically. Figure 3(d) shows that
$S_{z}$ reaches 0.5$\hbar$. Under our laser field, the structure of $\rm
C_{60}$ is well maintained, far below the Lindemann criterion lindemann1910
for melting. This criterion has been a major obstacle for a prior study
juraschek2019 , where the atomic displacements for BaO and LiNbO3 can be as
large as 0.1 $\rm\AA$. Therefore, an experimental test is difficult.

The top figure of Fig. 4 shows atomic displacements projected on the $xy$
plane for five atoms on the front pentagon. The scales for the $x$ and $y$
axes are increased by 100 for an easy view. One can see the displacement is
well below the Lindemann criterion. This paves the way to experimental
testing. What is even more interesting is that besides these internal
rotations, under thermal excitation, the entire molecule spins rapidly,
although along arbitrary axes at room temperature. The angular momentum is
$L=I\Omega$, where $I$ is the moment of inertia of $\rm C_{60}$ and $\Omega$
is the angular speed. Assuming $\rm C_{60}$ spins along the $z$ axis through
two pentagons (see Fig. 1), the moment of inertia is
$I_{z}=m\sum_{i=1}^{60}(x_{i}^{2}+y_{i}^{2})$, where $m$ is the mass of carbon
atom and $x_{i}$ and $y_{i}$ are the coordinates of atom $i$. Note that the
moment of inertia is almost identical along other axes because $\rm C_{60}$ is
highly spherical. An early estimate puts $\Omega$ between 1.5/ps and 2.8/ps
zhang1991 . According to the latest theoretical estimate bubenchikov2019 , its
average angular frequency is 0.34 rad/ps. Since our present study only
contains one single $\rm C_{60}$ and has no intermolecular interaction between
several fullerenes, we are unable to verify their average angular frequency.
Instead, we use their number and find $|{\bf J}|$ for $\rm C_{60}$ to be
$322\hbar$. This angular momentum is from the global rotation of $\rm C_{60}$.
Because our Hamiltonian in Eq. 2 only depends on the distance between
neighboring atoms, the rapid global rotation of C60 does not enter our
calculation directly and only implicitly through the laser field polarization.
However, due to the high symmetry of C60, different polarizations yield almost
identical results. Therefore, the global rotation must be treated separately
in our formalism. To see the thermal effect of the global rotation on the spin
angular momentum, we set the initial velocities of the carbon atoms to
$\sqrt{k_{B}T/m}$, where $k_{B}$ is the Boltzmann constant and $m$ is the
carbon mass. We allow a random distribution of the initial velocity
directions. We use four temperatures, $T=0$, 10, 150 and 300 K.

The results are shown in the bottom figures of Fig. 4. Figure 4(a) is our
result with 0 K, but with a small laser field amplitude of 0.01 $\rm V/\AA$.
The solid, dotted and long-dashed lines denote $S_{x}$, $S_{y}$, and $S_{z}$,
respectively, and are the same for the rest of the figures. We see that only
$S_{z}$ has a sizable value. But as we increase temperature to 10 K, we find
that all the other components are increased to 0.5 $\hbar$. The amplitude is
comparable among all the three components. At 150 K, the spin increases to
$5\hbar$. At room temperature of 300 K, Fig. 4(d) shows the spin angular
momentum increases to $10\hbar$. We also check whether our initial
configuration of velocity direction affects our results. Among all the
configurations we investigate, we do not see any qualitative difference. They
should be observable experimentally. Spinning $\rm C_{60}$ could be a natural
phonon angular momentum generator. As a large group of other fullerenes are
available and even its endohedral forms already exist, such as N@$\rm C_{60}$
and P@$\rm C_{60}$, our finding opens a new route to phonon angular momentum.
The future study should focus on how to harvest such a huge momentum for
technological applications.

In conclusion, we have proposed a route to generate phonon orbital and spin
angular momenta from $\rm C_{60}$, without resorting to more complicated
synthetic methods zhang2015 ; juraschek2019 ; nova2017 ; holanda2018 . We show
that a single laser pulse can transfer a significant amount of angular
momentum to $\rm C_{60}$. The phonon angular momentum that $\rm C_{60}$
receives shows a strong dependence on laser helicity. The circularly polarized
light injects more momentum than the linearly polarized light. We demonstrate
that the total angular momentum change follows the total energy change
faithfully, thus establishing that even in the time domain PAM is a valid
concept and can be used to characterize the system property. The orbital
angular momentum is normally larger than the spin counterpart, and it also has
a different dependence on time. Similar to DECP zeiger1992 , the orbital has a
cosine function dependence, while the spin has a sine function dependence. We
find that the atomic displacement is spiral and its amplitude is far below the
Lindemann criterion, which has been a big obstacle to a prior study
juraschek2019 . The angular momentum of the entire $\rm C_{60}$ reaches 322
$\hbar$ at room temperature. Since endohedral fullerenes such as N@$\rm
C_{60}$ and P@$\rm C_{60}$ are readily available, our study points out a large
group of materials suitable for phonon angular momentum generation. It is
expected that our finding will motivate experimental and theoretical
investigations in this field and beyond.

###### Acknowledgements.

This work was solely supported by the U.S. Department of Energy under Contract
No. DE-FG02-06ER46304. Part of the work was done on Indiana State University’s
high performance Quantum and Obsidian clusters. The research used resources of
the National Energy Research Scientific Computing Center, which is supported
by the Office of Science of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231. ∗<EMAIL_ADDRESS>https://orcid.org/0000-0002-1792-2701

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Figure 1:  (a) $\rm C_{60}$ structure. The empty circles denote sixty carbon
atoms. The laser polarization (the vector) is in the $xy$ plane (the front
pentagon) and controlled by the helicity $\epsilon$. (b) Energy level scheme
for $\rm C_{60}$. The double-arrow denotes the transition between the second
lowest unoccupied molecular orbital (LUMO+1) and highest occupied molecular
orbital (HOMO).  Figure 2:  (a) Total energy change as a function of time
under a 60-fs $\sigma^{+}$ pulse. Here $\epsilon=1$. The laser field amplitude
is 0.01 $\rm V/\AA$, and the photon energy is $\hbar\omega=2.76$ eV. Thin
solid, thick solid, and dashed lines denote the system total energy, lattice
kinetic and potential energy, respectively. The lattice kinetic energy is
multiplied by 100 to have an easy view. To demonstrate the energy
conservation, the system is not damped. The red thin line around 0 is the
laser electric field. The oscillation period in the lattice potential is 69.3
fs, which exactly matches the frequency of the Raman-active $A_{g}(1)$ mode.
(b) Total angular momentum J under $\sigma^{+}$ (solid line), $\sigma^{-}$
(dotted line) and $\pi$ pulses (dashed line). Only the strongest component of
J is shown. Linearly polarized light transfers a small angular momentum. The
left- and right-circularly polarized light transfer opposite angular momenta.
All the pulses have the same laser parameters as (a). (c), (d) and (e) show
the final $x$, $y$ and $z$ components of J as a function of ellipticity
$\epsilon$.  Figure 3:  (a) Absorbed energy $\Delta E$ as a function of laser
field amplitude $A_{0}$. The photon energy and pulse duration are the same as
Fig. 2. (b) Total angular momentum as a function of laser field amplitude
$A_{0}$, which matches the total energy change. This demonstrates that J is
physical. (c) The orbital $L_{z}$ and spin $S_{z}$ angular momenta as a
function of time. $S_{z}$ is very small when $\hbar\omega=2.76$ eV. In the
figure, it is multiplied by 10. (d) The orbital $L_{z}$ and spin $S_{z}$
angular momenta as a function of time with $\hbar\omega=0.185$ eV. $S_{z}$
reaches a comparable magnitude as $L_{z}$. The dependence of $S_{z}$ on time
follows a sine function, while that of $L_{z}$ follows a cosine function, very
similar to DECP zeiger1992 .

Figure 4: (Top) Traces of displacements of five carbon atoms on the front
pentagon in the earlier stage of laser excitation. The change is only 0.01
$\rm\AA$, which is well below the Lindemann criterion lindemann1910 . The
arrow on the top left denotes the rotation of entire $\rm C_{60}$. (Bottom)
Phonon spin angular momentum at (a) 0 K, (b) 10 K, (c) 150 K and (d) 300 K,
where the solid, dotted and long-dashed lines denote $S_{x}$, $S_{y}$ and
$S_{z}$, respectively. Its magnitude becomes larger as temperature increases.