Theory-independent randomness generation from spatial symmetries

2025-05-06 0 0 1.01MB 23 页 10玖币

Theory-independent randomness generation from

spatial symmetries

Caroline L. Jones1,2, Stefan L. Ludescher1,2, Albert Aloy1,2, and Markus P. Müller1,2,3

1Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria

2Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Vienna, Austria

3Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada

December 12, 2024

We characterize how the response of physical

systems to spatial rotations constrains the prob-

abilities of events that may be observed. From a

foundational point of view, we show that the set

of quantum correlations in our scenarios can be

derived from rotational symmetry alone, with-

out assuming quantum physics. This shows that

important predictions of quantum theory can

be derived from the structure of space, demon-

strating that semi-device-independent scenarios

can be utilized to shed light on the founda-

tions of physics. From a practical perspective,

these results allow us to introduce semi-device-

independent protocols for the generation of se-

cure random numbers based on the breaking of

spatial symmetries. While experimental imple-

mentations will rely on quantum physics, the se-

curity analysis and the amount of extracted ran-

domness is theory-independent and certiﬁed by

the observed correlations only. That is, our pro-

tocols rely on a physically meaningful assump-

tion: a bound on a theory-independent notion

of spin.

1 Introduction

Quantum ﬁeld theory and general relativity, as they cur-

rently stand, describe two distinct classes of physical

phenomena: probabilities of events on the one hand,

and spacetime geometry on the other. Large eﬀorts are

currently underway to construct a theory of quantum

gravity that would describe both classes of phenomena

and their interaction in a uniﬁed way. Given the dif-

ﬁculties in this endeavour, one may start with a more

modest, but nonetheless illuminating approach: ana-

lyze how probabilities of detector clicks and properties

of spacetime interact, and what constraints they impose

on one another. Here, we propose to use semi-device-

independent (semi-DI) quantum information protocols

to study this interrelation.

DI and semi-DI approaches [1–9] treat devices in an

Caroline L. Jones: CarolineLouise.Jones@oeaw.ac.at,

Stefan L. Ludescher: Stefan.Ludescher@oeaw.ac.at,

CLJ and SLL contributed equally to this work.

Figure 1: Setup. A ﬁxed but arbitrary state is generated in the

preparation device P, which is rotated by an angle αx∈ {0, α}

relative to the measurement device Maccording to an input

setting x∈ {1,2}. The state is then sent to M, where a

measurement yields one of two outcomes b∈ {±1}.

experiment as “black boxes”: no assumptions (or only

very mild ones) are made about the inner workings

of the devices, and the analysis relies on the observed

input-output statistics alone. While Bell and other DI

black-box scenarios have previously been used to study

the foundations of quantum theory [10,11], here we sug-

gest to “put the boxes into space and time”.

Speciﬁcally, we consider the prepare-and-measure

scenario sketched in Fig. 1, which can be used to gener-

ate random numbers that are secure against eavesdrop-

pers with additional classical information [9,12–16]. We

deﬁne a class of semi-DI quantum random number gen-

erators based on an assumption about how the transmit-

ted system may respond to spatial rotations. Crucially,

this semi-DI assumption does not rely on the validity

of quantum theory, since it is representation-theoretic

in nature and hence applies to all possible probabilistic

theories. We show that the exact shape of the set of

quantum correlations in this setup appears to emerge

as a direct consequence of the symmetries of spacetime,

which also entails the security of our protocol against

post-quantum eavesdroppers.

2 The setup

We consider a semi-DI random number generator simi-

lar to the one described in [9,17], given by the prepare-

and-measure scenario depicted in Fig. 1. The goal is to

arXiv:2210.14811v3 [quant-ph] 12 Dec 2024

generate statistics P(b|x)that certify that even exter-

nal eavesdroppers with additional (classical) knowledge

cannot predict b. As in standard DI quantum informa-

tion, the security of semi-DI protocols does not require

any assumptions on the inner-workings of the devices,

but it requires some constraint on the physical system

that is communicated between the devices [5,9,18]. This

has often been implemented with a bound on the di-

mension of the Hilbert space of the transmitted system,

restricting the communication to qubits or qutrits, as

in [5,12,18,19], although this is arguably not very well-

motivated for non-idealized physical scenarios. An al-

ternative scheme was provided in [9,17], in which the

mean value of some observable H(such as the energy of

the transmitted system) was constrained. This formu-

lation, however, requires trust in the valid characteri-

zation of the observable H, including, for example, the

assumption of a speciﬁc gap above the ground state. In

fact, the physical meaning of H(say, as the generator

of time translations) plays no direct role in their analy-

sis. Here, instead, we propose semi-DI assumptions on

quantities like spin or energy, which anchor the secu-

rity of the resulting protocols on properties of spacetime

physics that are directly related to the interpretation of

these quantities. Not only are these assumptions ar-

guably physically well-motivated, but they can also be

formulated without assuming the validity of quantum

theory, as we will show below. This is in contrast to

dimension bounds or assumptions on the expectation

values of observables, which rely crucially on the valid-

ity of quantum theory.

3 Quantum boxes

Let us start by describing the setup in terms of quan-

tum theory, which we will later generalize to a theory-

agnostic description. We consider two devices (Fig. 1).

The ﬁrst device prepares some quantum state ρ1and

takes an input x∈ {1,2}. The experimenter either does

nothing to the device (i.e. applies R0=1if x= 1), or

rotates it by an angle αaround a ﬁxed axis relative to

the other device (i.e. applies the rotation Rα, if x= 2).

After the rotation, the physical system is prepared and

sent to the second device. The second device produces

an outcome b∈ {±1}, and is described by a POVM

(positive operator-valued measure) {Mb}. Minimal as-

sumptions are made about the devices [20], such that

ρ1and Mbare treated as unknown and may ﬂuctuate

according to some shared random variable λ.

While we allow such shared randomness (see Eq. (7)

below), we do not allow shared entanglement between

preparation and measurement devices, which is a stan-

dard assumption in the semi-DI context [21]. Disallow-

ing this, and demanding that the full preparation device

is rotated, prevents the rotation from being applied only

to a part of the emitted system, which in turn prevents

the appearance of detectable relative phases like (−1)

for a 2π-rotation of spin-1/2fermions.

Well-known arguments (e.g. in [22, Sec. 13.1]) im-

ply that fundamental symmetries, such as the rotations

Rα, must act as unitary transformations Uαon Hilbert

space, furnishing a projective representation of the sym-

metry group (here SO(2)). All ﬁnite-dimensional pro-

jective representations of SO(2) can be written in the

form

Uα=

j=−J

eijα1nj,(1)

where jruns over either integers or half-integers, and

nj∈ {0,1,2. . .}. The assumption about the response

of the system to rotations is implemented via an up-

per bound Jon the absolute value of these labels. For

details see Appendix B.

Fixing some J∈ {0,1

2,1,3

2, . . .}introduces an as-

sumption on the physical system that is sent from the

preparation to the measurement device, namely, on its

possible response to spatial rotations. This is what

makes our scenario semi-DI, and what replaces the more

common assumption on the Hilbert space dimension of

the transmitted system. It is important to note that we

do not ﬁx the numbers nj, thus allowing for the num-

ber of copies to vary, i.e. the Hilbert space dimension is

not bounded by this. The number Jupper-bounds the

spin or angular momentum quantum number associated

with the physical system that is sent from the prepara-

tion to the measurement device. For example, if we

have a single particle of spin J, then Uα= exp(iαZJ),

where ZJ= diag(J, J −1,...,−J)is the spin-Jrepre-

sentation of the Pauli Zmatrix, such that nj= 1 for

j=−J, −J+ 1, . . . , J. However, since the njare arbi-

trary, the representation (1) is allowed to be reducible,

which includes the case of composite systems. For ex-

ample, if the measurement probes helicity with a po-

larizer, then sending a single photon corresponds to a

scenario with J= 1, and Nphotons to J=N[25].

Moreover, every J′> N will serve as a valid upper

bound.

Our mathematical formulation does not presuppose

that the SO(2)-representation must arise from spatial

rotations: it could also arise for some other reason, e.g.

from periodicity of time evolution. However, the spe-

cial case that the preparation device is physically ro-

tated in space is a paradigmatic instance in which the

group symmetry is manifestly imposed from special co-

variance [24].

We are interested in the possible correlations between

outcome band setting xthat can be obtained under

an assumption on Jvia Eq. (1) in the quantum case.

Let us for the moment assume that the initial state ρ1

is a pure state ρ1=|ϕ1⟩⟨ϕ1|, then |ϕ2⟩=Uα|ϕ1⟩is

prepared on input x= 2, and the observable M=M1−

M−1characterizes the measurement procedure. If we

consider all possible pure states |ϕx⟩and observables M

arising from POVMs {Mb}b∈{−1,+1}in this way, then

QJ,α :={(E1, E2)|Ex=⟨ϕx|M|ϕx⟩,|ϕ2⟩=Uα|ϕ1⟩}

(2)

is the set of all possible correlations arising in our sce-

nario, and Ex=P(+1|x)−P(−1|x)characterizes the

bias of the outcome toward ±1for a given x. In [9]

it was shown that when the states that may be sent

in a general prepare-and-measure scenario have overlap

γ≥ |⟨ϕ1|ϕ2⟩|, the set of possible correlations is charac-

terized by the inequality

2p1 + E1p1 + E2+p1−E1p1−E2≥γ. (3)

We show in Section Athat for our scenario,

γ= min |⟨ϕ1|ϕ2⟩| =cos(Jα)if |Jα|<π

0if |Jα| ≥ π

.(4)

The bound γdescribes the smallest possible overlap of

any initial state with its rotation by α, given that the

absolute value of its spin is at most J. From [9], it

follows that (3) and (4) deﬁne some set of correlations

QJ,α (see Fig. 2), of which we know that our set of

interest is a subset: QJ,α ⊆e

QJ,α. In Appendix C, we

show that the two sets are in fact identical: the extremal

boundary of e

QJ,α can be realized via rotations of the

family of states (|j⟩+eiθ|−j⟩)/√2, hence QJ,α =e

QJ,α.

The set QJ,α grows with Jα until Jα =π/2, at which

point a |ϕ1⟩exists such that |ϕ2⟩=Uα|ϕ1⟩is orthogo-

nal to it. If |ϕ1⟩and |ϕ2⟩are perfectly distinguishable,

there exist (even deterministic) strategies to generate

all conceivable correlations.

Anticipating the generation of private randomness as

discussed further below, we deﬁne classical correlations

as convex combinations of deterministic behaviors, i.e.

Eλ:= (E1, E2)∈ {±1} × {±1}, that again satisfy the

maximum spin Jbound:

CJ,α :={E=X

p(λ)Eλ|Eλ∈QJ,α,Eλ∈ {±1}×{±1}},

(5)

where {p(λ)}λis a probability distribution. If Jα <

π/2, the states are not perfectly distinguishable, and so

correlations are limited to Eλ= (±1,±1); alternatively,

if Jα ≥π/2, the states can be perfectly distinguishable,

(+1,+1)(-1,+1)

(+1,-1)(-1,-1)

Figure 2: The quantum sets QJ,α (dark blue) and the classical

sets CJ,α (dark red; line E1=E2), and the quantum and

classical relaxed sets Qδ

J,α and Cδ

J,α for δ∈ {0.15,0.3}. We set

J= 1 and α= 0.66 in this ﬁgure.

and so Eλ= (±1,∓1) are also possible correlations.

Convex combinations of the former case gives the set

CJ,α ={(E1, E2)|−1≤E1=E2≤1}, whilst the latter

case gives all possible correlations.

So far only pure states have been considered. How-

ever, it turns out that this is suﬃcient, as the set of

mixed state correlations, deﬁned by

Q′

J,α :={(E1, E2)|Ex= tr(Mρx), ρ2=Uαρ1U†

α},(6)

coincides precisely with QJ,α. Clearly QJ,α ⊆ Q′

J,α,

and the converse Q′

J,α ⊆ QJ,α can be proven by purify-

ing arbitrary states ρusing an ancilla system, without

adding any spin (for details, see D). Thus, the set QJ,α

is convex, which means that it also describes scenarios

where preparation ρ1and measurements Mbﬂuctuate

according to some shared random variable λdistributed

∼p(λ), i.e.

P(b|α) = X

p(λ)tr(Mb(λ)Uαρ1(λ)U†

α)(7)

(where the input x∈ {0, α}is chosen independently

from λ). So far we have assumed that the constraint on

the maximum spin Jholds exactly and in every run of

the experiment. However, in a more realistic scenario,

one may want to grant room for imperfections. This

can be taken into account by trusting only that the

constraint strictly holds with probability 1−δ, with

0≤δ < 1, but for probability δthe system might carry

arbitrarily high spin. This leads to the relaxed quantum

set

Qδ

J,α = (1 −δ)QJ,α +δ[−1,+1] ×[−1,+1](8)

depicted in Fig. 2. Similarly, replacing Qby Cin this

expression deﬁnes the classical relaxed set Cδ

J,α. For a

full characterization of the relaxed quantum and classi-

cal sets, see Appendix E, where we also discuss types of

experimental uncertainties for which these sets are phys-

ically relevant. For example, we show that for coherent

states, where the photon number nfollows a Poisson dis-

tribution on Fock space, the relaxed quantum set Qδ

J,α

with δ=O(η1/4)contains the relevant set of possible

correlations, with η:= Prob(n > N)giving the proba-

bility of a constraint on J(= N)failing (which tends to

zero exponentially in N).

4 Generating private randomness

Adapting the results of [17], we can show that corre-

lations in QJ,α outside of the classical set admit the

generation of private randomness. Consider an eaves-

dropper Eve with classical (but no quantum) side infor-

mation who tries to guess the value of b. Alice, who

uses the setup of Fig. 1to generate private random

outcomes b, will in general not have complete knowl-

edge of all variables λ∈Λof relevance for the exper-

iment, which is expressed in Eq. (7) by P(b|x)being

the mixture Pλp(λ)P(b|x, λ). Eve, however, may have

additional relevant information λ(in addition to know-

ing the inputs x). It is straightforward to see that

if p(λ)>0, then Eve cannot perfectly predict b(i.e.

0< P (b|x, λ)<1) if the observed correlations Eare

outside of the classical set CJ,α, as long as the semi-DI

assumption is also satisﬁed for every given value of λ.

In order to generate private randomness in our sce-

nario, Alice would like to guarantee that the conditional

entropy H(B|X, Λ) = −Pb,x,λ p(b, x, λ) log2p(b|x, λ)

is large, quantifying Eve’s diﬃculty to predict b.

Since H(B|X, Λ) = Pλp(λ)H(Eλ)where H(E) :=

−1

2Pb,x

1+bEx

2log 1+bEx

2, the amount of conditional

entropy H⋆that Alice can guarantee if she observes

correlations E= (E1, E2), i.e. H(B|X, Λ) ≥H⋆, is de-

termined by the optimization problem

H⋆= min

{p(λ),Eλ}X

p(λ)H(Eλ)

subject to X

λ:Eλ∈Qω

J,α

p(λ)≥1−ε

and X

p(λ)Eλ=E.(9)

That is, H⋆tells us the number of certiﬁed bits of pri-

vate randomness against Eve, under the assumption

that the transmitted systems have spin at most J—

or, rather, when this assumption holds approximately

(up to some ω), with high probability (1 −ε). This

quantity is non-zero, H⋆≡H⋆

ε,ω,α >0, whenever the

observed correlations are outside of the relaxed classical

set, E̸∈ Cε

J,α. For ε=ω= 0, this optimization prob-

lem is equivalent to the one in [17, Sec. 3.2] for the case

that there is, in the terminology of that paper, no max-

average assumption (see Appendix F). For determining

the numerical value of H⋆

0,0,α, we thus refer the reader

to [17]. Furthermore, as we show in Appendix G, we

have a robustness bound for H⋆

ε,ω,α, which reads

H⋆

0,0,α ≥H⋆

ε,ω,α

≥H⋆

0,0,α+c(ε+ω)+ log(1 −ε)−εlog(2/ε)

1−ε,

where c= 2 cot(Jα)/J. Thus, for small ε, ω > 0, the

number of certiﬁed random bits can still be well approx-

imated by using the results of [17] for ε=ω= 0.

5 Rotation boxes

We now drop the assumption that quantum theory

holds, and consider the most general form the proba-

bilities P(b|α)may take that is consistent with the ro-

tational symmetry of the setup while implementing our

spin bound. As discussed later in more detail in Sec-

tion 5.1, to every prepare-and-measure scenario, we can

associate a convex set of states in a real vector space (in

quantum theory, these are the density operators in the

space of Hermitian matrices). Covariance implies [22]

that spacetime symmetries (and hence their subgroup

SO(2)) have linear representations on this space, nec-

essarily characterized by a maximum charge (“spin”) J.

As shown later in Section 5.1, it follows that

P(b|α) =

k=0 c(b)

kcos(kα) + s(b)

ksin(kα),(10)

with suitable coeﬃcients c(b)

k,s(b)

k. In quantum the-

ory in particular, if P(b|α)satisﬁes Eq. (7) and Uα

satisﬁes the semi-DI assumption Eq. (1), then it is of

the form (10). Conversely, we show [24] that every

“rotation box” P(b|α)of the form (10), yielding valid

outcome probabilities between 0and 1, comes from

a representation of SO(2) on some (in general non-

quantum) probabilistic theory with maximal spin J.

Since P(+1|α) + P(−1|α)=1, the set of possible spin-

Jquantum and rotation boxes respectively can be de-

noted by

QJ:= {α7→ P(+1|α)|P(b|α)is of the form (7)},

RJ:= {α7→ P(+1|α)|P(b|α)is of the form (10)},

where, for QJ, we assume that Uαis of the form (1).

We have just seen that QJ⊆ RJ. Trivially, Q0=R0

is the set of constant probability functions, and it can

be shown that Q1/2=R1/2(see Appendix I). How-

ever, in [24], we show that QJ⊊RJ, i.e. that there are

more general ways to respond to spatial rotations than

allowed by quantum theory, if J≥3/2.

5.1 The physical meaning of rotation boxes

Natural extensions of quantum theory are often phrased

within the language of generalized probabilistic theo-

ries (GPTs) [29–32]. In particular, all possible con-

sistent statistical descriptions of prepare-and-measure

scenarios can be described by a GPT system [33]. A

GPT system Aconsists of vector space VA(here taken

to be ﬁnite-dimensional), a state space ΩA⊂VA, an

eﬀect space EA⊂V∗

Aand a set of transformations

TA⊂ L(VA). In summary, preparation procedures are

described by states ω∈ΩA, outcomes of measurements

by eﬀects e∈ EA, and transformations Tby linear maps

on T∈ TAsuch that (e, T ω)is the probability to obtain

the corresponding outcome, following the preparation

and transformation procedures. Quantum systems over

Cnare special cases of GPT systems, with VAthe space

of Hermitian complex n×nmatrices, ΩAthe set of den-

sity matrices, EAthe set of POVM elements, and TAthe

completely positive trace-preserving maps.

Now, assuming the rotational covariance of physics,

similar arguments as in the quantum case [22] imply

that there must be a representation of SO(2) on the

state space. First considering QT, the most general rep-

resentation acting on a Hilbert space is given by equa-

tion (1), which induces a representation on the real vec-

tor space of Hermitian matrices (and therefore also on

the state space of density matrices) Uα(ρ) := UαρU†

α.

In a suitable basis, the superoperator Uαhas the block

matrix form [24]

Uα=1⊕

k=1

1mk⊗cos (kα)−sin (kα)

sin (kα) cos (kα).(11)

This must be true because every representation of SO(2)

on a ﬁnite-dimensional real vector space is of this form.

To say that a quantum system carries a representation

of SO(2) of this form, with m2J̸= 0, is equivalent to say-

ing that the all outcome probabilities tr(MUαρU†

α)are

trigonometric polynomials in α(i.e. of the form (10)),

and the maximal degree over all states ρand POVM el-

ements Mis equals 2J. These are two equivalent ways

of saying that we have a quantum spin-Jsystem.

We can now drop the assumption that quantum the-

ory holds, and say that a GPT system is a spin-Jsystem

if it carries a representation of SO(2) as transformations

Tαthat can be decomposed as in (11). Similarly as in

quantum theory, this is equivalent to saying that all out-

come probabilities (e, Tαω)are trigonometric polynomi-

als in α, of maximal degree 2J. Moreover, all rotation

box probabilities P(b|α)of Equation (10) can be seen

as arising from some spin-JGPT system [24].

The “post-quantum number” Jbehaves in similar

ways as its quantum counterpart. For example, placing

two independent rotation boxes P1and P2side by side

gives a resulting box P(b1, b2|α) := P1(b1|α)P2(b2|α)

with J=J1+J2. This is in line with particle physics

intuition by hinting at Jbeing related to the number of

constituents or “size” of the physical system.

5.2 Agreement of correlation sets

If we consider only two possible inputs, x∈ {1,2}, with

corresponding rotations by 0and α(which is a ﬁxed

angle), the resulting set of rotation box correlations is

RJ,α :={(E1, E2)|E1=P(+1|0)−P(−1|0),

E2=P(+1|α)−P(−1|α), P as in (10)}.(12)

Obviously QJ,α ⊆ RJ,α, but we can say more:

Theorem 1. For every ﬁxed angle α, the quantum set

coincides with the rotation box set, i.e. QJ,α =RJ,α.

Proof. Clearly QJ,α ⊆ RJ,α. We use [26, Chapter 4,

Thm. 1.1]: If Tis a trigonometric polynomial of degree

nwith −1≤T(x)≤1∀x, then

T′(x)2+n2T(x)2≤n2.(13)

Suppose that Pdeﬁnes some spin-Jrotation box corre-

lation, i.e. P(+|α)is a trigonometric polynomial of de-

gree at most 2J, taking values in the interval [0,1] for

all α. Deﬁne T(α):=P(+|α)−P(−|α)=1−2P(+|α),

which is a trigonometric polynomial of degree n= 2J

with −1≤T(α)≤1. Rewrite (13) as T′(x)≤

2Jp1−T(x)2and set Ex:= T(0) and Ex′:= T(αε),

then

αε=Zαε

dα ≥Zαε

T′(α)dα

2Jp1−T(α)2

2JZEx′

p1−y2

2J(arcsin Ex′−arcsin Ex),

where we have substituted y=T(α). It follows that

2|arcsin E2−arcsin E1| ≤ Jα. (14)

For Jα ≥π/2, the set RJ,α contains all possible corre-

lations, as in the quantum case. For Jα < π/2, taking

the cosine of both sides of (14) reproduces, after some

elementary manipulations (see Appendix J), precisely

the conditions of the quantum set, as in (3) and (4),

hence (E1, E2)∈ QJ,α, and so RJ,α ⊆ QJ,α.

This shows that the set of quantum correlations in

our setup can be understood as a consequence of the in-

terplay of probabilities and spatial symmetries, without

assuming the validity of quantum theory. Notably, [27]

also identify a general polytope Gthat characterizes the

set of correlations under an abstract informational re-

striction [28] when no assumption is made on the un-

derlying physical theory, and in the simplest case of

two inputs, this polytope agrees with the set of achiev-

able quantum correlations. Here, however, we show that

a physically well-motivated assumption reproduces the

curved boundary of the set of quantum correlations ex-

actly, for all J. Moreover, Theorem 1 implies that the

amount of certiﬁable randomness H⋆remains correct

even if the validity of quantum theory is not assumed.

To the best of our knowledge, there has not been a

description of a semi-DI prepare-and-measure scenario

with this property in earlier work.

6 Post-quantum security

The equality RJ,α =QJ,α implies that the semi-DI

protocol above is secure against post-quantum eaves-

droppers. While Alice observes quantum correlations

E∈ QJ,α, i.e. of the form (7), it is conceivable that

these are actually mixtures of beyond-quantum rotation

boxes Eλ∈ Rω

J,α such that E=Pλp(λ)Eλ, where Eve

may have access to beyond-quantum physics and know

the value of λ. To see how many bits of private random-

ness H⋆Alice can guarantee against Eve in this case,

the optimization problem (9) has to be altered by re-

laxing the condition on Eλ∈ Qω

J,α to Eλ∈ Rω

J,α, i.e.

by only demanding that every transmitted system is, up

to probability ε, approximately a (not necessarily quan-

tum) rotation box of maximal spin J. However, since

Rω

J,α =Qω

J,α, the optimization problem and hence H⋆

are unaﬀected by this. Moreover, the deﬁnition of the

classical set CJ,α as the mixtures of the possible deter-

ministic correlations is the same, regardless of whether

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Theory-independentrandomnessgenerationfromspatialsymmetriesCarolineL.Jones1,2,StefanL.Ludescher1,2,AlbertAloy1,2,andMarkusP.Müller1,2,31InstituteforQuantumOpticsandQuantumInformation,AustrianAcademyofSciences,Boltzmanngasse3,A-1090Vienna,Austria2ViennaCenterforQuantumScienceandTechnology(VCQ),Facult...

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Theory-independent randomness generation from spatial symmetries.pdf

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Theory-independent randomness generation from spatial symmetries

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