Certifying randomness in quantum state collapse Liang-Liang Sun1Xingjian Zhang2Xiang-Zhou1Zheng-Da Li34

2025-09-29 2 0 651.37KB 27 页 10玖币

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Certifying randomness in quantum state collapse

Liang-Liang Sun,1Xingjian Zhang,2Xiang-Zhou,1Zheng-Da Li,3,4

Xiongfeng Ma,2Jingyun Fan, 3,4†Sixia Yu,1‡

1Hefei National Laboratory for Physical Sciences at the Microscale and

Department of Modern Physics, University of Science and Technology of China,

Hefei, Anhui 230026, China

2Center for Quantum Information, Institute for Interdisciplinary

Information Sciences, Tsinghua University, Beijing, China

3Shenzhen Institute for Quantum Science and Engineering and Department of Physics,

Southern University of Science and Technology, Shenzhen, 518055, China

4Guangdong Provincial Key Laboratory of Quantum Science and Engineering,

Southern University of Science and Technology, Shenzhen, 518055, China,

†E-mail: fanjy@sustech.edu.cn

‡E-mail:yusixia@ustc.edu.cn

The unpredictable process of state collapse caused by quantum measurements

makes the generation of quantum randomness possible. In this paper, we ex-

plore the quantitative connection between the randomness generation and the

state collapse and provide a randomness veriﬁcation protocol under the as-

sumptions: (I) independence between the source and the measurement devices

and (II) the Lüders’ rule for collapsing state. Without involving heavy math-

ematical machinery, the amount of generated quantum randomness can be

directly estimated with the disturbance effect originating from the state col-

lapse. In the protocol, we can employ measurements that are trusted to be

arXiv:2210.16632v2 [quant-ph] 17 Aug 2023

non-malicious but not necessarily be characterized. Equipped with trusted

and characterized projection measurements, we can further optimize the ran-

domness generation performance. Our protocol also shows a high efﬁciency

and yields a higher randomness generation rate than the one based on uncer-

tainty relation. We expect our results to provide new insights for understand-

ing and generating quantum randomness.

1 Introduction

Randomness is ubiquitous in modern society. In particular, it plays an indispensable role in

cryptography. Such a resource is absent within the deterministic Newtonian physics. On the

contrary, there is an ample supply of intrinsic randomness in a quantum world. Many quantum

properties, such as nonlocality, uncertainty principle, and contextuality, can ensure the pre-

sentation of quantum randomness and have been harnessed to devising quantum randomness

generators (QRGs) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].These properties deal with var-

ious scenarios where speciﬁc functioning of quantum devices are required, for example, the

system dimension [15, 16, 17, 18], indistinguishability of non-orthogonal quantum states [19],

and energy of system [20, 21, 22]. These requirements are the assumptions or the expense of

the revelent QRNGs. Among all quantum randomness generation schemes, device-independent

(DI) protocols enjoy the highest security with almost the only assumption on the correctness

of quantum physics [3], however, which is extremely challenging in the experiment and could

achieve only very low rates of randomness generation. The other extreme are fully trusted

quantum randomness generators, where more randomness is easily extracted at the expense of

fully trusting the inner working of physical devices, which is most unscure but generate random

number at a high rate. The semi-DI protocols are the compromise between security and rate

of generation, in which the central problem is ﬁnding protocols of operationally simple, high

randomness generating rate, and fewer security assumptions.

In this paper, we exploit the common knowledge on QRNG: no matter what quantum prop-

erty it involves, state collapse induced by measurements, which is the solely unpredictable

process in quantum theory, has to be present. We explore the possibility of directly verify-

ing quantum randomness with state collapse. By using the disturbance effect accessible in a

sequence of incompatible measurements, we provide a conﬁrmative answer to the question:

we establish a quantitative connection between randomness generation, state collapse, and the

disturbance effect. We employed a prepare-and-measure QRNG scenario to demonstrate this

connection. It involves an untrusted source of quantum states and two quantum measurements

performed in sequence, which is readily implementable on photonic experimental platforms.

With a few reasonable assumptions on the device’s functioning, the protocol can employ a gen-

eral unknown general measurement trusted to be non-malicious and also allows for optimizing

the performance using completed trusted and characterized projection measurements. In vari-

ous contexts, quantum randomness generated via our protocol can be directly estimated without

involving heavy mathematical machinery. Thus, we provide an efﬁcient RNG protocol as well

as a quantitative account for the fundamental connection between the key concepts, namely,

quantum randomness and state collapse.

The rest of the paper is structured as follows. In section I, we brieﬂy review measures of

quantum randomness. In section II, we introduce the set-up of our protocol. In section III, we

establish the connection between disturbance and quantum randomness against a classical ad-

versary. We show that our the performance of our QRNG protocol can be optimized when more

information about measurements is at hand. In section IV, we use the protocol to estimate the

quantum randomness against classical and the quantum adversaries in the asymptotic limit of

an inﬁnite data size. In section V, we compare our result with the protocol based on uncertainty

relation.

2 Quantum Randomness Measures

In information theory, quantum randomness evaluation can be formalized in an adversarial sce-

nario [23]. Consider a user, Alice, and an adversary, Eve, share particles in a joint state ρAE .

A local measurement MAperformed on the subsystem of Alice alters the entire state from

ρAE to ρ′

AE . Because of the presence of Eve’s side information, Alice’s measurement results

are not completely private. Depending on Alice’s measurement and Eve’s adversary strategies,

different entropic measures may be applied to quantify the amount of private randomness. In a

generic single-shot case, one applies the conditional min-entropy as the randomness measure.

Generally, Eve may utilise the full knowledge of her system, and the conditional min-entropy

is deﬁned as

min(A|E) = −infσEDmax(ρ′

AE ∥idA⊗σE),(1)

where id denotes the identity operator, σEis a normalized state on Eve’s system, and Dmax(ρ∥σ)

is the maximum relative entropy,

Dmax(ρ∥σ) = inf{λ∈R:ρ≤2λσ}.

In certain contexts, the potential side information has a classical nature. Operationally, this

corresponds to the case where Eve carries out a measurement on her system and use the mea-

surement outcome as her guess. Then, the conditional min-entropy degenerates to the following

quantity,

min(A|E) = −infσEDmax(ρ′′

AE ∥id ⊗σE),(2)

where ρ′′

AE is the post-measurement state after Alice’s and Eve’s local measurements. Depend-

ing on whether Eve’s side information is characterised by a quantum state or a classical random

variable, we call the entropic measures in Eq. (1) and (2) as conditioned on a quantum adversary

and a classical adversary, respectively.

From the adversarial perspective, quantum randomness is conversely associated with the

maximum probability that Eve can correctly guessing the outcomes on Alice’s side, which we

call the guessing probability. For the case where Alice performs measurement MA={Mi}on

a pure state |ϕ⟩, the best adversarial strategy for Eve is simply guessing the outcome with the

maximum probability, given by

GA|ϕ:= maxip(i|A;ϕ),

where p(i|A;ϕ) = ⟨ϕ|Mi|ϕ⟩. For a mixed state ρ, Eve can utilise her side information for a

better guess. In the case of a classical adversary, the guessing probability is given by

GA|ρ= max{rn,ϕn}Pnrn·GA|ϕn.

where the optimization is taken over all pure state decompositions ρ=Pnrn|ϕn⟩⟨ϕn|. The

conditional min-entropy in Eq. (2) has the following equivalent expression,

min(A|ρ) = −log GA|ρ.(3)

When Alice repeats projective measurements in basis {|i⟩} independently and identically

for sufﬁcient times, the conditional entropy Eq. (1) asymptotically converges to [24, 25, 26]

min,asy(A|E) = S(ρ∥∆(ρ)),(4)

where ∆(ρ) := Pi|i⟩⟨i|ρ|i⟩⟨i|and the relative entropy S(ρ∥∆(ρ)) := tr ρ[log ρ−log ∆(ρ)].

The conditional Eq.(2) asymptotically converges to [24, 25, 26]

min,asy(A|E) = minrn,ϕnPnrn·H(pϕn),(5)

with pϕn={p(i|A, ϕn)}and H(·)being the Shannon entropy.

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CertifyingrandomnessinquantumstatecollapseLiang-LiangSun,1XingjianZhang,2Xiang-Zhou,1Zheng-DaLi,3,4XiongfengMa,2JingyunFan,3,4†SixiaYu,1‡1HefeiNationalLaboratoryforPhysicalSciencesattheMicroscaleandDepartmentofModernPhysics,UniversityofScienceandTechnologyofChina,Hefei,Anhui230026,China2CenterforQua...

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Certifying randomness in quantum state collapse Liang-Liang Sun1Xingjian Zhang2Xiang-Zhou1Zheng-Da Li34

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