PHYSICAL REVIEW A 89, 032312 (2014)
Effects of relaxed assumptions on semi-device-independent randomness expansion
Yu-Kun Wang,
1,2
Su-Juan Qin,
1,*
Ting-Ting Song,
1
Fen-Zhuo Guo,
3
Wei Huang,
1
and Hui-Juan Zuo
1,4
1
State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China
2
State Key Laboratory of Integrated Services Network, Xidian University, Xian 710071, China
3
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
4
Mathematics and Information Science College, Hebei Normal University, Shijiazhuang 050024, China
(Received 18 December 2013; published 11 March 2014)
Semi-device-independent randomness expansion is the protocol which can generate and certify perfect free
random numbers, without making detailed assumptions on the internal working of quantum devices, except the
bound on the dimension of the Hilbert space. The two-dimensional (2D) quantum witness violation has been
used to generate guaranteed fresh randomness from untrusted devices in the prepare-and-measure configuration.
In this paper we first show that the 2D quantum witness violation can be simulated by a classical deterministic
process by relaxing the assumption of the dimension. A deterministic process to simulate the generation of
random numbers is given, under the hidden variables model, with just the high dimensional measurement device
and one qubit communication. This makes the generated number essentially deterministic, which is said to be
pseudorandom. Then we also show that when we relax the freedom of choice, there is the exact region for which
quantum correlations can be simulated by the classical process in the task of randomness expansion.
DOI: 10.1103/PhysRevA.89.032312 PACS number(s): 03.67.Dd, 03.67.Ac, 05.40.−a
I. INTRODUCTION
The emergence of quantum theory in the early 20th century
led to a revolution in many areas of physics. One of its
main features is the introduction of intrinsic randomness,
originating from the very nature of the theory. True random
data is a resource very vital in many applications, such
as computer simulations, statistical sampling, gambling, and
cryptography protocols. As we know, an important assumption
in the quantum key distribution (QKD) protocol is that
the random resource used in the protocol is unbiased and
completely unaccessible to the adversary. For this reason,
the random number generators (RNGs) should be among the
first commercially available devices. However, there is no true
randomness in the classical world, since any classical system
admits in principle a deterministic description and any random
classical process can be seen as the probabilistic combination
of deterministic classical processes. The observed randomness
of the classical process is due to a lack of knowledge about its
fundamental description.
Quantum theory, on the other hand, is intrinsically random,
and it has been taken as a tool to certify and quantify the
presence of the true randomness recently. Inspired by the
previous work of quantum information processing based on
nonlocal correlations of the entanglement particles [1–3],
Colbeck et al. [4,5] proposed the true random-number-
expansion protocol based on the Greenberger-Horne-Zeilinger
correlation test, and Pironio et al. proposed the protocol
certified by the Bell-CHSH inequality violation [6,7]. The
expansion protocol based on entanglement forms the device-
independent (DI) scheme because its implementation requires
no assumption about the internal working of the devices in
both the state preparation and measurement. The randomness
of their experimental results can be certified only by the GHZ
paradox and the Bell inequality violation. They are supposed
*
qsujuan2003@163.com
to be the test for a nonlocal correlations between the outcomes
of pairs of events, which are seen as black boxes producing
a classical output when given a classical input. However,
the DI protocol has much more complicated experimental
setups, and most commercial true RNGs are based on one-
way protocols. Then, based on the two-dimensional (2D)
quantum witness violation, Li et al. [8,9] proposed a semi-
device-independent (SDI) random-number expansion protocol
without entanglement. And the security of the proposed SDI
protocols [8,9] is based on the configuration of n →1 quantum
random access codes (QRACs) [10–12]. It requires two black
boxes: one is responsible for the state preparation, and the other
is the state measurement. The preparation box can encode n
classical bits into one qubit in a way that the measurement box
can decode any one of the n classical bits with the success
probability strictly greater than 1/2 when it receives the coded
qubit. It still makes no assumption about the internal working
of the state preparation and measurement device, except for
the following three basic rules: (1) freedom of choice, the
state preparation, and the measurement settings in the test
should be chosen at randomly and independently; (2) 2D
system will be considered in the protocol, i.e., the prepared
state ρ
x
∈ C
2
; and (3) operators M
y
act on a 2D Hilbert
space, and the inner space of the measurement black box
cannot do high dimensional measurement. The above three
assumptions are a crucial resource in the real expansion.
In this paper we will analyze what will happen for the
security of the SDI randomness expansion when we relax
them. It is known, the 2D witness violation can be simulated
by a classical deterministic process [10–12] when we relax
assumptions 2 and 3. However, it needs the prepared state and
the measurement device to all be a high dimensional system.
In this paper we give a deterministic process to simulate the
generation of random numbers through the hidden variables
model with just the high dimensional measurement device,
under one qubit communication. This can be fooled into
thinking that the observed outputs are not predetermined, but it
is essentially deterministic in practice. Then we will show how
1050-2947/2014/89(3)/032312(6) 032312-1 ©2014 American Physical Society