A dimension witness provides a device-independent certification of the minimal dimension required to reproduce the observed data without imposing assumptions on the functioning of the devices used to generate the experimental statistics. In this paper, we provide a family of Bell expressions where Alice and Bob perform $2^{n-1}$ and $n$ number of dichotomic measurements, respectively, which serve as the device-independent dimension witnesses of Hilbert space of $2^m$ dimensions with $m=1,2,...,2^{\lfloor n/2\rfloor }$. The family of Bell expressions considered here determines the success probability of a communication game known as the $n$-bit parity-oblivious random access code. The parity obliviousness constraint is equivalent to the preparation noncontextuality assumption in an ontological model of an operational theory. For any given $n\ge 3$, if such a constraint is imposed on the encoding scheme of the random access code, then the local bound of the Bell expression reduces to the preparation noncontextual bound. We provide explicit examples for the $n=4\text{and}5$ case to demonstrate that the relevant Bell expressions certify the qubit and two-qubit system, and for the $n=6$ case to demonstrate that the relevant Bell expression certifies the qubit, two-qubit, and three-qubit systems. We further demonstrate the sharing of quantum preparation contextuality by multiple Bobs sequentially to examine whether the number of Bobs sharing the preparation contextuality is dependent on the dimension of the system. We provide explicit examples of $n=5$ and 6 to demonstrate that the number of Bobs sequentially sharing the contextuality remains the same for any of the $2^m$-dimensional systems.

中文翻译：

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使用贝尔表达式族对希尔伯特空间维度进行设备无关的认证

尺寸见证人提供了不依赖于设备的最小尺寸证明，该最小尺寸可用于再现观察到的数据，而无需对用于生成实验统计数据的设备的功能进行假设。在本文中，我们提供了一系列Bell表达式，其中Alice和Bob执行$2^{ñ-\mathrm{1个}}$ 和 $ñ$ 二分法测量的数量，分别作为设备的希尔伯特空间的设备无关维度的见证 $2^{米}$ 尺寸与 $米=\mathrm{1个}，2，...，2^{\lfloor ñ/2\rfloor }$。这里考虑的Bell表达式族确定了称为$ñ$位奇偶性可忽略的随机访问代码。奇偶性遗忘约束等效于操作理论的本体模型中的准备非上下文假设。对于任何给定$ñ\ge 3$如果对随机访问码的编码方案施加了这样的约束，则Bell表达式的局部边界减小为准备非上下文边界。我们为$ñ=4\text{和}5$ 证明相关的Bell表达式证明了qubit和two-qubit系统的情况，对于 $ñ=6$以证明相关的Bell表达式证明了qubit，2 qubit和3 qubit系统。我们进一步证明了多个鲍勃依序共享量子准备上下文，以检查共享准备上下文的鲍勃的数量是否取决于系统的维度。我们提供以下示例$ñ=5$ 和6以证明顺序共享上下文的Bob的数量对于任何一个 $2^{米}$维系统。