Bell’s Inequality Violation on MATLAB Somi Yun, Fakhar Zaman, Junaid ur Rehman, and Hyundong Shin Department of Electronic Engineering, Kyung Hee University, Korea Email: [email protected] Abstract—Bell’s inequality provides a framework to objectively A A A A decide the locality of a system. In this article, we demonstrate the violation of Bell’s inequality on IBM quantum devices and + = ≥ Matlab. Our main objective in Matlab simulations is to charac- terize the robustness of Bell’s inequality under different types B C B C B C B C of noise models. We find parameter thresholds for depolarizing and amplitude damping channels below which violation of Bell’s inequality can be ensured for the maximally entangled states. Fig. 1: A visual proof of Bell’s inequality : P (A; ∼ B) + P (B; ∼ C) ≥ P (A; ∼ C) using Venn diagram. I. INTRODUCTION Quantum mechanics and theory of relativity are two im- portant pillars of modern physics. At the early stages of characteristics are called entangled states which are the basis development of quantum mechanics, many researchers argued of quantum mechanics. that the quantum mechanics is either incomplete (phenomenon In this paper, we demonstrate the violation of Bell’s inequal- of hidden variables) or incorrect. Einstein and his colleagues ity on a simulator created through MATLAB App Designer (famously called EPR) argued for the incompleteness of quan- which models the gate-level noise of IBM quantum device tum mechanics based on the principle of locality. On the other with depolarizing and amplitude damping channels. We adjust hand, Bohr defended the integrity of quantum mechanics by modelling parameters p and η which represent the levels of stating nonlocality as a feature of quantum mechanics [1]. quantum noise and find the limits of each parameters for Bell’s According to locality, two particles that are spatially distant inequality violated. We also show violation of Bell’s inequality cannot directly affect each other. In order for one particle to on IBM quantum device and validate the results of simulator. affect another, the space between the two must be mediated. Conversely, in nonlocality, two distant particles can construct II. VIOLATION OF BELL’S INEQUALITY one quantum state and are influenced by actions on each other. Although both sides came up with plausible arguments, but EPR and Bohr’s debate unfolded only through thought The Bell’s inequality experimentally describes the theory of experiments and logic did not escape the realm of abstract local hidden variables and quantum mechanics as follows: philosophical debate. In the real world, there is no way to examine thought experiments. Later, Bell proposed physical P (A; ∼ B) + P (B; ∼ C) ≥ P (A; ∼ C) (1) experiments to check the validity of thought experiments being argued. Bell hypothesized that Einstein’s theory of hidden Fig. 1 shows the Venn diagram of Bell’s inequality. However, variables was correct under local realism with locality as the quantum entangled states can not be explained by the the basic premise. He proposed an inequality which must be classical structure that follows the principle of locality [5], satisfied for the local systems but could be easily violated [6]. The singlet state jφi is the most representative example for nonlocal systems [2], [3]. The theory of local hidden of violating Bell’s inequality where jφi is defined as follows: variables—each object has unique attributes, but we simply cannot explain them because we do not know the hidden vari- 1 jφi = p (j01i − j10i): (2) ables which is one way of interpreting quantum mechanics. 2 The knowledge of hidden variables makes it possible to predict physical quantities as in classical physics. From Fig. 2, A is a spin up event along z-axis, B is the event In quantum mechanics, Bell’s inequality is violated due of spin up along 45◦ from z-axis and C is the event of spin to two landmark features of quantum physics–– nonlocality up along x-axis. Thus P (A; ∼ B) is probability of particle- and coherence. Quantum coherence is based on the idea that 1 in event A and particle-1 is not in event B, P (B; ∼ C) is particles such as electrons are described by a wave function probability of particle-1 in event B and particle-1 is not in [4]. If wave-like property of a particle is split in two, then the event C and P (A; ∼ C) is probability of particle-1 in event two waves may coherently interfere with each other in such A and particle-1 is not in event C [7], respectively. Since the a way as to form a correlated composite state. Among the initial state is singlet the probability that particle-1 is in event quantum states, special states having coherence and nonlocal B (resp. C) is same as particle-2 is not in event B (resp. C). A Singlet state Bell test B 0 H Z H | i 45◦ 45◦ 0 X Rz(π/4) H C | i (a) P (A; ∼ B) 0 H Z Rz(π/4) H | i Fig. 2: Spin directions of event A, B and C . 0 X Rz(π/2) H | i To demonstrate the violation of Bell’s inequality, we apply (b) P (B; ∼ C) measurements (projection operators) to calculate the probabil- ities as follows 0 H Z H | i P (A; ∼ B) = hφj ΠA ⊗ ΠB jφi 1 1 (3) 0 X Rz(π/2) H = − p | i 4 4 2 P (B; ∼ C) = hφj ΠB ⊗ ΠC jφi (c) P (A; ∼ C) 1 1 (4) = − p Fig. 3: Circuits of violation of Bell’s inequality that can 4 4 2 be divided into two parts: preparation of the singlet state P (A; ∼ C) = hφj ΠA ⊗ ΠC jφi as the initial state and the Bell test. (a) P (A; ∼ B) is the 1 (5) = ; probability of finding 00 in the circuit (a) measurement result, 2 (b) P (B; ∼ C) is the probability of finding 00 in the circuit where (b) measurement result, (c) P (A; ∼ C) is the probability of finding 00 in the circuit (c) measurement result. 1 Π = (1 + Z) (6) A 2 1 1 ΠB = 1 + p (X + Z) (7) A. Gate Models 2 2 1 Quantum noise can be characterized by different quantum Π = (1 + X) ; (8) C 2 channels, e.g., depolarizing and amplitude damping channel [8]. Depolarizing channel maps a state ρ onto a convex and X (Z) denotes single qubit Pauli x (z) operator. From combination of ρ and the maximally mixed state: equation (3)-(5) 1 1 N (ρ) = (1 − p) ρ + pπ; (10) P (A; ∼ B) + P (B; ∼ C) = − p D 2 2 2 (9) where p is the depolarizing noise parameter and π denotes the P (A; ∼ C) : maximally mixed state, respectively. However, the results of Therefore, it violates the Bell’s inequality. IBM quantum implementation suggest the noise of nonunital nature. Therefore, we additionally apply amplitude damping III. BELL’S INEQUALITY VIOLATION ON MATLAB channel to characterize errors of non ideal gates. Fig. 3 shows the schematic of MATLAB App Designer to The amplitude damping channel is a channel which pro- demonstrate the violation of Bell’s inequality. Initially, we duces the asymmetric decay of the excited state j1i to the ground state j0i with probability η. The action of the channel prepare the singlet state by using ideal gates. We use Rz gates to determine the spin direction particle-1 and particle-2 on the two-qubit input state ρ is followed by performing the Hadamard gate H and performing 1 1 measurement in computational basis where R is the rotation X X y z NA (ρ) = (Ki ⊗ Kj) ρ (Ki ⊗ Kj) ; (11) matrix around z-axis. In practical, either the Bell’s inequality j=0 i=0 is violated or not depends on the noise level in quantum gates. In order to adjust gate error rates, we use MATLAB App where η is the damping parameter and the Kraus operators Designer to simulate the non ideal gates and create circuits. K0 and K1 for the one-qubit amplitude damping channel are 1:0 TABLE I: Gate errors 0:8 S1 Nonideal gate Modeling parameter 0:6 IBM 0:4 S2 Pauli-Z p = 0.008 p Probability 0:2 Pauli-X = 0.162 CNOT p = 0.0167 0:0 00 01 10 11 Rz p = 0.02172 H p = 0.00244, η = 0.05776 State (a) P (A; ∼ B) 1:0 η 0:8 S1 0:6 IBM 0:4 S2 Probability 0:2 0:0 00 01 10 11 0.03414 State 0.21 (b) P (B; ∼ C) p2 O 1:0 0:8 S1 IBM 0:6 0.1591 0:4 S2 Probability 0:2 p1 0:0 00 01 10 11 Fig. 5: The tetrahedron region of Bell’s inequality violation. State (c) P (A; ∼ C) noise level in R gates, respectively. We find the tetrahedron Fig. 4: Measurement statistics from MATLAB App Designer z region of Bell’s inequality violation (see Fig. 5) where the four and IBM quantum device. S is the simulation with the 1 corners of the tetrahedron are (0,0,0), (0.1591,0,0), (0,0.21,0) noiseless singlet state and the noisy Bell test. S is the 2 and (0,0,0.03414), respectively. Fig. 5 shows the parameter simulation with the noisy singlet state and the noisy Bell test space for the noise models we use. The tetrahedron shown which works similar to IBM. here is an approximate depiction for the parameters’ range for the violation of Bell’s inequality.