⇦Chinese Physics Letters, 2023, Vol. 40, No. 10, Article code 100501 A Hierarchy in Majorana Non-Abelian Tests and Hidden Variable Models Peng Qian (钱鹏)1 and Dong E. Liu (刘东)2,1,3,4* Affiliations 1Beijing Academy of Quantum Information Sciences, Beijing 100193, China 2State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China 3Frontier Science Center for Quantum Information, Beijing 100184, China 4Hefei National Laboratory, Hefei 230088, China Received 2 July 2023; accepted manuscript online 28 August 2023; published online 22 September 2023 *Corresponding author. Email: dongeliu@mail.tsinghua.edu.cn Citation Text: Qian P and Liu D E 2023 Chin. Phys. Lett. 40 100501    Abstract The recent progress of the Majorana experiments paves a way for the future tests of non-Abelian braiding statistics and topologically protected quantum information processing. However, a deficient design in those tests could be very dangerous and reach false-positive conclusions. A careful theoretical analysis is necessary so as to develop loophole-free tests. We introduce a series of classical hidden variable models to capture certain key properties of Majorana system: non-locality, topologically non-triviality, and quantum interference. Those models could help us to classify the Majorana properties and to set up the boundaries and limitations of Majorana non-Abelian tests: fusion tests, braiding tests and test set with joint measurements. We find a hierarchy among those Majorana tests with increasing experimental complexity.
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 DOI:10.1088/0256-307X/40/10/100501 © 2023 Chinese Physics Society Article Text The building blocks of topological quantum computation[1-3] are non-Abelian anyons, which are proposed to show novel non-Abelian braiding statistics.[3-6] A simple type of non-Abelian anyon, i.e., Majorana zero mode (MZM), was theoretically proposed[7-15] in many physically realizable systems. The latest advancements in experimental studies[16-37] have substantially contributed to our understanding, although there remains contentious interpretation surrounding the observation of Majorana resonances. The gradual, experiment-based step-by-step construction of Majorana devices[38,39] is essential; however, these developments present a promising platform for examining non-Abelian braiding statistics and actual quantum information processing. The most popular schemes proposed for validating non-Abelian statistics are braiding tests and fusion tests.[3,40-44] Moreover, joint-measurement, C-NOT gate, non-trivial T-gate test and Bell non-locality test[45,46] are believed to be necessary for quantum information processing. Here, we focus on relatively simple properties: Majorana non-local behaviors, topologically non-triviality, and quantum interference and entanglement in the near future. Among them, the realistic experimental schemes for controlling Majorana systems were theoretically proposed;[40,44] and some key proposals have already been tested in the laboratory for Majorana quantum information processing.[47-54] In the hidden variable theories, the probabilities appear when we measure physical quantities due to the lack of the knowledge or statistical approximation in the underlying complex deterministic theory,[55-65] which reproduces certain behaviors in quantum mechanics. The experimental activities in the Bell inequality test eventually rule out local hidden variable explanations in several physical systems.[57,66-77] However, this is not sufficient to say that the behaviors observed in other systems, e.g., Majorana platforms, are of quantum mechanics. Furthermore, the hidden variable theory provides some significant viewpoints for investigating in quantum information process and fundamental aspects of the micro world.[78-84] For our purpose, if we want to validate Majorana behaviors seriously for future quantum information applications, it is reasonable to treat the system as a black-box and perform the test subsequently. In that sense, potential hidden variable theories need to be carefully considered and excluded. Otherwise, a deficient test design may reach a positive but incorrect conclusion. For example, braiding and measurements in Majorana systems can generate topologically protected Clifford gates including entanglement gates, which can also be realized in certain hidden variable theories.[64] Therefore, it is natural to ask whether we can find certain hidden variable theories such that each theory can capture certain properties of Majorana systems but can not capture other tests' results; and the hope is to set up the limitations and boundaries of each Majorana test by using these theories. In this work, we propose a couple of classical hidden variable (HV) models, which are designed to capture and/or distinguish certain key quantum mechanical properties of Majorana systems: simplest Majorana non-local behaviors, topologically non-triviality and quantum interference. We would show what properties can be seen in a particular test; and examine if the test results can be captured by both the theories, which help us to set up the boundaries and limitations of the corresponding test. Based on this philosophy, we show: (1) fusion tests only capture certain Majorana non-local behaviors but cannot capture others; (2) braiding tests show non-Abelian statistics indicating topologically non-triviality, but not necessarily quantum interference; (3) test set with joint measurements is helpful and necessary to capture quantum interference in Majorana systems. Finally, we conclude that there exists a hierarchy among the Majorana non-Abelian tests: fusion test, braiding test, and test set with joint measurements. Theoretical Framework. Now, we introduce a couple of HV models which are inspired from famous Spekkens' model.[64] Classical HV Theory I. We assume that: (1) we have incomplete knowledge about the system states for some unknown complexity, (2) we get an outcome and observe incomplete knowledge after specific operations called measurements, (3) we cannot get access to their hidden complexity.
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Fig. 1. (a)–(c) The HV model I, where the number from $1$ to $2n$ labels the position of boxes from left to right. (d) A four-box system (with total parity even) for fusion. If we get even parity result after box-1,2 measurement, and continue to apply box-1,2 measurement, the state will be unchanged. However, if we apply a box-2,3 measurement in the second step, we will have two possible outcomes with equal probability.
(i) State Formalism. We consider a classical theory with a box representation, where each box corresponds to an MZM as shown in Fig. 1; and the box can be either filled or empty. We assume that we can only know the parity of any (pair of) two boxes after measurement. For example, we measure the combination of the first and second boxes and then get the result even. We can only know that the two boxes are either both filled or empty, but cannot determine the exact state of each box. This uncertainty is due to the unobserved classical complexity corresponding to the hidden variables. (ii) Measurement. We show three kinds of 2-box measurements (box 1,2, box 2,3, and box 1,3) in Fig. 1. Each measurement returns two-box parity: even or odd. We also assume that the parity of the all-box system is fixed similarly to the Majorana systems. After box-1,2 measurement as an example, we could either obtain an even-parity result $\overbrace{\square\square}$ with a pair of square symbols or obtain an odd parity result $\overbrace{\bigcirc\bigcirc}$ with a pair of circle symbols; we use a bracket to label post-measurement box-pair called “connections”. Considering a four-box (with total parity even) example shown in Fig. 1(d), box-1,2 measurement will reach either an even-even state: $\overbrace{\square\square}\overbrace{\square\square}$, or an odd-odd state: $\overbrace{\bigcirc\bigcirc}\overbrace{\bigcirc\bigcirc}$ with equal probability. If we repeat the previous measurement, we will get the same result; if the measurement is different from the previous one, we will get an uncertain result.
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Fig. 2. The HV model II. (a) Two equivalent forms of state description: the lower means the back. (b) The standard configuration of a six-box state: the connection of box 1,6 is both the rightmost and frontmost as definition, and connection of box 3,5 is relative right and front to box 2,4. For non-standard configurations, we can apply crossover along with a parity change. (c) Both box-1,3 and box-2,4 measurements lead to standard configurations. (d) Braiding procedures $P^2_{23}$ and $P_{23}P_{32}$. The braiding is confined in the plane perpendicular to the paper, or in the light-blue plane shown in the bracket.
Classical HV Theory II: To include topology, we take into account the relative positions of connections if two connections have overlaps as shown in Fig. 2(a): (1) the front-back order (front connections are on the top of back ones) of the connections has a non-trivial consequence; (2) we also define the right-left order of connections, where the connection is on the right as long as one box in the pair is on the right of all boxes in other box pairs. (i) Standard States. For the order of connections discussed above, we can choose any order. However, for the convenience of discussion, we define a standard configuration with fixed order: the right connection is the front connection [e.g. in the upper part of Fig. 2(b)]. For non-standard cases, we need to apply crossovers (exchange positions of two overlapping connections) to reach a standard configuration. Each crossover will induce a parity change [an example shown in Fig. 2(b)]. In the text, we sometimes use an up-down bracket to describe an equivalent front-back position, e.g., in Fig. 2(a). (ii) Measurement. We only consider measurements in standard configurations, i.e., “standard measurement rule”. For an unidentified state, measurements [e.g., both box-2,4 and box-1,3 measurements in Fig. 2(c)] always result in standard configurations. On the other hand, if we know the connections of boxes, we will only apply the standard rule for the standard cases. If we are in a non-standard form, we will need to apply crossovers to reach a standard one. (iii) Braiding. In our theory, braiding corresponds to the position exchange of boxes in the plane shown in Fig. 2(d). Considering different directions of braiding, the exchange of box $n_1$ and box $n_2$ is labeled by $P_{n_1n_2}$ for anticlockwise exchange if $n_1 < n_2$ (for clockwise exchange if $n_1>n_2$). In an example shown in Fig. 2(d), we apply two paths of braiding operations on the same initial state $\overbrace{\square\square}\overbrace{\square\square}$. The first path contains two successive $P_{23}$, where the first $P_{23}$ takes the state to a non-standard form and we apply a crossover to reach the standard one, then applying the second $P_{23}$ to get $\overbrace{\bigcirc\bigcirc}\overbrace{\bigcirc\bigcirc}$. In the second path, the first part is the same while we apply $P_{32}$ in the next step and we get a non-standard form with a knot. To reach a standard configuration, we apply a crossover to remove the knot and to obtain the form $\overbrace{\square\square}\overbrace{\square\square}$. Majorana Non-Abelian Tests. Let us now consider a variety of Majorana tests, and check if their key properties, e.g., Majorana non-local behaviors, topologically non-triviality, and quantum interference, can be captured by our classical HV theories I and II. (I) Fusion Test. The non-Abelian anyon system follows special fusion processes which describe the outcomes after anyon combinations.[3] Majorana zero modes belong to the Ising non-Abelian anyon model, which includes three types of anyons: the vacuum $I$, non-Abelian anyon $\sigma$ (i.e., Majorana), and the fermion $\psi$. A pair of $\sigma$ combines to fuse into either a vacuum or a fermion: $\sigma\times\sigma\rightarrow I+ \psi$. With more anyons, we have multiple ways to fuse them together, where the quantum states describing fusion transformation can be written as \begin{align} |a,b\rightarrow i\rangle|i,c\rightarrow d\rangle=\sum_{j}(F^{d}_{abc})^{i}_{j}|b,c\rightarrow j\rangle|a,j\rightarrow d\rangle, \tag {1} \end{align} where $|a,b\rightarrow i\rangle|i,c\rightarrow d\rangle$ indicates a state with particular fusion channel with anyons $a$ and $b$ first fusing to $i$, and then $i$ and $c$ fusing to $d$. The matrix $(F^{d}_{abc})^{i}_{j}$ describes the transformation under different fusion channels. For Ising anyons, \begin{align} F^{\sigma}_{\sigma\sigma\sigma}=\frac{1}{\sqrt 2}\left( \begin{array}{cc} 1 & 1 \\ 1 & -1 \\ \end{array} \right).\nonumber \end{align} Now, let us first focus on our HV theory I. We have \begin{align} ⬡⬡⬡⬡\rightarrow\overbrace{\square\square}\overbrace{\square\square}\rightarrow (\overbrace{\square\overbrace{\square\square}\square}+\overbrace{\bigcirc\overbrace{\bigcirc\bigcirc}\bigcirc})/2 , \tag {2} \\ ⬡⬡⬡⬡\rightarrow\overbrace{\bigcirc\bigcirc}\overbrace{\bigcirc\bigcirc}\rightarrow (\overbrace{\square\overbrace{\square\square}\square}+\overbrace{\bigcirc\overbrace{\bigcirc\bigcirc}\bigcirc})/2 . \tag {3} \end{align} Here, the first arrow in Eqs. (2) and (3) indicates the fusion (measurement) of box 1,2, which yields either fixed even parity (vacuum $I$) or odd parity (fermion $\psi$); the following box-2,3 measurement (second arrow) can have two possibilities: box 2 and box 3 first fuse into even parity or odd parity. We see that the measurements in the HV model are analogous to the fusion, and reach the same measurement statistics as those in the Ising anyon model described by the above $F$ matrix. Considering a proposed fusion test in Ref. [41], the topological superconducting device can realize two different 4-MZM fusion routes: (1) First create two pairs of MZMs and then fuse in the same pair as their creation. (2) First create two pairs of MZMs and then fuse in the different pairs. These two fusion routes give different measurement statistics. As shown in Fig. 3, it is simple to check that the corresponding fusion procedures in our HV model generate the same measurement outcome statistics as those in Majorana cases. Our HV model can fully capture the measurement outcome after fusion-only processes in Majorana systems (see part I in the Supplemental Information (SI)[85] for more examples). This HV theory I does not include any topological consideration. Therefore, there is no doubt that the fusion tests can only capture certain Majorana non-locality behaviors but not including topological non-triviality.
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Fig. 3. Fusion process of our HV model reproducing the Majorana fusion processes in Ref. [41]. In the right path steps 1–3: the 4-box state splits into two pairs, both 2-box measurements obtain even, and then the measurement in the same box pair (box 1,2) obtains deterministic even parity result. In the left path steps 4–6: the 4-box state splits into two different pairs (box 1,4 and box 2,3) with even parity after measurement, and then the box-1,2 measurement gives us an equal probability of even or odd result.
(II) Braiding and Topology. Non-Abelian anyons show nontrivial behaviors under the braiding process, and the exchange operation of anyons $a$ and $b$ can be captured by the operator $B_{\rm ab}$, where these braiding operations can be written as $B_{ij}\propto1+\gamma_{i}\gamma_{j}$ for Majorana modes.[3] Let us consider whether our HV theory I can capture braiding relations. For example, braiding box 2,3 of initial state $\overbrace{\square\square}\overbrace{\square\square}$ requires the exchange of boxes in different box pairs, as in different fusion channels of the Majorana theory. After this braiding, we can reach $\rlap{\overbrace{\phantom{\square\square\square}}}\square\overbrace{\square\square\square}$ (no connection order is considered here), then the box-1,3 measurement (corresponding to $\sigma_{y}$ measurement in the quantum case[85]) results in even parity. However, the $\sigma_{y}$ measurement of quantum state $B_{23}|0\rangle$ reaches odd. Therefor, we fail to capture the braiding process without topological consideration. Now let us look at HV theory II. Considering the same process on the initial state $\overbrace{\square\square}\overbrace{\square\square}$, we reach the state $\overbrace{\bigcirc\bigcirc}\overbrace{\bigcirc\bigcirc}$ as shown in the first path in Fig. 2(d), which is the same as the quantum case. We further examine the Hadamard gates which can be realized using a sequence of Majorana braidings $H=B_{12}B_{23}B_{12}$ in the quantum anyon model as shown in Fig. 4(a). For our HV theory II in Fig. 4(b), we also successively braid box 1,2 ($P_{12}$), box 2,3 ($P_{23}$) and box 1,2 ($P_{12}$) and then apply a crossover to get a standard form. We can simply show any measurement here to give the same results as those in the Majorana Hadamard gate. For the complete cases please see part II of the SI.[85] Let us consider the “successive braiding”, i.e., a non-Abelian braiding test[40] with four Majorana modes $\gamma_1,\gamma_2,\gamma_3,\gamma_4$, where both $\gamma_1,\gamma_2$ and $\gamma_3,\gamma_4$ pair are initialized in even parity. We apply multiple braiding procedures ($n$ times) of $\gamma_2$ and $\gamma_3$ in an anti-clockwise way. We focus on the probability to obtain the odd result in the $(\gamma_3 \gamma_4)$ parity measurement at different $n$'s. Braiding once, the parity of $\gamma_3,\gamma_4$ has $50\%$ chance to be odd in measurement. After two successive braidings, we get the definite odd parity; after three successive braidings, we will get an equal probability of both results; after four successive braidings, we get the definite even parity. As the number $n$ increases, the probability will repeat this pattern with period 4. In our HV theory II, as shown in Fig. 4(c), we get the same measurement consequence as those in the quantum case.
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Fig. 4. (a) Hadamard gate in Majorana-based quantum computation. (b) Hadamard gate in our HV theory II wherein the second step and last step have been applied in crossovers to change into a standard configuration. (c) Example of non-Abelian braiding in the four boxes case in our HV theory II. Suppose that in the initial state, box 3,4 are in even parity. The table shows the possibility $p$ to obtain the odd parity in box-3,4 measurement, where the label $n$ describes the number of successive braiding procedures of box 2 and box 3.
In either the Majorana model or our HV theory II, the objects (MZMs or boxes) describing a specific fusion channel are all paired. Therefore, the braiding of two objects involves at most two pairs that are of four MZMs or boxes. Considering the braiding of two boxes among $2n$ boxes, this procedure only changes the topology of two connections (of two box-pairs) involved in the braiding, but can not change the relative positions between these two connections and the other $n$-2 connections. Therefore, any single braiding step in more box cases involves only four relevant boxes. Examples for braiding in more-MZM systems can be found in part II of the SI.[85] In summary, the braidings can be well described in our HV theory II with topology considerations; and this theory goes beyond the fusion tests and shows the unique topological-nontrivial property of braiding operations. (III) Joint-Measurement and Quantum Interference. Let us assume that there are two Majorana qubits, where each one is encoded into four MZMs; and the single-qubit Pauli measurements can be realized by fermion-parity measurement of different Majorana pairs within each qubit.[3] The joint-measurement can be realized by the fermion-parity measurement of four MZMs, in which two MZMs are from one qubit and the other two are from the other qubit.[3] For joint-measurement $ZZ$, the state $\frac{|0\rangle+|1\rangle}{\sqrt{2}}\otimes \frac{|0\rangle+|1\rangle}{\sqrt{2}}$ will collapse onto $(|00\rangle+|11\rangle)/\sqrt{2}$ if the measurement outcome is even, and thus generate entanglement between the two qubits. In our HV theory II with two “qubits” (eight boxes), we can measure the four-box joint-parity, in which two boxes are from the first qubit and the other two are from the second qubit. For example, if the box 1,4 and box 2,3 are initially even for both qubits, the joint-parity measurement of box 1,2 in the first qubit and box 1,2 in the second qubit gives the following projection: \begin{align} \overbrace{\square\overbrace{\square\square}\square}|&\overbrace{\square\overbrace{\square\square}\square}\rightarrow\notag\\ &\overbrace{\square\square}\overbrace{\square\square}|\overbrace{\square\square}\overbrace{\square\square} +\overbrace{\bigcirc\bigcirc}\overbrace{\bigcirc\bigcirc}|\overbrace{\bigcirc\bigcirc}\overbrace{\bigcirc\bigcirc}. \tag {4} \end{align} Here, we assume that the measurement outcome is even, and the plus sign indicates two classical uncertain states with equal probability. This is an analog to joint-measurement $ZZ$ in the quantum theory. Meanwhile, for another example with 6 MZMs in the initial state $|0_{12}0_{35}0_{46}\rangle$, we first joint-measure MZM $\gamma_{2}\gamma_{3}\gamma_{4}\gamma_{5}$ and assume even result, we get $(|0_{16}0_{23}0_{45}\rangle+|0_{16}1_{23}1_{45}\rangle)/\sqrt{2}$. Then the measurement of MZM $\gamma_{2}\gamma_{5}$ will reach a definite even result $|0_{16}0_{25}0_{34}\rangle$ due to the cancelation of quantum interference. However, as shown in Fig. 4(d), the HV theory II fails to capture this process and reaches $(\overbrace{\square\overbrace{\square\square}\overbrace{\square\square}\square}+ \overbrace{\square\overbrace{\bigcirc\bigcirc}\overbrace{\bigcirc\bigcirc}\square})/2$. We can look at another example in the Majorana system, i.e., the CNOT gate of two-MZM qubit which involves both joint-measurement and two-box measurement. We show in the part III of the SI[85] that our HV theory II can not capture CNOT gate process. It is clear that any HV theory cannot capture the complete signature of quantum interference. Therefore, we conclude that, to reveal certain signatures of quantum interference, the combination of joint-measurement and two-box measurement are helpful and necessary. Discussions. There are some other tests for more properties of Majorana systems. (1) Braiding and measurement only cannot generate T-gate, which however can be generated by preparing a noisy magic state ancilla along with state distillation.[86] T-gate test along with other Clifford operations specifies quantum property which could generate arbitrary quantum states. (2) Bell nonlocality tests clearly complete the Majorana system validation but require a fully universal quantum gate-set including T-gate to break Bell bound. However, the braiding and measurement only cannot break the Bell bound.[87-89] In the future, we need either to fabricate more sophisticated devices along with complicated experimental procedures or to propose novel and simple test schemes. Acknowledgement. This work was supported by the National Natural Science Foundation of China (Grant No. 11974198), the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302400), and the Tsinghua University Initiative Scientific Research Program. References Fault-tolerant quantum computation by anyonsTopological quantum computationNon-Abelian anyons and topological quantum computationOn the theory of identical particlesSuperselection sectors with braid group statistics and exchange algebrasNon-Abelian Statistics of Half-Quantum Vortices in p -Wave SuperconductorsPaired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effectUnpaired Majorana fermions in quantum wiresSuperconducting Proximity Effect and Majorana Fermions at the Surface of a Topological InsulatorNon-Abelian Topological Order in s -Wave Superfluids of Ultracold Fermionic AtomsGeneric New Platform for Topological Quantum Computation Using Semiconductor HeterostructuresMajorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor HeterostructuresHelical Liquids and Majorana Bound States in Quantum WiresMajorana fermions in a tunable semiconductor deviceNew directions in the pursuit of Majorana fermions in solid state systemsSignatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire DevicesAnomalous Zero-Bias Conductance Peak in a Nb–InSb Nanowire–Nb Hybrid DeviceSuperconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossoverAnomalous Modulation of a Zero-Bias Peak in a Hybrid Nanowire-Superconductor DeviceObservation of Majorana fermions in ferromagnetic atomic chains on a superconductorExponential protection of zero modes in Majorana islandsMajorana bound state in a coupled quantum-dot hybrid-nanowire systemMajorana Zero Mode Detected with Spin Selective Andreev Reflection in the Vortex of a Topological SuperconductorEvidence for Majorana bound states in an iron-based superconductorZero-energy vortex bound state in the superconducting topological surface state of Fe(Se,Te)Robust and Clean Majorana Zero Mode in the Vortex Core of High-Temperature Superconductor ( Li 0.84 Fe 0.16 ) OHFeSe Evidence of topological superconductivity in planar Josephson junctionsTopological superconductivity in a phase-controlled Josephson junctionQuantized Conductance of Majorana Zero Mode in the Vortex of the Topological Superconductor (Li0.84 Fe0.16 )OHFeSeNearly quantized conductance plateau of vortex zero mode in an iron-based superconductorPlateau Regions for Zero-Bias Peaks within 5% of the Quantized Conductance Value 2 e 2 / h Conductance-Matrix Symmetries of a Three-Terminal Hybrid DeviceClosing of the induced gap in a hybrid superconductor-semiconductor nanowireParametric exploration of zero-energy modes in three-terminal InSb-Al nanowire devicesInAs-Al hybrid devices passing the topological gap protocolNonlocal conductance spectroscopy of Andreev bound states in gate-defined InAs/Al nanowiresLocal and Nonlocal Transport Spectroscopy in Planar Josephson JunctionsNext steps of quantum transport in Majorana nanowire devicesRecent progress on Majorana in semiconductor-superconductor heterostructures—engineering and detectionFlux-controlled quantum computation with Majorana fermionsMilestones Toward Majorana-Based Quantum ComputingMajorana fermion exchange in quasi-one-dimensional networksCoulomb-assisted braiding of Majorana fermions in a Josephson junction arrayScalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modesOn the Problem of Hidden Variables in Quantum MechanicsProposed Experiment to Test Local Hidden-Variable TheoriesRealization of Microwave Quantum Circuits Using Hybrid Superconducting-Semiconducting Nanowire Josephson ElementsSemiconductor-Nanowire-Based Superconducting QubitHigh-Kinetic-Inductance Superconducting Nanowire Resonators for Circuit QED in a Magnetic FieldDirect Microwave Measurement of Andreev-Bound-State Dynamics in a Semiconductor-Nanowire Josephson JunctionDestructive Little-Parks Effect in a Full-Shell Nanowire-Based TransmonMagnetic-Field-Resilient Superconducting Coplanar-Waveguide Resonators for Hybrid Circuit Quantum Electrodynamics ExperimentsParity-Protected Superconductor-Semiconductor QubitPhoton-assisted tunnelling of zero modes in a Majorana wireA Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I"Relative State" Formulation of Quantum MechanicsExperimental Test of Bell's Inequalities Using Time- Varying AnalyzersBell's inequality holds for all non-product statesHidden variables and the two theorems of John BellDynamical reduction modelsQuantum computing and hidden variablesResearch on hidden variable theories: A review of recent progressesNonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility TheoremEvidence for the epistemic view of quantum states: A toy theoryLocal hidden–variable models for entangled quantum statesViolation of Bell's Inequality under Strict Einstein Locality ConditionsRobust Long-Distance Entanglement and a Loophole-Free Bell Test with Ions and PhotonsProposal for a Loophole-Free Bell Test Using Homodyne DetectionNo extension of quantum theory can have improved predictive powerExperimental high-dimensional two-photon entanglement and violations of generalized Bell inequalitiesLoophole-free Bell inequality violation using electron spins separated by 1.3 kilometresLoophole-free Bell test using electron spins in diamond: second experiment and additional analysisEvent-Ready Bell Test Using Entangled Atoms Simultaneously Closing Detection and Locality LoopholesBell inequalities and entanglement in solid-state devicesSpin-current noise and Bell inequalities in a realistic superconductor-quantum dot entanglerViolating Bell’s inequality with remotely connected superconducting qubitsMacroscopic entanglement and violation of Bell’s inequalities between two spatially separated quantum dots in a planar photonic crystal systemOn the reality of the quantum stateInformation processing in generalized probabilistic theoriesRealism and the physical worldEinstein, Incompleteness, and the Epistemic View of Quantum StatesEntropy and information causality in general probabilistic theoriesQuantum-Bayesian coherenceQuantum protocols within Spekkens' toy modelUniversal quantum computation with ideal Clifford gates and noisy ancillasFault-tolerant quantum random-number generator certified by Majorana fermionsA Practical Phase Gate for Producing Bell Violations in Majorana WiresUbiquitous Nonlocal Entanglement with Majorana Zero Modes
[1] Kitaev A Y 2003 Ann. Phys. 303 2
[2] Freedman M, Kitaev A, Larsen M, and Wang Z 2003 Bull. Amer. Math. Soc. 40 31
[3] Nayak C, Simon S H, Stern A, Freedman M, and Sarma S D 2008 Rev. Mod. Phys. 80 1083
[4] Leinaas J M and Myrheim J 1977 Il Nuovo Cimento B (1971-1996) 37 1
[5] Fredenhagen K, Rehren K H, and Schroer B 1989 Commun. Math. Phys. 125 201
[6] Ivanov D A 2001 Phys. Rev. Lett. 86 268
[7] Read N and Green D 2000 Phys. Rev. B 61 10267
[8] Kitaev A Y 2001 Phys. Usp. 44 131
[9] Fu L and Kane C L 2008 Phys. Rev. Lett. 100 096407
[10] Sato M, Takahashi Y, and Fujimoto S 2009 Phys. Rev. Lett. 103 020401
[11] Sau J D, Lutchyn R M, Tewari S, and Sarma S D 2010 Phys. Rev. Lett. 104 040502
[12] Lutchyn R M, Sau J D, and Sarma S D 2010 Phys. Rev. Lett. 105 077001
[13] Oreg Y, Refael G, and von Oppen F 2010 Phys. Rev. Lett. 105 177002
[14] Alicea J 2010 Phys. Rev. B 81 125318
[15] Alicea J 2012 Rep. Prog. Phys. 75 076501
[16] Mourik V, Zuo K, Frolov S M, Plissard S R, Bakkers E P A M, and Kouwenhoven L P 2012 Science 336 1003
[17] Deng M, Yu C, Huang G, Larsson M, Caroff P, and Xu H 2012 Nano Lett. 12 6414
[18] Churchill H O H, Fatemi V, Grove-Rasmussen K, Deng M T, Caroff P, Xu H Q, and Marcus C M 2013 Phys. Rev. B 87 241401
[19] Finck A D K, van Harlingen D J, Mohseni P K, Jung K, and Li X 2013 Phys. Rev. Lett. 110 126406
[20] Nadj-Perge S, Drozdov I K, Li J, Chen H, Jeon S, Seo J, MacDonald A H, Bernevig B A, and Yazdani A 2014 Science 346 602
[21] Albrecht S M, Higginbotham A P, Madsen M, Kuemmeth F, Jespersen T S, Nygård J, Krogstrup P, and Marcus C 2016 Nature 531 206
[22] Deng M, Vaitiekėnas S, Hansen E B, Danon J, Leijnse M, Flensberg K, Nygård J, Krogstrup P, and Marcus C M 2016 Science 354 1557
[23] Sun H H, Zhang K W, Hu L H, Li C, Wang G Y, Ma H Y, Xu Z A, Gao C L, Guan D D, Li Y Y, Liu C, Qian D, Zhou Y, Fu L, Li S C, Zhang F C, and Jia J F 2016 Phys. Rev. Lett. 116 257003
[24] Wang D F, Kong L Y, Fan P et al. 2018 Science 362 333
[25] Machida T, Sun Y, Pyon S, Takeda S, Kohsaka Y, Hanaguri T, Sasagawa T, and Tamegai T 2019 Nat. Mater. 18 811
[26] Liu Q, Chen C, Zhang T, Peng R, Yan Y J, Wen C H P, Lou X, Huang Y L, Tian J P, Dong X L, Wang G W, Bao W C, Wang Q H, Yin Z P, Zhao Z X, and Feng D L 2018 Phys. Rev. X 8 041056
[27] Fornieri A, Whiticar A M, Setiawan F et al. 2019 Nature 569 89
[28] Ren H C, Pientka F, Hart S et al. 2019 Nature 569 93
[29] Chen C, Liu Q, Zhang T, Li D, Shen P, Dong X, Zhao Z X, Zhang T, and Feng D 2019 Chin. Phys. Lett. 36 057403
[30] Zhu S Y, Kong L Y, Cao L et al. 2020 Science 367 189
[31] Wang Z Y, Song H D, Pan D, Zhang Z T, Miao W T, Li R D, Cao Z, Zhang G, Liu L, Wen L J, Zhuo R, Liu D E, He K, Shang R, Zhao J, and Zhang H 2022 Phys. Rev. Lett. 129 167702
[32] Ménard G C, Anselmetti G L R, Martinez E A, Puglia D, Malinowski F K, Lee J S, Choi S, Pendharkar M, Palmstrøm C J, Flensberg K, Marcus C M, Casparis L, and Higginbotham A P 2020 Phys. Rev. Lett. 124 036802
[33] Puglia D, Martinez E A, Ménard G C, Pöschl A, Gronin S, Gardner G C, Kallaher R, Manfra M J, Marcus C M, Higginbotham A P, and Casparis L 2021 Phys. Rev. B 103 235201
[34] Wang J Y, van Loo N, Mazur G P, Levajac V, Malinowski F K, Lemang M, Borsoi F, Badawy G, Gazibegovic S, Bakkers E P A M, Quintero-Pérez M, Heedt S, and Kouwenhoven L P 2022 Phys. Rev. B 106 075306
[35] Aghaee M, Akkala A, Alam Z et al. (Microsoft Quantum) 2023 Phys. Rev. B 107 245423
[36] Pöschl A, Danilenko A, Sabonis D, Kristjuhan K, Lindemann T, Thomas C, Manfra M J, and Marcus C M 2022 Phys. Rev. B 106 L241301
[37] Banerjee A, Lesser O, Rahman M A, Thomas C, Wang T, Manfra M J, Berg E, Oreg Y, Stern A, and Marcus C M 2023 Phys. Rev. Lett. 130 096202
[38] Zhang H, Liu D E, Wimmer M, and Kouwenhoven L P 2019 Nat. Commun. 10 5128
[39] Cao Z, Chen S, Zhang G, and Liu D E 2023 Sci. Chin. Phys. Mech. & Astron. 66 267003
[40] Hyart T, van Heck B, Fulga I C, Burrello M, Akhmerov A R, and Beenakker C W J 2013 Phys. Rev. B 88 035121
[41] Aasen D, Hell M, Mishmash R V, Higginbotham A, Danon J, Leijnse M, Jespersen T S, Folk J A, Marcus C M, Flensberg K, and Alicea J 2016 Phys. Rev. X 6 031016
[42] Clarke D J, Sau J D, and Tewari S 2011 Phys. Rev. B 84 035120
[43] van Heck B, Akhmerov A, Hassler F, Burrello M, and Beenakker C 2012 New J. Phys. 14 035019
[44] Karzig T, Knapp C, Lutchyn R M, Bonderson P, Hastings M B, Nayak C, Alicea J, Flensberg K, Plugge S, Oreg Y, Marcus C M, and Freedman M H 2017 Phys. Rev. B 95 235305
[45] Bell J S 1966 Rev. Mod. Phys. 38 447
[46] Clauser J F, Horne M A, Shimony A, and Holt R A 1969 Phys. Rev. Lett. 23 880
[47] de Lange G, van Heck B, Bruno A, van Woerkom D J, Geresdi A, Plissard S R, Bakkers E P A M, Akhmerov A R, and DiCarlo L 2015 Phys. Rev. Lett. 115 127002
[48] Larsen T W, Petersson K D, Kuemmeth F, Jespersen T S, Krogstrup P, Nygård J, and Marcus C M 2015 Phys. Rev. Lett. 115 127001
[49] Samkharadze N, Bruno A, Scarlino P, Zheng G, DiVincenzo D P, DiCarlo L, and Vandersypen L M K 2016 Phys. Rev. Appl. 5 044004
[50] Hays M, de Lange G, Serniak K, van Woerkom D J, Bouman D, Krogstrup P, Nygård J, Geresdi A, and Devoret M H 2018 Phys. Rev. Lett. 121 047001
[51] Sabonis D, Erlandsson O, Kringhøj A, van Heck B, Larsen T W, Petkovic I, Krogstrup P, Petersson K D, and Marcus C M 2020 Phys. Rev. Lett. 125 156804
[52] Kroll J, Borsoi F, van der Enden K, Uilhoorn W, de Jong D, Quintero-Pérez M, van Woerkom D, Bruno A, Plissard S, Car D, Bakkers E, Cassidy M, and Kouwenhoven L 2019 Phys. Rev. Appl. 11 064053
[53] Larsen T W, Gershenson M E, Casparis L, Kringhøj A, Pearson N J, McNeil R P G, Kuemmeth F, Krogstrup P, Petersson K D, and Marcus C M 2020 Phys. Rev. Lett. 125 056801
[54] van Zanten D M T, Sabonis D, Suter J et al. 2020 Nat. Phys. 16 663
[55] Bohm D 1952 Phys. Rev. 85 166
[56] Everett H 1957 Rev. Mod. Phys. 29 454
[57] Aspect A, Dalibard J, and Roger G 1982 Phys. Rev. Lett. 49 1804
[58] Gisin N 1991 Phys. Lett. A 154 201
[59] Mermin N D 1993 Rev. Mod. Phys. 65 803
[60] Bassi A and Ghirardi G 2003 Phys. Rep. 379 257
[61] Aaronson S 2005 Phys. Rev. A 71 032325
[62] Genovese M 2005 Phys. Rep. 413 319
[63] Leggett A J 2003 Found. Phys. 33 1469
[64] Spekkens R W 2007 Phys. Rev. A 75 032110
[65] Augusiak R, Demianowicz M, and Acín A 2014 J. Phys. A 47 424002
[66] Weihs G, Jennewein T, Simon C, Weinfurter H, and Zeilinger A 1998 Phys. Rev. Lett. 81 5039
[67] Simon C and Irvine W T M 2003 Phys. Rev. Lett. 91 110405
[68] García-Patrón R, Fiurášek J, Cerf N J, Wenger J, Tualle-Brouri R, and Grangier P 2004 Phys. Rev. Lett. 93 130409
[69] Colbeck R and Renner R 2011 Nat. Commun. 2 411
[70] Dada A C, Leach J, Buller G S, Padgett M J, and Andersson E 2011 Nat. Phys. 7 677
[71] Hensen B, Bernien H, Dréau A E et al. 2015 Nature 526 682
[72] Hensen B, Kalb N, Blok M et al. 2016 Sci. Rep. 6 30289
[73] Rosenfeld W, Burchardt D, Garthoff R, Redeker K, Ortegel N, Rau M, and Weinfurter H 2017 Phys. Rev. Lett. 119 010402
[74] Chtchelkatchev N M, Blatter G, Lesovik G B, and Martin T 2002 Phys. Rev. B 66 161320
[75] Sauret O, Martin T, and Feinberg D 2005 Phys. Rev. B 72 024544
[76] Zhong Y P, Chang H S, Satzinger K, Chou M H, Bienfait A, Conner C R, Dumur É, Grebel J, Peairs G, Povey R et al. 2019 Nat. Phys. 15 741
[77] Yao P and Hughes S 2009 Opt. Express 17 11505
[78] Pusey M F, Barrett J, and Rudolph T 2012 Nat. Phys. 8 475
[79] Barrett J 2007 Phys. Rev. A 75 032304
[80] Leggett A J 2008 Rep. Prog. Phys. 71 022001
[81] Harrigan N and Spekkens R W 2010 Found. Phys. 40 125
[82] Barnum H, Barrett J, Clark L O, Leifer M, Spekkens R, Stepanik N, Wilce A, and Wilke R 2010 New J. Phys. 12 033024
[83] Fuchs C A and Schack R 2013 Rev. Mod. Phys. 85 1693
[84] Disilvestro L and Markham D 2017 Phys. Rev. A 95 052324
[85]See the Supplemental Information for more details.
[86] Bravyi S and Kitaev A 2005 Phys. Rev. A 71 022316
[87] Deng D L and Duan L M 2013 Phys. Rev. A 88 012323
[88] Clarke D J, Sau J D, and Sarma S D 2016 Phys. Rev. X 6 021005
[89] Romito A and Gefen Y 2017 Phys. Rev. Lett. 119 157702