Certified Quantum Measurement of Majorana Fermions

TL;DR

This paper introduces a quantum self-testing protocol to certify Majorana fermion measurements, demonstrating anti-commutativity and quantum contextuality, with robustness to experimental errors and fidelity bounds.

Contribution

It provides a novel, robust protocol for certifying Majorana fermions through measurement statistics and contextuality witnesses, advancing experimental verification methods.

Findings

Ideal measurement statistics imply anti-commutativity of Majorana operators

The protocol is robust to experimental errors with linear fidelity bounds

Violation of the contextuality inequality confirms quantum nature of measurements

Abstract

We present a quantum self-testing protocol to certify measurements of fermion parity involving Majorana fermion modes. We show that observing a set of ideal measurement statistics implies anti-commutativity of the implemented Majorana fermion parity operators, a necessary prerequisite for Majorana detection. Our protocol is robust to experimental errors. We obtain lower bounds on the fidelities of the state and measurement operators that are linear in the errors. We propose to analyze experimental outcomes in terms of a contextuality witness $W$, which satisfies $\langle W \rangle \le 3$ for any classical probabilistic model of the data. A violation of the inequality witnesses quantum contextuality, and the closeness to the maximum ideal value $\langle W \rangle=5$ indicates the degree of confidence in the detection of Majorana fermions.

Tables

Table 1. Table 1: Mapping between the (logical) 4 4 4 -dimensional and (physical) 8 8 8 -dimensional spaces (fermion parity assingnments for P 36 subscript 𝑃 36 P_{36} , P 12 subscript 𝑃 12 P_{12} and P 45 subscript 𝑃 45 P_{45} and three qubits representations). For each logical state the upper row corresponds to even parity, while the lower to odd parity. Here | Φ ± ⟩ = | ↑ ↑ ⟩ ± | ↓ ↓ ⟩ 2 ket subscript Φ plus-or-minus plus-or-minus ket ↑ absent ↑ ket ↓ absent ↓ 2 \ket{\Phi_{\pm}}=\frac{\ket{\uparrow\uparrow}\pm\ket{\downarrow\downarrow}}{\sqrt{2}} and | Φ ± ′ ⟩ = | ↑ ↓ ⟩ ± | ↓ ↑ ⟩ 2 ket subscript superscript Φ ′ plus-or-minus plus-or-minus ket ↑ absent ↓ ket ↓ absent ↑ 2 \ket{\Phi^{\prime}_{\pm}}=\frac{\ket{\uparrow\downarrow}\pm\ket{\downarrow\uparrow}}{\sqrt{2}} .

Logical Qubits	Fermion Parities	Physical Qubits
$|00⟩$	$|+,+,+⟩$	$|\downarrow ⟩|{\Phi }_{-}⟩$
	$|-,+,+⟩$	$|\downarrow ⟩|{\Phi }_{-}^{\prime }⟩$
$|01⟩$	$|-,+,-⟩$	$-|\downarrow ⟩|{\Phi }_{+}⟩$
	$|+,+,-⟩$	$-|\downarrow ⟩|{\Phi }_{+}^{\prime }⟩$
$|10⟩$	$|-,-,+⟩$	$|\uparrow ⟩|{\Phi }_{-}^{\prime }⟩$
	$|+,-,+⟩$	$-|\uparrow ⟩|{\Phi }_{-}⟩$
$|11⟩$	$|+,-,-⟩$	$-|\uparrow ⟩|{\Phi }_{+}^{\prime }⟩$
	$|-,-,-⟩$	$|\uparrow ⟩|{\Phi }_{+}⟩$

Table 2. Table 2: (Top) Peres-Mermin-like magic squares using Majorana fermion parity operators and related three physical qubits. Operators in the same row or column commute. (Bottom) The “emergent” operators realize Peres-Mermin magic squares in the (four-dimensional) even-parity ( 𝒫 = + 1 𝒫 1 {\cal P}=+1 ) or odd-parity ( 𝒫 = − 1 𝒫 1 {\cal P}=-1 ) subspaces.

$P_{36}$	$P_{25}$	$P_{14}$
$P_{12}$	$P_{34}$	$P_{56}$
$P_{45}$	$P_{16}$	$P_{23}$

Table 3. Table 3: (Left) The six unknown operators and five contexts. (Right) The six “emergent” logical two-qubit Pauli operators.

$A_{12}$	$A_{34}$	$A_{56}$
$A_{45}$	$A_{16}$	$A_{23}$

Table 4. Table 5: Expectation values of product of operators in rows ( ℛ i subscript ℛ 𝑖 {\cal R}_{i} ) and columns ( 𝒞 i subscript 𝒞 𝑖 {\cal C}_{i} ) for different initial states.

Initial state	Expectation Values
$|a_{36}^{\left(0\right)},a_{25}^{\left(0\right)},a_{14}^{\left(0\right)}⟩$	${ℛ}_1$	${ℛ}_2$	${𝒞}_1$	${𝒞}_2$	${𝒞}_3$
$|+,+,+⟩$	$-1$	$-1$	$-1$	$-1$	$-1$
$|+,+,-⟩$	$+1$	$+1$	$+1$	$+1$	$-1$
$|+,-,+⟩$	$+1$	$+1$	$+1$	$-1$	$+1$
$|+,-,-⟩$	$-1$	$-1$	$-1$	$+1$	$+1$
$|-,+,+⟩$	$+1$	$+1$	$-1$	$+1$	$+1$
$|-,+,-⟩$	$-1$	$-1$	$+1$	$-1$	$+1$
$|-,-,+⟩$	$-1$	$-1$	$+1$	$+1$	$-1$
$|-,-,-⟩$	$+1$	$+1$	$-1$	$-1$	$-1$

Equations

$${\gamma }_j^{†}={\gamma }_j^{},\text{mbox}and\{{\gamma }_j^{},{\gamma }_k^{}\}={\gamma }_j^{}{\gamma }_k^{}+{\gamma }_k^{}{\gamma }_j^{}=2{\delta }_{jk}.$$

$${\gamma }_{2m-1}^{}$$

$${\gamma }_{2m}^{}$$

$$P_{jk}=i{\gamma }_j^{}{\gamma }_k^{},1\le j<k\le 6.$$

$$\displaystyle\{P_{36},P_{25},P_{14}\}\ ,\ \{P_{12},P_{34},P_{56}\}\ ,\ \{P_{45},P_{16},P_{23}\}\ ,\$$

$$\displaystyle\{P_{36},P_{12},P_{45}\}\ ,\ \{P_{25},P_{34},P_{16}\}\ ,\ \{P_{14},P_{56},P_{23}\}\ ,\$$

$$\displaystyle\{P_{35},P_{16},P_{24}\}\ ,\ \{P_{46},P_{25},P_{13}\}\ ,\ \{P_{12},P_{35},P_{46}\}\ ,\$$

$$\displaystyle\{P_{56},P_{24},P_{13}\}\ ,\ \{P_{46},P_{15},P_{23}\}\ ,\ \{P_{34},P_{26},P_{15}\}\ ,\$$

$$\displaystyle\{P_{36},P_{24},P_{15}\}\ ,\ \{P_{13},P_{26},P_{45}\}\ ,\ \{P_{35},P_{26},P_{14}\}\ .\$$

$$⟨\Psi |\prod _{r\in {\mathcal{R}}_i}{A}_r|\Psi ⟩$$

$$⟨\Psi |\prod _{r\in {\mathcal{C}}_i}{A}_r|\Psi ⟩$$

$$⟨\Psi |\prod _{r\in {\mathcal{R}}_i}{A}_r|\Psi ⟩$$

$$\pm ⟨\Psi |\prod _{r\in {\mathcal{C}}_i}{A}_r|\Psi ⟩$$

$$F(|\hat{\Psi }⟩,|\Psi ⟩)$$

$$F({\hat{A}}_r,A_r)$$

$${\hat{Q}}_a=U_{\mathcal{S}\mathcal{E}}^{†}(\text{mathds}{1}_{\mathcal{S}}\otimes |a⟩{⟨a|}_{\mathcal{E}})U_{\mathcal{S}\mathcal{E}},$$

$$U_{\mathcal{S}{\mathcal{E}}_r}|\Psi ⟩\otimes {|a⟩}_r$$

$$\begin{gathered} ⟨\Phi |{⟨a|}_r{U}_{\mathcal{S}{\mathcal{E}}_r}^{†}{U}_{\mathcal{S}{\mathcal{E}}_r}|\Psi ⟩{|b⟩}_r \\ =\sum _{c=0}^1(-1{)}^{(a+b)c}⟨\Phi |\sqrt{Q_{r,a\oplus c}}\sqrt{Q_{r,b\oplus c}}|\Psi ⟩. \end{gathered}$$

$$⟨\Phi |\sum _{c=0}^1{Q}_{r,c}|\Psi ⟩=⟨\Phi ||\Psi ⟩,$$

$$\bra{\Phi}\big{(}\sqrt{Q_{r,a}}\sqrt{Q_{r,b}}-\sqrt{Q_{r,b}}\sqrt{Q_{r,a}}\,\big{)}\ket{\Psi}=0,$$

$$\text{Tr}({\hat{Q}}_{r,a}(\rho \otimes |0⟩{⟨0|}_r))$$

$$=$$

$$U_{{\cal S}{\cal E}_{r}}=\left(\begin{array}[]{cc}\sqrt{Q_{r,0}}&\sqrt{Q_{r,1}}\\ \sqrt{Q_{r,1}}&-\sqrt{Q_{r,0}}\\ \end{array}\right),\ \hat{Q}_{s,b}=\left(\begin{array}[]{cc}Q_{s,b}&0\\ 0&Q_{s,b}\\ \end{array}\right).$$

$$\prod _{r\in {\mathcal{R}}_i}{A}_r|\Psi ⟩$$

$$\prod _{r\in {\mathcal{C}}_i}{A}_r|\Psi ⟩$$

$$A_{12}{A}_{34}|\Psi ⟩$$

$$A_{34}|\Psi ⟩$$

$$\begin{gathered} A_{12}{A}_{16}|\Psi ⟩=A_{12}{A}_{34}|\Psi ⟩=A_{56}|\Psi ⟩ \\ =-A_{23}|\Psi ⟩=-A_{16}{A}_{12}|\Psi ⟩=-A_{16}{A}_{12}|\Psi ⟩. \end{gathered}$$

$$V^{\prime }=\mathrm{span}\{|\Psi ⟩,A_{12}|\Psi ⟩,A_{16}|\Psi ⟩,A_{12}{A}_{16}|\Psi ⟩\}.$$

$$A_{56}|\Psi ⟩$$

$$A_{56}{A}_{12}|\Psi ⟩$$

$$A_{56}{A}_{16}|\Psi ⟩$$

$$A_{56}{A}_{12}{A}_{16}|\Psi ⟩$$

$$[A,B]=0$$

$$\{A,B\}=0$$

$$[{\mathbb{A}}_{12}{\mathbb{A}}_{16},{\mathbb{A}}_{45}{\mathbb{A}}_{34}]=0.$$

$${\mathbb{A}}_{12}{\mathbb{A}}_{16}|\alpha ,\beta ⟩$$

Full text

I Introduction

Topological qubits offer promising basic units for quantum information processing due to their inherent resilience against decoherence Alicea (2012). Majorana fermions Majorana (1937) are candidates for realizing such topological qubits, and the ability to braid them is the focus of several recent investigations. Theoretically, Majorana fermions emerge from the interplay between the existence of a topologically non-trivial vacuum and a, typically, symmetry-protected physical boundary (or defect). They are realized as zero-energy modes or quasi-particle excitations of certain quantum systems. Recent experimental efforts to detect and control Majorana zero-energy modes in topological superconducting nanowires provide a step towards realizing non-Abelian braiding and, thus, topological computation. Several experimental groups have reported evidence of Majorana zero-energy modes, such as an observation of a zero bias conductance peak or Shapiro steps in superconducting nanowires et al. (2012); Rokhinson et al. (2012). The evidence, however, remains indirect and it is unclear what would constitute proof of the existence of Majorana fermions Liu et al. (2012). Moreover, interpretation of what embodies a Majorana excitation, and its physical realization, in a closed particle number-conserving many-body topological superfluid deepens the mystery Ortiz et al. (2014); Ortiz and Cobanera (2016).

Even if one had strong evidence that a system is in a topological superfluid phase with emerging Majorana fermions, in order to reap the advantages of the topological approach to quantum computing, one must be confident that the measurements performed actually implement ideal quantum operations with high fidelity. This is especially important for proposals where gates are performed by parity measurements and anyonic teleportation, rather than physical braiding Bonderson et al. (2009). In this paper, we present a protocol to certify quantum measurements of observables and states using only the statistics of measurements outcomes, while making no assumptions about the underlying physics in the experimental apparatus. Our technique represents an extension of what is known as self-testing in quantum information Mayers and Yao (2004); Acín and et al. (2007); Colbeck and Kent (2011). In particular, we are interested in currently proposed platforms utilizing fermionic parity measurements et al. (2017). In this way, and given experimental data, one hopes to argue for the consistency of that data with the existence of Majorana fermions.

In the quantum information literature, self-testing refers to the action of uniquely determining a quantum state, up to a certain notion of equivalence. Unlike tomography, self-testing is based solely on the statistics of measurement outcomes, with minimal assumptions about the measurement operators. These quantum self-testing protocols are more stringent than the well-known Bell tests Popescu and Rohrlich (1992). While violation of a Bell inequality for a bipartite system establishes that its quantum state is entangled, it cannot certify, for instance, that its quantum state is maximally entangled Romito and Gefen (2017). Self-testing protocols typically assume that the physical system has a Hilbert space with a natural local tensor product structure. For self-testing a fermionic system, however, we have to relax this assumption. In our scenario, involving 6 Majorana fermion modes and 6 parity operators, a minimal assumption is compatibility of observables sharing no common Majorana mode. A successful certification implies that the experimentally measured observables anti-commute exactly the way ideal fermionic parity operators should. We demonstrate that ideal statistics imply emergence of an invariant four-dimensional tensor-product subspace (encoding two logical qubits) out of a putative Majorana fermion non-tensor-product state space, and the ideal state is a Bell state up to local unitary equivalence. An observation of the ideal statistics in our protocol would constitute substantive evidence of the existence of Majorana fermions. This is so, since ideal statistics implies anti-commutativity of a Majorana fermion and its parity operator, a definite smoking gun for Majorana fermion detection. Experiments, however, suffer from imperfections, and any practical certification protocol should include the effect of non-ideal quantum measurement devices and procedures. We have obtained lower bounds on state and operator fidelities, linear in the error, that constitute rigorous statements on robustness of the self-testing protocol for detection of Majorana fermions.

The paper is organized as follows. Preliminary background concepts and strategy for self-testing Majorana fermion parities are discussed in Sec. II. In particular, in Sec. II.1, we map Majorana fermion parity operators to two-qubit Pauli operators, and construct maximal sets of compatible measurements, so-called contexts. In Sec. II.2, we introduce the notion of quantum self-testing. Section III describes our particular measurement scenario and contains a summary of our main results, which are two theorems, proved later in Sections IV and V. Specifically, we prove rigidity of the measurement scenario in Sec. IV, and address the robustness to small experimental errors in Sec. V. Finally, in Sec. VI we summarize main findings and analyze our fermion parity certification protocol from the standpoint of a contextuality witness $W$. We suggest a possible experimental setup and propose to analyze experimental data validating a contextuality inequality involving such $W$. We also emphasize the generality of our approach and its potential application to other quantum measurements involving phenomena such as braiding. An accessible discussion, addressed to experimentalists, of what an ideal statistics situation means in the context of self-testing fermion parities is presented in Appendix A. Several technical details, important to appreciate the mathematical and physical implications of our results, are included in the the Appendices B, C, and D.

II Background

II.1 Majorana Fermions

Majorana fermion modes are potential blueprint qubits for topological computation. Consider $6$ Majorana modes belonging to $6$ different quantum wires or vortices. Those modes are defined by Majorana operators $\gamma_{j}^{\;}$ for $j=1,\ldots,6$, which satisfy the Majorana algebra

[TABLE]

The complex $\dagger$-closed algebra generated is $\dagger$-isomorphic to the complex $8\times 8$ matrices, so its irreducible representations on a Hilbert space all can be identified with a Jordan-Wigner representation on $3$ two-level (qubit) systems. Explicitly, one such representation maps

[TABLE]

where $\sigma^{m}_{\tau}$, $m=1,2,3$ and $\sigma_{\tau}=\sigma_{x},\sigma_{y},\sigma_{z}$, are Pauli matrices and we have chosen a particular sign convention without physical consequences.

In the following, we confine ourselves to the $15$ physically measurable “parity” observables

[TABLE]

The total parity ${\cal P}=-i\prod_{j}\gamma_{j}$, commutes with every other parity observable, partitions the full Hilbert space into even (${\cal P}=+1$) and odd (${\cal P}=-1$) parity subspaces. These subspaces are invariant under the action of any parity operator and are isomorphic to logical two-qubit subspaces as illustrated by the mapping of Table 1. We use $X$, $Y$, $Z$ to denote logical Pauli operators acting on these two-qubit subspaces.

We say a set of fermion parity measurements are compatible if the corresponding parity operators are mutually commuting. There are exactly $15$ maximal sets of compatible measurements, which are given by

[TABLE]

We can select the first $6$ of those sets and form a $3\times 3$ table which works like a Peres-Mermin magic square Peres (1990a); Mermin (1990) up to a unitary equivalence in both even and odd parity subspaces, as illustrated in Table 2.

II.2 Quantum Self-testing

A self-testing protocol aims to certify that both an unknown state $\ket{\Psi}$ and a set of unknown measurements are equivalent to an ideal, usually entangled, state $|\hat{\Psi}\rangle$ and a set of ideal measurements. Importantly, the certification does not rely on any assumptions about the state and measurements, other than the assumption that certain pairs of measurement operators commute. The protocol involves repeatedly performing different sets of pairwise commuting measurements. If the ideal measurement statistics are obtained, then the state and measurements are uniquely determined, up to some notion of equivalence. This was first observed by Popescu and Rohrlich Popescu and Rohrlich (1992), who proved that any state that maximally violates a particular Bell inequality (the CHSH inequality) is equivalent to a singlet state of two qubits. The equivalence is up to a local isometry, because the measurement statistics are unaffected by a local change of basis and by the existence of an auxiliary subsystem on which the measurements act trivially. The notion of self-testing was formalized by Mayers and Yao Mayers and Yao (2004), and since then, self-testing protocols for many other states and measurement scenarios McKague (2014); Wu et al. (2014); Kaniewski (2016); Coladangelo et al. (2017); Kalev and Miller (2017); Breiner et al. (2019) have been discovered. Such protocols are often called device-independent because they rely only on the statistics of measurement outcomes, and not on any physical assumptions about the measurement apparatus.

Two important notions in the self-testing literature are that of rigidity and robustness. A measurement scenario is rigid if achieving the ideal expectation values uniquely determines the state and measurements, up to a local isometry. In any real experiment, however, the ideal statistics will not be achieved exactly due to errors in the state preparation and measurements. Thus, any practical self-testing protocol must include a robustness statement. Robustness implies that the state and measurements are still determined approximately if the statistics deviate from the ideal case by a small amount. There are fewer known robustness results for measurements than for states Kaniewski (2017). Our main results are a rigidity theorem and a robustness theorem for Majorana fermion parity operators.

Our results differ from previous self-testing results in a few respects. First, self-testing scenarios typically involve two or more parties whose measurement operators commute due to a locality assumption. The locality can be physically enforced, for example, by requiring the measurements made by different parties to be spacelike separated. In the scenario we consider, there is no natural notion of locality. Therefore, we do not assume that the full Hilbert space ${\cal H}$ factors as a tensor product. Nonetheless, as we show, if the measurement operators have the ideal expectation values, then there is a natural tensor product decomposition. The unknown state $\ket{\Psi}$ is maximally entangled with respect to this emergent tensor product structure. Second, robust self-testing statements are often formulated in terms of an extraction map, which acts on a joint system comprised of the unknown Hilbert space and a reference Hilbert space with a known dimension. In this formulation, a robustness statement asserts that there exists such an extraction map, such that the output state of the reference system has high fidelity with the ideal state McKague et al. (2012a, b). Our theorems avoid using an extraction map and instead directly construct a four-dimensional subspace of ${\cal H}$. In the rigid case, we show that the subspace contains $\ket{\Psi}$ and is invariant under the action of each of the measurement operators. In the case of errors we define an ideal state and ideal operators on the subspace and we lower bound the fidelities of the actual state and measurement operators.

III Statement of Results

We consider an experimental setup ideally involving 6 Majorana modes and 15 parity operators. However, we do not assume Majorana fermion parity operators from the outset as our aim is to infer Majorana behavior solely from the statistics of measurement outcomes. We assume that a quantum system is prepared in some unknown state $\rho$. Since any mixed state has a pure state extension, we can take $\rho=\ket{\Psi}\bra{\Psi}$ to be pure without loss of generality. We also assume a set of unknown measurements, each of which is given by a two-element positive operator-valued measure (POVM) $Q_{r}=\{Q_{r,0},Q_{r,1}\}$. Here $r\in\{1,\ldots,15\}$ labels the measurement and for all $r$, $Q_{r,0}+Q_{r,1}=\mathds{1}$ and $Q_{r,a}\geq 0$, with $a=0,1$. We assume that $[Q_{r,a},Q_{s,b}]=0$ whenever $r$ and $s$ correspond to parities having no Majorana modes in common. We emphasize that no other assumptions about the state or measurements are made. In particular, we do not assume the dimension of ${\cal H}$ or that ${\cal H}$ factors as a tensor product.

Before proving our main results, we first show that the POVM elements can be taken to be orthogonal projectors without loss of generality. In general, by Neumark’s dilation theorem Peres (1990b), any POVM can be realized by a projective measurement on an extended Hilbert space. In our case, we further require that the projectors obtained by dilation have the same pairwise commutativity structure that was assumed of the POVMs. This is accomplished by the following proposition, which we prove in Sec. IV.

Proposition 1**.**

Let $\{Q_{r,a}\}$ be a set of two-outcome POVM elements. Then there exist projectors $\hat{Q}_{r,a}$ satisfying $\Tr(\hat{Q}_{r,a}(\rho\otimes\ket{0}\bra{0}_{r}))=\Tr(Q_{r,a}\rho)$ for all density operators $\rho$, and $[\hat{Q}_{r,a},\hat{Q}_{s,b}]=0$ whenever $[Q_{r,a},Q_{s,b}]=0$.

Having extended the POVMs to projective measurements, we can define the Hermitian operators $A_{r}=2\hat{Q}_{r,0}-\mathds{1}$. Note that $A_{r}^{2}=\mathds{1}$, and so each $A_{r}$ is unitary and has eigenvalues in $\{-1,+1\}$. Such operators are called Hermitian involutions. These operators can be visualized as edges on $K_{6}$, the complete graph on 6 vertices, as shown in Fig. 1. The vertices correspond to Majorana modes, and two operators commute if their associated edges do not share a vertex. When convenient, we will use a double index as in $A_{jk}$ to denote the operator associated with edge $(j,k)$. The maximal sets of commuting observables are given by perfect matchings on $K_{6}$.

Our self-testing theorems apply to any set of six parities corresponding to a cycle subgraph $G\subseteq K_{6}$. For concreteness, we take $G$ to be the cycle whose edge set is $E=\{(1,2),(2,3),(3,4),(4,5),(5,6),(1,6)\}$. We refer to a maximal set of commuting parity operators in $G$ as a context. We arrange the six unknown operators into a 2-by-3 table where the operators in each row and column form a context (see Table 3 (Left)).

Any two operators $A_{r}$ and $A_{s}$ not in the same row or column correspond to edges $r$ and $s$ that are adjacent in $G$, which we denote $r\sim s$. The ideal fermionic parity operators corresponding to adjacent edges anti-commute. Since 6 Majorana modes acting on a given parity (even or odd) sector encode 2 logical qubits, the ideal operators can be any set of 6 logical two-qubit Pauli operators with the ideal commutation and anti-commutation relations. For concreteness, we fix a basis in which the ideal operators are as in Table 3 (Right).

Let ${\cal R}_{i}$ and ${\cal C}_{i}$ be the sets of edges in the $i$th row and column, respectively. The ideal expectations in our self-testing protocol are the following expectation values of products of observables in each context:

[TABLE]

The ideal expectations are achieved by the ideal state $|\hat{\Psi}\rangle=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11}$. We remark that our particular definition of the ideal expectations is a choice of convention. A similar rigidity result for a different ideal state follows from any similar set of ideal expectations where an odd number of contexts have an expectation value of $-1$.

Let ${\cal A}$ be the algebra generated by $\{A_{r}:r\in G\}$, and let $V\subseteq{\cal H}$ be the subspace defined by $V=\mathrm{span}\{A\ket{\Psi}:A\in{\cal A}\}$. Let $P$ be the projector onto $V$, and let ${\mathbb{A}}_{r}=PA_{r}P$.

Theorem 1** (Rigidity of Majorana Parities).**

If the ideal expectations are satisfied, then V is a 4-dimensional subspace and $\{{\mathbb{A}}_{r},{\mathbb{A}}_{s}\}=0$, for all $r,s\in G$ with $r\sim s$. Furthermore, the state $\ket{\Psi}$ satisfies $\prod_{r\in{\cal C}_{i}}A_{r}\ket{\Psi}=\ket{\Psi}$ for $i\in\{1,2\}$.

The proof of the theorem given in the next section. As a consequence of Theorem 1, a basis for $V$ can be chosen in which the operators on the (Left) in Table 3 equal those on the (Right) of the same Table 3, and $\ket{\Psi}=|\hat{\Psi}\rangle$.

In practice, experimental measurements do not satisfy the ideal expectations due to imperfections in the state preparation and measurements. We say that the ideal expectations are satisfied to within error $\epsilon$ if

[TABLE]

where the minus sign in the second line is used for the third column only. In the presence of errors, the subspace $V$ is no longer invariant under the action of the operators $A_{r}$. However, the protocol is still robust in the following sense. There exists an ideal subspace $\hat{V}$ of dimension 4, along with an ideal state $|\hat{\Psi}\rangle$ and ideal operators $\hat{A}_{r}$ whose fidelities with respect to the actual state and operators are close to 1, within errors linear in $\epsilon$. Here the state fidelity is $F(|\hat{\Psi}\rangle,\ket{\Psi})=|\langle\hat{\Psi}\ket{\Psi}|^{2}$, and the operator fidelity is defined as $F(\hat{A}_{r},A_{r})=\frac{1}{4}\mathrm{Tr}(\hat{A}_{r}A_{r})$. Formally, we have the following:

Theorem 2** (Protocol Robustness).**

If the ideal expectations are satisfied within error $\epsilon$, then there exists $\hat{V}\subseteq{\cal H}$, with $\mbox{dim}(\hat{V})=4$, Hermitian involutions $\hat{A}_{r}:\hat{V}\rightarrow\hat{V}$ for each $r\in G$ such that $\{\hat{A}_{r},\hat{A}_{s}\}=0$ if $r\sim s$ and $[\hat{A}_{r},\hat{A}_{s}]=0$ otherwise, and a state $\hat{\ket{\Psi}}\in\hat{V}$ such that $\prod_{r\in{\cal C}_{i}}\hat{A}_{r}\hat{\ket{\Psi}}=\hat{\ket{\Psi}}$ for $i\in\{1,2\}$, and such that they satisfy

[TABLE]

where $\epsilon_{0}=14\epsilon$, $\epsilon_{1}=0$, $\epsilon_{2}=25\epsilon/2$, and $\epsilon_{3}=(\sqrt{2}+\sqrt{14}+\sqrt{44})^{2}\epsilon/2\simeq 69.5\epsilon$.

The proof of the above theorem is given in Sec. V, with some details deferred to Appendix C and D. The perfect fidelity of the first column operators is due to a choice of basis. For simplicity, we have chosen our error bounds to be equal. Our results can be generalized to the case of unequal errors for different contexts, however, we do not carry out this analysis here.

IV Rigidity of Majorana Fermion Parity Measurements

In this section we first prove Proposition 1, and then we prove Theorem 1 with the help of some lemmas.

IV.1 From POVM to Projective Measurements

Given any POVM $\{Q_{a}\}$, $a=0,\ldots,d-1$, acting on system ${\cal S}$, Neumark proved Peres (1990a) that there exists a projective measurement $\{\hat{Q}_{a}\}$ on an extended system ${\cal S}\otimes{\cal E}$, with $\Tr(\hat{Q}_{a}(\rho\otimes\ket{0}\bra{0}_{{\cal E}}))=\Tr(Q_{a}\rho)$ for all density operators $\rho$ on ${\cal S}$, where the dimension of the extension ${\cal E}$ equals the number of elements $d$ in the POVM. The projectors are of the form

[TABLE]

where $U_{{\cal S}{\cal E}}^{\;}$ is a unitary on the extended Hilbert space.

In the case of multiple POVMs $Q_{r}$, where $r$ labels the POVM with elements $\{Q_{r,a}\}$, the POVMs can be extended to projectors $\{\hat{Q}_{r,a}\}$ by adding several ancillas ($\otimes_{r}{\cal E}_{r}$), one for each POVM. The situation is depicted in Fig. 2. Our task is to prove that for $d=2$, $[\hat{Q}_{r,a},\hat{Q}_{s,b}]=0$ whenever $[Q_{r,a},Q_{s,b}]=0$.

Proof of Proposition 1.

Define the projector $\hat{Q}_{r,a}$ on the extended system ${\cal S}\otimes{\cal E}_{r}$ corresponding to POVM element $Q_{r,a}$ according to Eq. (4), with

[TABLE]

where $\ket{a}_{r}$ is the basis-vector of the extension ${\cal E}_{r}$ with $a\in\{0,1\}$ and $\oplus$ denotes addition mod 2. We first show that $U_{{\cal S}{\cal E}_{r}}$ is unitary. Indeed, we have

[TABLE]

If $a=b$, then the last line above equals

[TABLE]

whereas if $a\neq b$, it equals

[TABLE]

where we’ve used $Q_{r,b}=\mathds{1}_{{\cal S}}-Q_{r,a}$ and thus $\sqrt{Q_{r,a}}$ commutes with $\sqrt{Q_{r,b}}$. Therefore, $U_{{\cal S}{\cal E}_{r}}$ is unitary. Next, using the cyclic property of the trace, we compute

[TABLE]

Finally, for $s\neq r$, in a block matrix representation with respect to the $\ket{a}_{r}$ basis,

[TABLE]

Therefore, $[Q_{r,a},Q_{s,b}]=0$ implies that $[U_{{\cal S}{\cal E}_{r}},\hat{Q}_{s,b}]=0$, and hence $[\hat{Q}_{r,a},\hat{Q}_{s,b}]=0$. ∎

IV.2 Rigidity of the State and Observables

We begin with a lemma that states that operators with adjacent edges anti-commute in their action on $\ket{\Psi}$. Since each $A_{r}$ has eigenvalues in $\{-1,+1\}$, the ideal expectations are satisfied only if $\ket{\Psi}$ is a $\pm 1$ eigenstate of the products of operators in each context.

[TABLE]

where again the minus sign in the last equation is for column $3$ only. Using the fact that $A_{r}^{2}=\mathds{1}$, and the commutativity of operators in each context, we can move operators freely between the left and right sides of the above equations. For example, the identities

[TABLE]

hold for row $1$ and column $2$, respectively.

Lemma 3**.**

Suppose the ideal expectations are satisfied. Then $\{A_{r},A_{s}\}\ket{\Psi}=0$, for $r\sim s$.

Proof.

We show that $\{A_{12},A_{16}\}\ket{\Psi}=0$. Making repeated use of identities such as the ones above, we compute

[TABLE]

By symmetry of the table, a similar argument shows that the same relation holds for any $A_{r}$ and $A_{s}$ with $r\sim s$. ∎

We now construct a subspace and show that it is invariant under the action of ${\cal A}$. Define $V^{\prime}\subseteq{\cal H}$ by

[TABLE]

Lemma 4**.**

$A_{r}V^{\prime}\subseteq V^{\prime}$* for all $r\in G$.*

Proof.

Since $A_{12}^{2}=\mathds{1}$, $A_{12}V^{\prime}\subseteq V^{\prime}$. To see that $A_{16}V^{\prime}\subseteq V^{\prime}$, note that Lemma 3 implies $A_{16}A_{12}\ket{\Psi}=-A_{12}A_{16}\ket{\Psi}$ and $A_{16}A_{12}A_{16}\ket{\Psi}=-A_{12}\ket{\Psi}$. We next check that $A_{34}V^{\prime}\subseteq V^{\prime}$. This follows from $A_{34}\ket{\Psi}=A_{16}\ket{\Psi}$, and from the fact that $A_{34}$ commutes with $A_{16}$ and $A_{12}$. By symmetry of the table, we also have that $A_{45}V^{\prime}\subseteq V^{\prime}$. It remains to show that $A_{56}$ and $A_{23}$ act invariantly on $V^{\prime}$. Explicity,

[TABLE]

Similarly, by symmetry, we also have $A_{23}V^{\prime}\subseteq V^{\prime}$. ∎

Having shown that $V^{\prime}$ is an invariant subspace, it follows that $V^{\prime}=V={\cal A}\ket{\Psi}$. We can now work with the operators restricted onto $V$. Let ${\mathbb{A}}_{r}=PA_{r}P$, with $P$ the projector onto $V$. Note that $AP=PAP$ for all $A\in{\cal A}$. The next Lemma states that commutativity and anti-commutativity of operators on a full Hilbert space is preserved under restriction onto a subspace.

Lemma 5**.**

Let $A$ and $B$ be Hermitian involutions, and let $P$ be a projector such that $AP=PAP$. Then

[TABLE]

Proof.

$(PAP)(PBP)=PAPBP=(PAP)^{{\dagger}}BP=(AP)^{{\dagger}}BP=PABP=\pm PBAP=\pm PBPAP=\pm(PBP)(PAP)$, with the plus or minus sign depending on whether $A$ and $B$ commute or anti-commute, respectively. ∎

We are now ready to prove Theorem 1.

Proof of Theorem 1.

We first determine the action on $V$ of the operators in ${\cal C}_{1}$ and ${\cal C}_{2}$. From Lemma 5, both ${\mathbb{A}}_{12}$ and ${\mathbb{A}}_{16}$ commute with both ${\mathbb{A}}_{34}$ and ${\mathbb{A}}_{45}$, and therefore

[TABLE]

Since ${\mathbb{A}}_{jk}^{2}=P$, ${\mathbb{A}}_{12}{\mathbb{A}}_{16}$ and ${\mathbb{A}}_{45}{\mathbb{A}}_{34}$ are unitary on $V$. Therefore, there exists an orthogonal basis for $V$ of simultaneous eigenstates of ${\mathbb{A}}_{12}{\mathbb{A}}_{16}$ and ${\mathbb{A}}_{45}{\mathbb{A}}_{34}$. Let $\ket{\alpha,\beta}$ be one such eigenstate satisfying

[TABLE]

and also satisfying $\bra{\alpha,\beta}\ket{\Psi}\neq 0$. Such a state exists since $\ket{\Psi}\in V$. From Lemma 3, $\{A_{12},A_{16}\}\ket{\Psi}=0$, and hence $\{{\mathbb{A}}_{12},{\mathbb{A}}_{16}\}\ket{\Psi}=0$ by Lemma 5. Similarly, $\{{\mathbb{A}}_{34},{\mathbb{A}}_{45}\}\ket{\Psi}=0$. These two equations imply $\alpha+\bar{\alpha}=0$ and $\beta+\bar{\beta}=0$, and thus $\alpha,\beta\in\{i,-i\}$, where $\bar{\alpha}$ denotes the complex conjugation of $\alpha$.

Now, define $\ket{\bar{\alpha},{\beta}}={\mathbb{A}}_{12}\ket{\alpha,\beta}$. Note that ${\mathbb{A}}_{12}{\mathbb{A}}_{16}({\mathbb{A}}_{12}\ket{\alpha,\beta})={\mathbb{A}}_{12}({\mathbb{A}}_{12}{\mathbb{A}}_{16})^{{\dagger}}\ket{\alpha,\beta}=\bar{\alpha}\,{\mathbb{A}}_{12}\ket{\alpha,\beta}$, and therefore

[TABLE]

Similarly, defining $\ket{\alpha,\bar{\beta}}={\mathbb{A}}_{45}\ket{\alpha,\beta}$, and $\ket{\bar{\alpha},\bar{\beta}}={\mathbb{A}}_{12}{\mathbb{A}}_{45}\ket{\alpha,\beta}$, we see that $\ket{\alpha,\beta}$, $\ket{\alpha,\bar{\beta}}$, $\ket{\bar{\alpha},\beta}$, and $\ket{\bar{\alpha},\bar{\beta}}$ are joint eigenstates of ${\mathbb{A}}_{12}{\mathbb{A}}_{16}$ and ${\mathbb{A}}_{45}{\mathbb{A}}_{34}$ with eigenvalues $(\alpha,\beta)$, $(\bar{\alpha},\beta)$, $(\alpha,\bar{\beta})$, and $(\bar{\alpha},\bar{\beta})$, respectively. These eigenstates are pairwise orthogonal, since $\alpha,\beta\in\{i,-i\}$. Therefore, $V$ is a 4-dimensional subspace. Next, note that

[TABLE]

Similar calculations applied to the remaining eigenstates of ${\mathbb{A}}_{12}{\mathbb{A}}_{16}$ and ${\mathbb{A}}_{34}{\mathbb{A}}_{45}$ show that $\{{\mathbb{A}}_{12},{\mathbb{A}}_{16}\}=0$ and $\{{\mathbb{A}}_{45},{\mathbb{A}}_{34}\}=0$. Therefore, there is a basis for $V$ in which

[TABLE]

We work in this basis for the remainder of the proof. The next step is to determine $\ket{\Psi}$. From ${\mathbb{A}}_{12}{\mathbb{A}}_{45}\ket{\Psi}=ZZ\ket{\Psi}=\ket{\Psi}$, and ${\mathbb{A}}_{34}{\mathbb{A}}_{16}\ket{\Psi}=XX\ket{\Psi}=\ket{\Psi}$, it follows that

[TABLE]

The final step is to determine ${\mathbb{A}}_{56}$ and ${\mathbb{A}}_{23}$. We begin with ${\mathbb{A}}_{56}$. Since $[A_{56},A_{12}]=0=[A_{56},A_{34}]$, Lemma 5 implies that $[{\mathbb{A}}_{56},{\mathbb{A}}_{12}]=0=[{\mathbb{A}}_{56},{\mathbb{A}}_{34}]$. Therefore, when expanded in a basis of two-qubit Pauli matrices, ${\mathbb{A}}_{56}$ can only have non-zero weight on $II$, $ZI$, $IX$, and $ZX$. However, ${\mathbb{A}}_{56}\ket{\Psi}={\mathbb{A}}_{12}{\mathbb{A}}_{34}\ket{\Psi}=ZX\ket{\Psi}$, which implies ${\mathbb{A}}_{56}=ZX$, since the states $\ket{\Psi}$, $XI\ket{\Psi}$, $IX\ket{\Psi}$, and $ZX\ket{\Psi}$ are pairwise orthogonal. Similarly, using $[{\mathbb{A}}_{23},{\mathbb{A}}_{45}]=0=[{\mathbb{A}}_{23},{\mathbb{A}}_{16}]$ and ${\mathbb{A}}_{23}\ket{\Psi}={\mathbb{A}}_{45}{\mathbb{A}}_{26}\ket{\Psi}$, it follows that ${\mathbb{A}}_{23}=XZ$.

∎

V Robustness to Errors

We now consider the situation where the ideal statistics are satisfied to within error $\epsilon$. We first prove an approximate version of Lemma  3.

Lemma 6**.**

Suppose the ideal expectations are satisfied to within error $\epsilon$. Then $\|\{A_{r},A_{s}\}\ket{\Psi}\|\leq 5\sqrt{2\epsilon}$ for all $r,s\in G$ with $r\sim s$.

Proof.

We show that $\|\{A_{12},A_{16}\}\ket{\Psi}\|\leq 5\sqrt{2\epsilon}$. For $r$ and $s$ in the same column,

[TABLE]

where in the first line, the plus sign is used for column 3, and the minus sign for columns 1 and 2. Similarly, for both rows of the table, with $r$, $s$, and $t$ in the same row,

[TABLE]

Therefore, by a chain of triangle inequalities, and using the fact that $\|U\ket{\Psi}\|=\|\ket{\Psi}\|$ for any unitary $U$,

[TABLE]

Using a similar argument for any $r\sim s$, one can prove $\|\{A_{r},A_{s}\}\ket{\Psi}\|\leq 5\sqrt{2\epsilon}$. ∎

By a corollary to Jordan’s lemma, which we prove in Appendix B, ${\cal H}$ decomposes as ${\cal H}=\bigoplus_{l}{\cal H}_{l}$, where each ${\cal H}_{l}$ is 4-dimensional and invariant under the action of $A_{12}$, $A_{16}$, $A_{34}$, and $A_{45}$. Since both of $A_{12},A_{16}$ commute with both of $A_{34},A_{45}$, each invariant subspace ${\cal H}_{l}$ in the Jordan decomposition factors as a tensor product of two qubits. Therefore, there is a basis for each subspace such that

[TABLE]

with $\theta_{l},\phi_{l}\in[-\frac{\pi}{2},\frac{\pi}{2}]$. Label this chosen basis for each ${\cal H}_{l}$ as $\{\ket{00_{l}},\ket{01_{l}},\ket{10_{l}},\ket{11_{l}}\}$.

Next, with respect to this Jordan decomposition, one can write $\ket{\Psi}$ as

[TABLE]

where each $\ket{\Psi_{l}}\in{\cal H}_{l}$ and $\sum p_{l}=1$. We define the ideal subspace $\hat{V}$ as the linear span

[TABLE]

We define the ideal operators to be logical Pauli product operators in the above basis. Specifically, $\hat{A}_{12}=ZI$, $\hat{A}_{34}=IX$, $\hat{A}_{56}=ZX$, $\hat{A}_{45}=IZ$, $\hat{A}_{16}=XI$, $\hat{A}_{23}=XZ$. Finally, we define the ideal state with respect to the above basis as

[TABLE]

Proof of Theorem 2.

By definition, $\{\hat{A}_{r},\hat{A}_{s}\}=0$ for $r,s\in G$ with $r\sim s$, and the ideal state satisfies $\prod_{r\in{\cal C}_{i}}\hat{A}_{r}\hat{\ket{\Psi}}=\hat{\ket{\Psi}}$ for $i\in\{1,2\}$.

We first calculate the state fidelity. Define $|\hat{\Psi}_{l}\rangle=\frac{1}{\sqrt{2}}(\ket{00_{l}}+\ket{11_{l}})$. Using the freedom to choose the overall phase in each subspace, we set $\langle\hat{\Psi}_{l}|\Psi_{l}\rangle\geq 0$. Therefore,

[TABLE]

It will be convenient to work in the $Y$ basis $\{\ket{0_{Y}0_{Y}},\ket{0_{Y}1_{Y}},\ket{1_{Y}0_{Y}},\ket{1_{Y}1_{Y}}\}$ within each Jordan subspace ${\cal H}_{l}$. Here $\ket{0_{Y}}=(\ket{0}+i\ket{1})/\sqrt{2}$, and $\ket{1_{Y}}=(\ket{0}-i\ket{1})/\sqrt{2}$. We expand $\ket{\Psi_{l}}$ as

[TABLE]

where $\sum_{a,b}|c^{l}_{ab}|^{2}=1$. In this basis, the ideal state is

[TABLE]

Therefore,

[TABLE]

In Appendix C, we prove that

[TABLE]

Thus, the state fidelity is bounded according to

[TABLE]

Next we bound the fidelity of the operators, starting with the first column. From Eq. (V) and the definition of the ideal operators, for $r\in{\cal C}_{1}$, $\Tr(\hat{A}_{r}A_{r})=4$ and so $F(\hat{A}_{r},A_{r})=1$. For the second column, Eq. (V) implies

[TABLE]

Combining Lemma 6 with Eq. (V),

[TABLE]

where the last line follows from $\theta_{l}\in[-\frac{\pi}{2},\frac{\pi}{2}]$ and therefore $\cos{\theta_{l}}\geq 0$. Combining the last two equations yields

[TABLE]

A similar calculation shows that $F(\hat{A}_{23},A_{23})\geq 1-\frac{25}{2}\epsilon$ also.

The final step is to bound the fidelities of the third column operators. We begin with $A_{56}$. The $\epsilon$ error in the first row implies

[TABLE]

and from the state fidelity, Eq. (V), we get

[TABLE]

We show in Appendix D that

[TABLE]

Applying the triangle inequality to Eqs. (14)-(16), and using $\hat{A}_{12}=A_{12}$,

[TABLE]

which implies

[TABLE]

Since $[A_{12},A_{56}]=0$ and $\hat{A}^{2}_{12}=P$, Lemma 5 implies $[\hat{A}_{12},{\mathbb{A}}_{56}]=0$. Therefore, when expanded in a basis of two-qubit Pauli matrices, ${\mathbb{A}}_{56}$ can only have $I$ or $Z$ acting on the first qubit. Since $\hat{A}_{12}\hat{A}_{34}=ZX$, only the $ZX$ component of ${\mathbb{A}}_{56}$ contributes to $\real\langle\hat{\Psi}|\hat{A}_{12}\hat{A}_{34}{\mathbb{A}}_{56}|\hat{\Psi}\rangle$, as any other component either gives zero or a purely imaginary number. Hence,

[TABLE]

and it follows that $F(\hat{A}_{56},A_{56},)\geq 1-\epsilon_{3}$. By a similar argument one can show that $F(\hat{A}_{23},A_{23})\geq 1-\epsilon_{3}$. ∎

VI Discussion and Outlook

We have shown that measurements of Majorana fermion parities can be self-tested. This fact provides a powerful tool for determining how consistent experimental data is with the existence of Majorana fermion modes. Experimentally, our protocol requires the ability to measure 6 observables $A_{jk}$, which ideally correspond to parities $P_{jk}$ between consecutive Majorana modes, as shown in Fig 3. The expectation value of any context (set of observables in a row or column of Table 3 (Left)) can be obtained by repeatedly preparing a specific initial state, measuring the operators in that context, and using Eqs. (18) and (19). Given data from the measurements in each context, we can express the result by an operator $W$, often called a contextuality witness[cite] in the quantum information literature,

[TABLE]

This witness obeys an inequality, $\langle W\rangle\leq 3$, for any classical assignments of outcomes to the measured observables, and is upper-bounded in quantum theory by $\langle W\rangle\leq 5$. The witness is not unique, as different choices of the initial state result in different combinations of signs of the terms in $W$, according to Table 5 in Appendix A.

For a fixed total parity (odd or even), $6$ Majorana fermions theoretically encode a two qubit subspace, where each qubit is encoded non-locally in $3$ Majorana modes in Fig. 3. Our results, Theorems 1 and 2, imply that the maximal value of $\langle W\rangle=5$ is obtained only if the observables corresponding to adjacent parities anti-commute. Furthermore, a small error certifies that each $A_{jk}$ has high fidelity with the corresponding Majorana parity operator. For example, in a basis where the parities in the first row are perfect, a $0.1\%$ error in the expectation value of each context implies upper bounds on the error of the second and third column operators of $1.25\%$ and $6.95\%$, respectively.

We emphasize that although our proposal is to self-test parities in Majorana modes, our protocol can be simulated in other physical systems via the Jordan-Wigner mapping given by Eq. (1) and Table 2. Examples include trapped ions, where Pauli product operators can be measured with global entangling gates and the use of an ancilla Leibfried and Wineland (2018), or also neutron beams entangled in energy, path and spin degrees of freedom Cabello et al. (2008); et al. .

We conclude with some open problems and suggestions for future work. The robustness bounds in our Theorem 2 are certainly not tight, which raises the question of how much they can be improved. It might be possible to obtain a stronger robustness statement using different methods such as those based on a semidefinite programming hierarchy et al. (2014, 2015), or linear operator inequalities Kaniewski (2016). How does our formulation of robustness by constructing an ideal subspace relate to the notion of robustness as measured by an extraction map? Finally, while our protocol certifies a single state and a set of measurements, gates in topological quantum computing are implemented by braiding. It is therefore desirable to extend the protocol to include a self-test of braiding operations.

We strongly believe that certification of quantum measurements in various physical scenarios is a promising technique for precision measurement and quantum validation. We anticipate generalizations of our current approach to many other situations of physical interest exploring the frontiers of quantum mechanics.

Acknowledgements.

This research was conducted in part under the PREP program with financial assistance from U.S. Department of Commerce, National Institute of Standards and Technology. Contributions to this article by workers at the National Institute of Standards and Technology, an agency of the U.S. Government, are not subject to U.S. copyright.

Appendix A Ideal Expectations and Statistics

Operators in each context of Table 2 realize complete sets of commuting observables (CSCO), allowing preparation of states with definite parity assignments. In the following discussion, we consider the experimental preparation of an initial state with a well-defined parity $a_{36}^{(0)}$, $a_{25}^{(0)}$ and $a_{14}^{(0)}$, $a_{jk}^{(0)}\in\{-1,+1\}$, corresponding to the claimed parity observable $A_{jk}$. We then measure the contexts (sets of operators in a row or column) of Table 3. As an illustration, the ideal probability distribution of measurement outcomes of all contexts for the initial state $a_{36}^{(0)}=+1,a_{25}^{(0)}=+1,a_{14}^{(0)}=-1$ is given in Table 4. From the statistics of measurement outcomes one can calculate the expectation value of the product of the operators in each context by using

[TABLE]

where $N(a_{r},a_{s})$ (or $N(a_{r},a_{s},a_{t})$) refers to the number of experimental outcomes with value $(a_{r},a_{s})$ (or $(a_{r},a_{s},a_{t})$). Ideal expectation values of all contexts for different initial states with all possible $a_{36}^{(0)},a_{25}^{(0)},a_{14}^{(0)}\in\{-1,+1\}$ is given in Table 5.

Appendix B Jordan’s Lemma

We prove a corollary, which we used in the main text, of what is known as Jordan’s Lemma in the quantum information literature. A particularly simple proof of Jordan’s Lemma appears in Ref. [et al., 2009], which we also include here for completeness.

Lemma 7** (Jordan’s Lemma).**

Let $A_{1}$ and $A_{2}$ be Hermitian involutions on a Hilbert space ${\cal H}$. Then ${\cal H}$ decomposes as a direct sum ${\cal H}=\bigoplus_{l}{\cal H}_{l}$, with $\dim{\cal H}_{l}\in\{1,2\}$, and $A_{1}$ and $A_{2}$ act invariantly on each ${\cal H}_{l}$.

Proof.

$A_{1}A_{2}$ is unitary since $(A_{1}A_{2})(A_{1}A_{2})^{{\dagger}}=A_{1}A_{2}A_{2}A_{1}=\mathds{1}$. Since $A_{1}A_{2}$ is unitary, there exists an orthonormal basis for ${\cal H}$ of eigenstates of $A_{1}A_{2}$. Let $\ket{\alpha}$ be any such eigenstate, where $A_{1}A_{2}\ket{\alpha}=\alpha\ket{\alpha}$. Define $\ket{\bar{\alpha}}=A_{1}\ket{\alpha}$. Then $A_{1}A_{2}\ket{\bar{\alpha}}=\bar{\alpha}\ket{\bar{\alpha}}$, since

[TABLE]

The span of $\ket{\alpha}$ and $\ket{\bar{\alpha}}$ is invariant under $A_{1}$, and also under $A_{2}$, since

[TABLE]

Thus any eigenstate of $A_{1}A_{2}$ defines an invariant subspace of dimension at most 2, and these eigenstates span ${\cal H}$, which completes the proof. ∎

Corollary 7.1**.**

For $k,k^{\prime}\in\{1,2\}$, let $A_{k}$ and $B_{k^{\prime}}$ be Hermitian involutions with $[A_{k},B_{k^{\prime}}]=0$. Then ${\cal H}$ decomposes as ${\cal H}=\bigoplus_{l}({\cal H}_{a_{l}}\otimes{\cal H}_{b_{l}})$, with ${\cal H}_{a_{l}}$ and ${\cal H}_{b_{l}}$ of dimension at most $2$, and $A_{k}=\bigoplus_{l}(A_{k_{l}}\otimes\mathds{1}_{l})$ and $B_{k^{\prime}}=\bigoplus_{l}(\mathds{1}_{l}\otimes B_{k^{\prime}_{l}})$.

Proof.

Note that $[A_{1}A_{2},B_{1}B_{2}]=0$. Thus, there exists an orthonormal basis for ${\cal H}$ of simultaneous eigenstates of $A_{1}A_{2}$ and $B_{1}B_{2}$. Let $\ket{\alpha,\beta}$ be any such eigenstate, where $A_{1}A_{2}\ket{\alpha,\beta}=\alpha\ket{\alpha,\beta}$ and $B_{1}B_{2}\ket{\alpha,\beta}=\beta\ket{\alpha,\beta}$. Define $\ket{\bar{\alpha},\beta}=A_{1}\ket{\alpha,\beta}$, $\ket{\alpha,\bar{\beta}}=B_{1}\ket{\alpha,\beta}$, and $\ket{\bar{\alpha},\bar{\beta}}=A_{1}B_{1}\ket{\alpha,\beta}$. Then $\mathrm{span}\{\ket{\alpha,\beta},\ket{\bar{\alpha},\beta},\ket{\alpha,\bar{\beta}},\ket{\bar{\alpha},\bar{\beta}}\}$ maps isomophically to $\mathrm{span}\{\ket{\alpha},\ket{\bar{\alpha}}\}\otimes\mathrm{span}\{\ket{\beta},\ket{\bar{\beta}}\}$. By the argument in the proof of Lemma 7, $A_{k}$ and $B_{k^{\prime}}$ act invariantly on the first and second tensor factors, and trivially on the second and first tensor factors, respectively. ∎

We remark that in the main text we assume that each Jordan subspace ${\cal H}_{l}$ has dimension 4. This is done without loss of generality, since we can extend any smaller dimensional subspace to $4$ dimensions, with all operators acting trivially on the extension.

Appendix C Derivation of Eqs. (10) and (11).

It will be convenient to work in the $Y$ basis $\{\ket{0_{Y}0_{Y}},\ket{0_{Y}1_{Y}},\ket{1_{Y}0_{Y}},\ket{1_{Y}1_{Y}}\}$ within each Jordan subspace ${\cal H}_{l}$. Here $\ket{0_{Y}}=(\ket{0}+i\ket{1})/\sqrt{2}$, and $\ket{1_{Y}}=(\ket{0}-i\ket{1})/\sqrt{2}$. We expand $\ket{\Psi_{l}}$ as

[TABLE]

where $\sum_{a,b}|c^{l}_{ab}|^{2}=1$. In this basis, the ideal state is

[TABLE]

Our first step is to bound $\sum_{l}p_{l}(|c_{00}^{l}|^{2}+|c_{11}^{l}|^{2})$. Applying Eq. (5) to ${\cal C}_{1}$ and ${\cal C}_{2}$, respectively, we obtain

[TABLE]

[TABLE]

Eq. (11) follows from the first of these equations. It also follows that

[TABLE]

Now,

[TABLE]

where in the last line we used the fact that $|x+y|^{2}\leq 2(|x|^{2}+|y|^{2})$ for any $x,y\in\mathbb{C}$. From Eqs. (20) and (21),

[TABLE]

Similarly,

[TABLE]

from which it follows that

[TABLE]

Adding Eqs. (C) and (C),

[TABLE]

We now need the following lemma, which is similar to Lemma 6.

Lemma 8**.**

Suppose the ideal expectations are satisfied to within error $\epsilon$. Then $\|A_{12}A_{34}\ket{\Psi}+A_{16}A_{45}\ket{\Psi}\|\leq 3\sqrt{2\epsilon}$.

Proof.

[TABLE]

where the last inequality follows from Eqs. (5) and (6). ∎

From the result of the lemma, we obtain

[TABLE]

and therefore,

[TABLE]

or,

[TABLE]

Adding Eqs. (24) and (26),

[TABLE]

where in the last inequality we used $\theta_{l}$,$\phi_{l}\in[-\frac{\pi}{2},\frac{\pi}{2}]$, and so $\cos{\theta_{l}}\geq 0$ and $\cos{\phi_{l}}\geq 0$. This last inequality is Eq. (10) in the main text.

Appendix D Derivation of Eq. (16)

By definition,

[TABLE]

Therefore,

[TABLE]

Summing over the Jordan subspaces, we get

[TABLE]

The second term in the last line is bounded by

[TABLE]

where we’ve used Eq. (11) and the same argument leading to Eq. (V), applied to $\phi_{l}$. Therefore,

[TABLE]

Using this bound we can say

[TABLE]

Therefore,

[TABLE]