Preserving entanglement during weak measurement demonstrated with a violation of the Bell-Leggett-Garg inequality

Weak measurement has provided new insight into the nature of quantum measurement, by demonstrating the ability to extract average state information without fully projecting the system. For single qubit measurements, this partial projection has been demonstrated with violations of the Leggett-Garg inequality. Here we investigate the effects of weak measurement on a maximally entangled Bell state through application of the Hybrid Bell-Leggett-Garg inequality (BLGI) on a linear chain of four transmon qubits. By correlating the results of weak ancilla measurements with subsequent projective readout, we achieve a violation of the BLGI with 27 standard deviations of certainty.

Since the inception of quantum mechanics, its validity as a complete description of reality has been challenged due to predictions that defy classical intuition 1 . For many years it was unclear whether predictions like entanglement and projective measurement represented real phenomena or artifacts of an incomplete model. Bell inequalities (BI) 2-6 provided the first quantitative test to distinguish between quantum entanglement and a yet undiscovered classical hidden variable theory. The Leggett-Garg inequality (LGI) 7 provides a similar test for projective measurement, and more recently has been adapted to include variable strength measurements [8][9][10][11][12][13][14] to study the process of measurement itself. Here we probe the intersection of both entanglement and measurement through the lens of the hybrid Bell-Leggett-Garg inequality (BLGI) 15,16 . By correlating data from ancilla-based weak measurements and direct projective measurements, we for the first time quantify the effect of measurement strength on entanglement collapse. Violation of the BLGI, which we achieve only at the weakest measurement strengths, offers compelling evidence of the completeness of quantum mechanics while avoiding several loopholes common to previous experimental tests [17][18][19][20] . This uniquely quantum result significantly constrains the nature of any possible classical theory of reality. Additionally, we demonstrate that with sufficient scale and fidelity, a universal quantum processor can be used to study richer fundamental physics.
The core assumption of a Bell inequality is that the measurements of two spatially separated objects cannot affect one another. Bell 2 designed a test for this locality assumption (which was later refined by Clauser, Horne, Shimony, and Holt into an inequality (CHSH) 3 ) involving correlated polarization measurements with varying rotation angles between two spatially separated photons. Entangled photons exhibit stronger correlations than what is possible classically, and can violate the inequality 4 .
To test such an inequality using two superconducting qubits, we replace polarizer rotations with qubit state rotations to map the desired measurement basis onto the ground (|0 ) and excited (|1 ) states of the system. For measurement rotations a (qubit 1) and b (qubit 2), the correlation amplitude between two measurements is given by E(a, b) = P (00) − P (10) − P (01) + P (11), where the term P (00) is the probability both qubits are in the ground state. The CHSH correlator combines four such two-qubit correlators with distinct pairs of angles (2) and is classically bounded by |CHSH| class. ≤ 2 for any local hidden variable theory. Entangled quantum states can violate this bound, with a fully entangled Bell state ideally saturating the quantum upper bound of |CHSH| quant. ≤ 2 √ 2 using the specific rotation angles a = 0, b = π/4, a = π/2, and b = 3π/4.
A similar but complementary test of quantum mechanics was proposed by Leggett and Garg 7 , in which the effects of quantum measurement allow larger correlations between successive measurements (e.g., at times t 1 < t 2 < t 3 ) than what is possible classically. Inequalities similar to the CHSH inequality can be constructed for these temporal sequences, and then violated by quantum systems 14 . The challenge is to rule out the possibility that such LGI violations arise from overly invasive intermediate measurements that classically perturb the system. To minimize the possibility of this "clumsiness" loophole 20 , variations have been explored that replace the intermediate measurements (e.g, time t 2 ) with null 21 or weak measurements [8][9][10][11][12]22,23 that minimize the quantum state disturbance while still extracting sufficient information on average. Unlike typical LGIs or CHSH BIs that require varying rotation angles, weak measurements permit all needed correlations to be determined simultaneously using a single experimental configuration Here we investigate the effects of weak measurement on an entangled state, and by doing so construct a more stringent test of quantum mechanics. This is accomplished by implementing a hybrid Bell-Leggett-Garg inequality (BLGI), as described by Dressel and Korotkov 16 and shown in Fig. 1. The algorithm consists of a CHSH-style experiment in which each Bell qubit is ancilla meas.
LGI: time correlations Figure 1. Schematic of the hybrid Bell-Leggett-Garg inequality and optical micrograph of the superconducting quantum device. The algorithm consists of two LGI weak measurement branches, bridged by the entanglement of the central Bell qubits. The Bell pair (β1,2) is initially prepared in the anti-symmetric singlet Bell state |Ψ − . Next each Bell qubit is rotated to its first measurement basis (a or b) and entangled with its ancilla qubit (α1,2). Finally, the Bell qubits are rotated and projectively read out in bases corresponding to angles a and b . By correlating the final projective read out and the weak ancilla measurements we calculate all four terms of a CHSH correlator simultaneously. measured twice in succession as for an LGI. These two LGI branches are bridged by the entanglement of the Bell pair, thus combining the spatial correlations of a BI with the temporal correlations of an LGI to measure the entire CHSH correlator in a single spatio-temporal experiment.
After preparing the Bell qubits in the anti-symmetric singlet state each Bell qubit (β 1,2 ) is rotated to its first measurement angle (a = 0, b = π/4) and then entangled with its ancilla qubit (α 1,2 ) to implement a tunable-strength measurement. Next, each Bell qubit is rotated to its final measurement angle (a = π/2, b = 3π/4), and all four qubits are read out. With this procedure the data for each measurement angle is encoded on a distinct qubit (a → α 1 , b → α 2 , a → β 1 , and b → β 2 ). The BLGI correlator then takes the form similar to Eq. (2) (4) where each term is calculated as in Eq. (1). As the ancilla measurement strength is decreased, we extract the same qubit information on average while only partially collapsing the Bell state. If the ancilla measurement is projective, the Bell state fully collapses, so |E(α 1 , α 2 )| = 1/ √ 2 and all other correlations vanish. For sufficiently weak measurements, however, the magnitudes of all four correlators approach the unperturbed Bell state values of 1/ √ 2 and C approaches 2 √ 2, violating the classical bound of 2.
We performed this experiment on a linear chain of Xmon transmon qubits, shown at the top of Fig. 1, with ground to excited state transition frequencies in the 4-6 GHz range 24 . Each qubit is individually addressed with a microwave control line which can be used for single qubit X or Y gates as well as a DC line for implementing Z-gates and frequency control. These control lines are used in conjunction to execute high fidelity two-qubit gates 25 for entanglement and ancilla measurement. The state of each qubit is measured independently using the dispersive shift 26 of a dedicated readout resonator. Resonators are frequency multiplexed 27 and readout with a broadband parametric amplifier 28 , which allows for fast high fidelity measurement. Further details of this device can be found in Refs 24 and 29.
The ancilla measurement protocol used in this experiment and shown in Fig. 2, is a modified version of the protocol demonstrated in an LGI violation from Groen et al. 9 . Initially, an ancilla qubit is Y -rotated by an angle 0 ≤ φ ≤ π/2 from its ground state to set the measurement strength. A control phase gate is then performed, causing a Z rotation of π/2 in the ancilla qubit depending on the target qubit's state. Finally a −Y rotation of π/2 is performed on the ancilla qubit to rotate into the correct measurement basis. The visibility of this measurement is then proportional to the distance of the ancilla state vector from the equator of the Bloch sphere, as shown in Fig. 2 (b). When φ = π/2 this operation becomes a control-NOT gate and implements a projective measurement. As φ → 0 the ancilla states become degenerate and no information is extracted.
As the final position of the ancilla state is dependent on the measurement strength, the ancilla readout is imperfectly correlated with the target qubit. That is, the visibility of an ancilla Z average, Z α sin(φ) Z τ , is compressed from the target Z average by a factor of approximately sin(φ). To reconstruct the target Z average from the ancilla Z average, we should thus rescale the signal by 1/ sin(φ). For more robust calibration, we used a data based rescaling to set the measured ground state (|0 ) average to 1, which ensures that the calibrated ancilla average is properly bounded by ±1, as shown in Fig. 2(c). Further details of this calibration can be found Figure 2. Weak measurement protocol. a, Full pulse sequence of the ancilla measurement algorithm used in the BLGI experiment. The measurement consists of a variable amplitude Y rotation by an angle φ which controls the strength of the measurement. This is followed by a CZ gate that entangles the ancilla qubit with the target qubit. Finally the ancilla is rotated by an angle −π/2 bringing it into the desired measurement basis. Two cases are compared, that of the target qubit in the ground (blue) or excited (red, π rotation) state. b, Bloch sphere representation of the ancilla qubit during the weak measurement protocol when the target qubit is in either the ground (blue) or excited (red) state. The Z averages of the ancilla and target qubit are correlated such that Z α = sin(φ) Z τ , where a full projective measurement corresponds to φ = π/2 and no measurement corresponds to φ = 0. c, Ancilla measurement of prepared target state before and after calibrating for measurement strength. We calibrate both curves by the scaling factor required to normalize the average 0 state curve. This is almost equivalent to dividing by sin(φ) but bounds the calibrated mean by ±1. In the calibrated case, the measured mean remains unchanged while the measured variance increases as φ decreases. The gold shaded region denotes angles at which weak measurement data can violate the BLGI while still being reliably calibrated in the supplemental information 29 . After this and other extensive system calibrations, we can begin our investigation in earnest. To ensure a smooth transition from the strong to weak measurement regime, we measure each two-qubit correlator in C vs. ancilla measurement strength φ. The data are plotted correlationqamplitude measurementqstrengthq(ϕ/π)q =qE(α 2 ,β 1 )-E(α 1 ,α 2 )-E(α 1 ,β 2 )-E(β 1 ,β 2 )q strong weak ±10σqerrorqbars quantumq classicalq Figure 3. Graph showing both experimental data (points) and theoretical predictions (lines) for the correlator C and its four terms vs. measurement strength φ. The horizontal gold line denotes the classical bound on C . The data set was taken by averaging together 200 traces in which each point was measured 3000 times for a total of 600,000 iterations per point. The error bars represent 10 standard deviations of the mean to demonstrate the scaling the ancilla measurement noise vs. φ. The magnitude of the correlations between each pair of qubits reveals the extent to which entanglement has been broken for each measurement strength.
in Fig. 3 along with theory curves generated by a quantum model that includes realistic environmental dephasing and readout fidelity 16 . Error bars for the violation data represent ±10 standard deviations of the mean to demonstrate the increase in noise with decreasing measurement strength. Given this variance, the largest violation we observe is 27 standard deviations at a measurement strength of φ = 0.15π. For projective angles φ ≈ π/2, the ancilla measurement results (E(α 1 , α 2 )) reflect the correlation expected from a collapsed Bell pair. As measurement strength is decreased, this correlation remains nearly constant while additional inter-qubit correlations (E(α 1 , β 2 ), E(β 1 , α 2 ), E(β 1 , β 2 )) emerge. Finally, C exceeds 2 even for reasonably large measurement angles (φ ∼ 0.2π) and saturates towards a value of 2.5 as the measurement strength approaches zero (which is our expected maximum after accounting for experimental imperfections).
The BLGI terms follow the theoretical model 29 very closely for all measurement strengths. This behavior, observed for the first time in experiment, reveals the continuous trade-off between the collapse of an entan-gled Bell state and the information gained from tunablestrength measurements. Each ancilla qubit, when calibrated, retains the same correlations for all measurement strengths, whereas each Bell qubit has its correlations damped through partial projection by its ancilla qubit 16 . The effect of partial projection can be seen in the difference in functional behavior between the Bell-ancilla (E(α 1 , β 2 ), E(β 1 , α 2 )) and the Bell-Bell (E(β 1 , β 2 )) correlator terms. In the Bell-ancilla terms, the correlations are suppressed solely for the Bell qubit but return as soon as measurement strength is decreased. In the Bell-Bell term, this effect is compounded as both qubits are being damped by partial ancilla projection, and correlations are slower to return. This gives E(β 1 , β 2 ) its distinct shape compared to the other correlators. The BLGI terms can be seen in greater detail in the supplement 29 .
The unique construction of the BLGI, allows us to avoid some of the more pervasive loopholes for traditional Bell or LG inequalities. The simultaneous measurement of all four CHSH terms in a single circuit allows us to avoid any configuration-dependent bias, such as the disjoint sampling loophole 19 . The unit detection efficiency in superconducting systems 6,29 similarly bypasses the fair sampling loophole 18 , which has hindered the investigation of related hybrid inequalities in optical systems 12,30 . Additionally, since the data from each ancilla qubit is only correlated with the data from the Bell and ancilla qubits on the remote LGI branch, we can substantially relax the usual LGI noninvasive measurement assumption to the standard locality assumption needed for a BI instead.
This locality assumption, fundamental to any Bell inequality, presumes no classical interactions between remote qubits occur during the correlation measurements. The close proximity of adjacent superconducting qubits on a chip implies that such an interaction cannot be ruled out. Thus, behavior that appears to be quantum could be the result of a fast classical interaction between hidden variables in the system. While we cannot completely rule out such a local hidden variable theory in this experiment, we can considerably restrict its form. Such a theory would have to conspire to mimic correlations resulting from both weak measurement and quantum entanglement, all while displaying the correct functional dependence on measurement strength. To explain these straightforward quantum mechanical predictions using only local classical interactions demands a significantly more complex classical model of reality.
In the course of this work we have shown, that for sufficiently weak measurements, the BLGI can be violated in a single simultaneous experiment. This violation quantifies the relationship between entanglement and measurement, or equivalently, restricts the form of any classical hidden variable theory. Moreover, this experiment demonstrates the potential of scaling to larger universal quantum processors. As we scale to larger multi-qubit systems, with the fidelity and control achieved here, we gain greater access to the rich physics at the heart of quantum mechanics.
Acknowledgments This work was supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office grant W911NF-10-1-0334. All statements of fact, opinion or conclusions contained herein are those of the authors and should not be con-strued as representing the official views or policies of IARPA, the ODNI, or the U.S. Government. Devices were made at the UC Santa Barbara Nanofabrication Facility, a part of the NSF-funded National Nanotechnology Infrastructure Network, and at the NanoStructures Cleanroom Facility. Author

SUPPLEMENTAL INFORMATION Bell and Leggett-Garg Inequalities
The CHSH correlator, designed by Clauser, Horne, Shimony, and Holt [2] as a refinement of the Bell inequality [1], provides a quantitative bound on classical hidden variable theories using correlated measurements between two spatially separated qubits. The correlator combines four different experimental configurations because it can be difficult to tell the difference between potentially classical (un-entangled) qubits and an entangled state in only one basis. With superconducting qubits, the measurement basis for each qubit is set using qubit rotations to map the desired state onto the ground (|0 ) and excited (|1 ) states of the system. For measurement rotations a (qubit 1) and b (qubit 2), shown in Fig. 1(a), the correlation amplitude is given by where P(00) is the probability both qubits are in the ground state. Given this equation we can see that both the Bell state |Φ + = (|00 + |11 )/ √ 2 and the prepared state |00 will have a correlation amplitude of 1 if a = b = 0. The difference only becomes clear when the detector angle of one qubit is rotated relative to the other. The behavior of E(a, b) vs detector rotation, described here as θ = a − b, is shown in Fig. 1(b) for both the classical and quantum case. If the two objects can be described separately, then E is only a linearly dependent on θ. If the two objects are entangled, then E is has a sinusoidal dependence on θ with the maximum difference occurring at θ = π/4.
To initially characterize the system we conducted a traditional CHSH experiment using the central Bell qubits (β 1,2 ). The relative measurement angles for each qubit were held fixed such that a = a + π/2 and b = b + π/2. We then varied θ = a − b from 0 to π, and measured each individual correlator as well as the sum given by For any two classical states measured at these angles, we should see a linear dependence of E(θ) and a bound on the the CHSH correlator of |CHSH| ≤ 2. Alternately, if the two qubits are in a maximally entangled Bell state, we should see sinusoidal behavior for E(θ) and a maximum CHSH value of 2 √ 2. The data, shown in Fig. 1(c), display the expected sinusoidal dependence for each individual term, with a maximum CHSH amplitude near θ = π/4. While this data shows a robust violation of the classical bound, it fails to reach the theoretical maxium bound of 2 √ 2. The maximum CHSH amplitude of ∼ 2.5 we see here is due to experimental imperfections which will be discussed later. This CHSH experiment provides the framework for the BLGI, as well as a benchmark for the maxium violation we should expect at the weakest measurement angles.
A complementary test of quantum mechanics is the LGI, which is similar to a Bell inequality but involves measurements separated in time rather than in space. Classical theories of measurement assume that the system is always in a definite state, and that an ideal measurement will not change the state of the system. In contrast, if one were to measure a quantum state in an orthogonal measurement basis, the act of measurement would project that object onto an eigenstate of the new basis. To distinguish one kind of system from the other, measurements are conducted in different bases at different times. For measurements conducted at times t 1 < t 2 < t 3 , we can construct correlators analogous to Eq. 1 but for different measurements of the same qubit, E(t i , t j ) = P (00) − P (10) − P (01) + P (11).
The inequality was originally composed of three distinct experiments. In the first experiment, the system measured protectively at time t 1 , followed by a final projective measurement at time t 3 . A second experiment is then carried out where an intermediate measurement in a different basis is conducted at time t 2 instead of time t 3 . The third experiment consists of only the measurements at times t 2 and t 3 . The LGI is then given by where E(t 1 , t 3 ) is the experiment in which no measurement is performed at time t 2 . Further details for LGIs can be found in the review article by Emary et al. [3].
The weak measurement techniques discussed in the main text were created to avoid the possibility of a "clumsy" measurement loophole [4]. When sequential measurements are performed on the same system, it is impossible to ensure that LGI violations are not due to overly invasive measurements perturbing the system in an unknown way. To minimize the effect of measurement, most LGIs replace the measurement at time t 1 with preparation of a known state, and the measurement at time t 2 with a null [5] or weak [6,7] measurement. These weak measurements [6,7] minimize back action on the system, while still extracting enough information to identify its state.
Using this technique, all the statistics of the LGI can be measured by conducting all three "measurements" in a single experimental configuration. To construct the Bell-Leggett-Garg inequality we combined a traditional CHSH experiment with this weak measurement technique. This allows us to measure all four terms of the CHSH correlator simultaneously in a single experiment.

BLGI Algorithm Assumptions and Loopholes
The fundamental assumptions of the hybrid Bell-Leggett-Garg inequality are those of local realism, which are familiar from the Bell inequalities: (i) If an object has several distinguishable physical states λ, then at any given time it occupies only one of them.
(ii) A measurement performed on one object of a spatially-separated pair cannot disturb the second object.
(iii) Measured results are determined causally by prior events.
Note that only assumption (ii) differs from the notion of macrorealism used in Leggett-Garg inequalities: it is weakened here to permit local invasiveness for sequential measurements in time made on the same object, while still forbidding spatially remote measurements from influencing each other. Note that the assumed physical state λ may be related to the quantum state, or may be a collection of more refined (but unspecified) hidden variables.
To these core assumptions we must append one more to permit noisy (i.e., realistic) detectors: (iv) Unbiased noisy detectors produce results that are correlated with the true object state λ on average.
This assumption can be understood as follows. The object state λ ideally determines each measurable property A(λ), but a physical detector (and environment) that interacts with the system will also have a distinct physical state ξ that may fluctuate noisily between realizations (e.g., from the coupling procedure). In such a case the detector will report a correspondingly fluctuating signal α(ξ) according to some response probability P A (ξ|λ) for obtaining the detector state ξ given each definite system state λ. For any sensible detector, these response probabilities will be fixed by the systematic and repeatable coupling procedure (such as our ancilla measurement circuit). To calibrate such a detector, we must then assume that averaging over many realizations of the detector noise will faithfully reflect information about each prepared system state λ (even if that state ultimately changes for subsequent measurements due to the coupling): Importantly, this equality formally states only what is usually assumed for an unbiased laboratory detector: that one can recover a meaningful system value A(λ) by averaging away any detector noise.
Now consider the Bell-Leggett-Garg correlation. A correlated pair of objects with the joint state λ is sampled from an ensemble with the distribution P (λ). (In our experiment, we prepare two qubits in a Bell state.) At a later time each object (k = 1, 2) is coupled to a detector (an ancilla qubit) that outputs a noisy signal α k calibrated to measure the bounded property A k (λ) ∈ however, for each λ the realizations of the output signal will average to the correct bounded value by assumption (iv). (We verify this assumption with the ancilla calibration measurements using definite preparations of 0 or 1 on the Bell qubits.) Finally, each object is measured with a second detector that outputs a signal b k for a similarly bounded property B k (ζ) ∈ [−1, 1] (we read out the qubits directly to obtain b k = ±1). From these four measured numbers, we then compute the CHSH-like correlator as a single number for each preparation The expanded ranges of the noisy signals α k generally produce a similarly expanded range for the correlator C for each preparation. Nevertheless, averaging C over many realizations of the detector noise ξ A k and ξ B k and system states λ will produce since postulate (ii) causes the joint distribution of the detector states to factor: Importantly, the joint probability P (ξ A k , ξ B k |λ) = P (ξ A k |λ)P (ξ B k |λ, ξ A k ) for each qubit k admits the dependence of the B k measurement on an invasive A k measurement that can alter the physical state λ. Despite any randomization of the results b k (ξ B k ) caused by such local invasiveness, however, the perturbed averages B k (λ) must still lie in the range [−1, 1] since each b k = ±1 There are, however, two notable ways that our derivation of the BLGI in Eq. (8) could fail.
First, the assumptions (i-iii) of local realism could fail, as in a standard Bell inequality. This is certainly possible in our case since the Bell qubits are neighbors on the same superconducting chip.
However, arranging for a locally realist model that accounts for the needed disturbance effects for the neighboring Bell qubits, the neighboring Bell-ancilla qubits, each remote pair of Bell-ancilla qubits, and the remote ancilla-ancilla qubits simultaneously is substantially more difficult (and therefore much less likely) than arranging for such disturbance in the usual Bell test on just two neighboring qubits. Moreover, our experiment verifies the detailed functional dependence of the quantum predictions as the weak measurement angle φ is varied, which further constrains any purported locally realist explanation. Thus our tested BLGI significantly tightens the locality loophole [9] compared to the usual Bell test performed on the same chip.
Second, the noisy detector assumption (iv) could fail due to hidden preparation noise ξ P not included in the state λ that systematically affects the detector output in both arms in a correlated way. In this case, the detector response would become noise-dependent P A (ξ|λ) → P A (ξ|λ, ξ P ) such that the calibration of Eq. (5) will be satisfied only after additionally averaging over ξ P .
Such correlated noise would prevent the detector distributions from factoring for each λ in Eq. (7), which formally spoils the inequality. However, in our experiment such a systematic bias due to correlated noise has been extensively checked during the characterization of the chip and the measurement calibration by deliberately preparing a variety of uncorrelated distributions P (λ) (i.e., different qubit states) and looking for spurious cross-correlations of the various qubit readout signals that would be expected in the presence of such hidden preparation noise. Hence, the failure of assumption (iv) additionally requires an unlikely preparation-conspiracy where every calibration check that has been done is somehow immune to the hidden detector-noise correlations.

Weak Measurement Calibration
As discussed in the main text the ancilla readout is imperfectly correlated to the Bell qubit's state. When measuring in the Z basis, Z α = sin(φ) Z β , shown in Fig. 2 (a) along with the ideal curves ± sin(φ). To calibrate this weak measurement we must first relate the measurement angle φ to microwave drive power, by fitting to a measurement of |1 state probability vs. π-pulse amplitude. The most straight forward calibration would then be to divide Z α by sin(φ) shown in the blue curves in Fig. 2 (b), but this method causes drift in the mean at the smallest angles.
This simple calibration fails because the raw data curves shown in Fig. 2 (a) converge to a value slightly below zero. This means that for the weakest measurements, the simple angle calibration will under-correct a |1 state measurement and over-correct a |0 state measurement. This overcorrection of the |0 state is problematic, since calibrated values for Z α not bounded by ±1 will possibly violate the inequality incorrectly. To prevent this, we instead use a data based calibration To apply this calibration to the correlator terms, we must first express them in terms of the measurement operator Z . In a superconducting system the state rotations are used to map the desired measurement basis onto the ground (|0 ) and excited (|1 ). For the ancilla measurement this is equivalent to mapping onto the Z measurement axis. Given probability P(1) of measuring the excited (|1 ) state, Z = 1−2P (1). After mapping state probabilities onto the Z measurement axis we can express the correlator as E(α, β) = Z α Z β . Expressed in this way we can see that for calibration factor cal(φ) ≈ 1/ sin(φ), E cal (α, β) = E(α, β) * cal(φ). Extending this to the BLGI we calibrate each term depending on the ancilla qubit being measured such that E(α 1 , β 2 ) → E(α 1 , β 2 ) * cal(φ 1 ), E(β 1 , α 2 ) → E(β 1 , α 2 ) * cal(φ 2 ), E(α 1 , α 2 ) → E(α 1 , α 2 ) * cal(φ 1 ) * cal(φ 2 ), and E(β 1 , β 2 ) remains unchanged.

Error Analysis and Pulse Sequence Optimization
While the algorithm and weak measurement scheme are simple in design, dependence on correlations between multiple qubits makes C sensitive to multiple error mechanisms. As all four qubits were operated away from the flux insensitive point they were more susceptible to dephasing effects. The amplitude of C vs. dephasing error per qubit is shown in Fig. 3(a). The violation amplitude is relatively robust to this error, and can sustain error rates of up to 30 percent while still exhibiting non-classical correlations. The second major error mechanism was reduced measurement visibility coming from T 1 energy decay or spurious |1 state population. The effect on C vs. single qubit measurement visibility is shown in Fig 3(b). The correlation amplitude is more sensitive to this reduced measurement visibility and is significantly degraded at even 90 percent. In both cases, the presence of errors not only lowers the maximum violation possible but the highest measurement strength at which a violation first occurs. As weaker measurement angles require finer calibration and provide noisier data, it is preferable to achieve a violation at the largest measurement strength possible.
(a) (b) percent phase error percent visibility measurement strength ϕ/π measurement strength ϕ/π BLGI correlator BLGI correlator Amplitude of C vs. measurement strength for various single qubit measurement visibility values. The system shows a greater sensitivity to this error mechanism and cannot tolerate a visibility much below 90 percent.  Given the sensitivity of C to various decoherence mechanisms it was important to reduce the BLGI pulse sequence time as much as possible for higher coherence. This is most notable during the weak measurement portion when we carry out simultaneous adiabatic CZ gates [10] between both ancilla-Bell pairs. Lastly, we introduced spin echo pulses in the middle of the algorithm which cancel out dephasing during the pulse sequence while simply transforming the original |Ψ + Bell state to a |Ψ − . To maximize measurement fidelity, we used a wide bandwidth parametric amplifier [11], to ensure a high signal to noise ratio and shorter readout time. A separate measurement at the beginning of the pulse sequence was used to herald [12] the qubits to the ground state, but this was a small (∼6 percent) effect. Lastly, we implemented numeric optimization of the adiabatic CZ gates using the ORBIT protocol [13] to fine tune parameters for the final data set. The full Pulse sequence and frequency placement of the qubits during the algorithm is shown in Fig. 4.
During the numeric optimization of the pulse sequence, single qubit phases can be adjusted slightly to increase correlations, leading to a larger violation. This is equivalent to changing the final rotation angle of the detectors slightly (∼ 3 degrees). The nature of the BLGI makes it immune to such rotations as loss of correlations in one correlator is naturally made up for in another. additionally, the initial detector rotation b was chosen based on the maximum of the original CHSH measurements. Due to differences in qubit coherence this does not necessarily occur at π/4, but at a slightly smaller angle. The individual BLGI correlator terms measured in this experiment along with theory curves accounting for these realistic rotations are plotted in Fig. 5. The behavior of each individual term depends on the type of qubits being correlated.

Sample fabrication and characterization
Devices are fabricated identically to Ref. [14], and extensive calibration was documented in Ref. [15].

Device parameters
The device parameters are listed in table I. Note that the coupling rate g is defined such that strength of the level splitting on resonance (swap rate) is 2g (Ref. [16]).