Experimental test of non-macrorealistic cat-states in the cloud

A quantum witness attempts to classify observations or experimental outcomes as arising from one of two possible classes of physical theories: those described by macrorealism, and those that are not (e.g., quantum theory). In this work, we experimentally implement a quantum witness on a set of"small cat states"(two-qubit entangled states) and"large cat states"(GHZ states with qubit number $n=4$ and $6$) using the IBM quantum experience with $5$ and $14$ qubits benchmarks, respectively. We also consider an alternative prepare-and-measure scenario to trade off the intermediate measurement with an additional assumption. With both approaches our results show that the small cat states are non-macrorealistic. In contrast, a six-qubit GHZ state does not violate the witness beyond a so-called disturbance condition, and thus may be understood in macrorealistic terms, whereas the four-qubit case remains ambiguous. Finally, we consider un-entangled superposition states of $n=2$, $3$, $4$, and $6$ qubits to demonstrate how the quantum witness can function as a dimensionality witness.


I. INTRODUCTION
The availability of public quantum computers via the so-called "IBM quantum experience" [1], promises both applications [2][3][4][5][6][7][8][9][10][11] and tests of fundamental physics [12][13][14]. In particular, as the number of qubits increases in such a noisy intermediate-scale quantum computer, it potentially allows for a rigorous study of the crossover between classical and quantum worlds [15], e.g., it may allow us to explore whether Schrödinger's cat is dead or alive [16]. Motivated by this question of whether quantum effects play a role on macroscopic scales, the Leggett-Garg inequality (LGI) [17,18] can be used to classify observations or experimental outcomes as arising from one of two possible classes of physical theories: those described by macrorealism, and those that are not (e.g., quantum theory). Here, a macrorealistic theory (or equivalently, for a given observation, a macrorealistic object) is one where the system properties are always well-defined (i.e., realism), and in which such properties can be observed in a measurement-independent manner (i.e., measurements just reveal pre-existing properties of the system, and do so in a way that do not change those properties).
Generally, it is thought that macrorealistic theories might apply when the dimension, particle number, or size of a system is increased, such that the behaviour of the system will tend to be macroscopic and can be observed without consequences. In contrast to spatial measurements on separate systems needed to test the Bell inequality [19,20], the LGI relies on temporal measurements on a single system, but with the added assumption that the system be macroscopic, and hence insensitive to the act of measurement.
In an earlier work [48] the IBM quantum experience was used to demonstrate a standard LGI violation with a single qubit combined with a test of measurement clumsiness, termed the adroitness [46]. In this work, we implement a quantum witness [22,49,50], which derives from the same assumptions of the LGI, on a set of "small cat states" (two-qubit entangled states) and "large cat states" (GHZ states) using the IBM Q 5 Tenerife and 14 Melbourne processors, respectively. We define "cat states" as superpositions of maximally polarized qubit configurations (see later) which maximize the "disconnectivity", or n-body irreducible entanglement, which serves as one potential measure of macroscopicity [17,18,[51][52][53][54]. By combining this witness with a disturbance parameter [42] we test the limits of macrorealism in a new regime.
In order to obtain the two-time correlation functions necessary to test the quantum witness, we use CNOT operations and ancillas to implement the intermediate measurement. This approach is denoted as the "directmeasure scenario". We perform a simulation of the noise and gate errors in this scenario with an instantaneousgate Lindblad master equation. In order to address the clumsiness loophole of the direct-measure scenario in the quantum witness [42], we employ a disturbance parameter which allows us to bound possible invasivity in the intermediate measurements by constructing circuits with and without the CNOT operations and ancillas. Although the quantum witness can be fully characterized with the direct-measure scenario, we also consider an alternative "prepare-and-measure scenario". In this approach one replaces intermediate measurements with state preparation, and hence one needs fewer qubits than the direct-measure scenario, as ancillas are not required.
Our tests show that small cat states clearly violate the quantum witness and are thus non-macrorealistic. On the other hand, as we increase the number of qubits involved in the cat state, the witness value is suppressed, suggesting that the IBM processor must be classified as macrorealistic for the six-qubit example studied. However, the four-qubit case remains ambiguous.
Finally, instead of preparing entangled states, we also consider unentangled states, i.e., direct products of superposition states, to demonstrate how the quantum witness can function as a dimensionality witness with n = 2, 3, 4, and 6 qubits [55]. In particular, in our results, we find that the maximal violation increases with n, showing that the dimensionality witness functions as expected. However, in this un-entangled case, the disconnectivity is low, and the macroscopic nature of the machine is less clear.
This paper is organized as follows. In Sec. II, we summarize the notions concerning the quantum witness. In Sec. III, we introduce the cat states and their corresponding theoretical prediction of the quantum witness. In Sec. IV, we introduce the quantum circuits which we implement in the IBM quantum experience. We briefly consider a simulation of the noise and gate errors with a instantaneous-gate Lindblad master equation. In order to address the clumsiness loophole, we test the disturbance condition with quantum numbers n = 1, 2, 4, and 6. In Sec. V, we show the quantum circuit for the prepare-and-measure scenario. In Sec. VI, the experimental results are presented. Finally, in Sec. VII, instead of considering entangled states, the quantum witness for the direct products of superposition states is implemented, and we show that in this case the quantum witness can be applied as a dimensionality witness.

II. QUANTUM WITNESS
The quantum witness considers two macroscopic observables, O 1 and O 2 , measured at two different times t 1 and t 2 with the corresponding outcomes i and j, respectively [49]. For any macrorealistic system, if we assume that the measurement outcomes are normalized, the system obeys realism, and the measurements are "non-invasive" [17,18], while the two-time correlation is classically related to the probabilities [49], where p j (t 2 ) is the probability of observing the outcome j at time t 2 , and p i (t 1 )p(j|i) (t1→t2) , is the two-time correlation function for observing the outcome i at time t 1 followed by the outcome j at time t 2 . Operationally, this means that at time t 1 one measures the observable O 1 on the system with results i, such that the system is then known to be in the state associated with i. The probability of this outcome is given by the probability distribution p i (t 1 ). Following this outcome, at time t 2 , the observable O 2 is performed on the system with results j. Given these definitions, the quantum witness [49] can be defined as with a sum over all possible outcomes i of the two-time correlation function. If W = 0, the state at time t 1 is said to be macrorealistic. Note that in Eq. (2), we select just outcomes where the system is found in the final state j, but the witness can be extended to include all outcomes if necessary [42]. Alternatively, one may, instead of directly measuring the two-time correlation functions, first run an experiment where the probabilities p i (t 1 ) are collected. Then one may run another experiment where one deterministically prepares the system in the state i, and measures p(j|i) (t1→t2) . This scenario, which we call 'prepareand-measure', replaces the non-invasive measurement assumption with an ideal-state preparation assumption, and a non-Markovian evolution assumption (see [49] and [56]).
In quantum theory, the measurement outcomes k are described by positive-operator valued measurements (POVMs) M k with properties M k ≥ 0 and k M k = 1 1. We can construct the two-time correlations of result j conditional on result i at a later time t 1 as where ρ 0 is an initial state and Φ(X) is a completepositive trace-preserving map that describes the time evolution of state X from t 1 to t 2 . It has been shown that the quantum witness, and thus the assumptions of macrorealism, can be experimentally violated by quantum mechanics [41,42,57].

III. SYSTEM AND IDEAL THEORETICAL RESULTS
One of the goals of the quantum witness is to identify if the macroscopic nature of a given system influences whether it behaves in a "quantum way" or in a macrorealistic fashion. While definitions of macroscopicity are myriad, Leggett himself suggested that a minimal starting point are the extensive difference and the disconnectivity [51,52]. The former compares the difference in magnitude of the observable outcomes to some fundamental length scale. The latter considers that if an object is composed of n 'particles', then that object should contain n-body irreducible entanglement. Recent experiments have attempted to maximize the extensive difference [42] with single qubits. If one considers an n-qubit system (as available in the IBM Quantum experience), the question arises of how important the disconnectivity is. In this paper, we primarily consider states which maximize the n-body disconnectivity.
At t 0 , we set the initial condition such that all n two level systems are initially prepared in a product state, namely |0 ⊗n . Then a unitary transformation U (n, θ) is performed on the system that generates an n-qubit entangled cat state at time t 1 , namely where |φ(n, θ) = cos θ 2 |0 ⊗n + sin θ 2 |1 ⊗n , with real coefficient θ (which for θ = π/2 and n > 2 are GHZ states). After time t 1 , the pure entangled states are 'evolved back' to the state |0 ⊗n , namely by performing the inverse unitary transformation U † . The combination of U and U † indirectly mimics [58] some aspects of the Rabi oscillations of a system restricted to states |0 ⊗n and |1 ⊗n , though it is not a full simulation of the dynamics of such a system for arbitrary n. Furthermore, we only consider the measurements that are projections onto the Pauli-Z basis (also known as "computational basis") for each qubit. For instance, M 0 = |0 0| ⊗n and M 1 = |1 1| ⊗n . Here, the indexes 0 and 1 on POVMs are short-hand notation for measuring all the qubits in the states |0 and |1 , respectively. In Fig. 1, we summarize the states and the measurements performed at times t 1 and t 2 .

FIG. 1.
We prepare n qubits on the state |0 ⊗n (blue) at time t0. A unitary U transfers the system into the entangled one |φ(n, θ) = cos θ 2 |0 ⊗n + sin θ 2 |1 ⊗n (red) at time t1. Then, an inverse unitary U † is performed to the entangled system, such that the system returns back to the state |0 ⊗n at time t2. The measurements M t 1 i , and M t 2 j are performed with results i and j at times t1 and t2, respectively.

IV. DIRECT-MEASURE SCENARIO
In the following, we experimentally test the "small cat states" for n = 2 with θ ∈ {0, π/8, 2π/8, 3π/8, 4π/8} using the processor IBM Q 5 Tenerife. For n = 4, and 6 ("large cat states"), the GHZ states are simply implemented by considering θ = π/2 with the 14-qubit processor IBM Q 14 Melbourne. We note that we restrict ourselves to a maximal qubit number of 6 due to requiring 6-ancilla qubits for measurements at time t 1 in the direct-measure scenario. While IBM Q 14 Melbourne has 14 qubits, one cannot perform CNOT gates between arbitrary qubits because the direction of a CNOT gate is limited by the physical processor design (see the physical structures in [1]), limiting us to 6 qubit in our cat state, and 6 ancilla qubits.
From the initial state |0 ⊗n , "cat states" can be easily obtained by performing the unitary transformation U . The unitary U can be decomposed into several parts. Firstly it contains the operation on the first qubit (with λ = ϑ = 0), followed subsequently by a series of CNOT gates between the first qubits and all others. The inverse operation U † is given by applying CNOT gates before again applying the U † 3 = U 3 (0, 0, −θ) gate on the first qubit. A schematic example for a two qubit system is shown in Fig. 2 (a). We note that if one were to directly implement the circuit in the figure on the IBM quantum experience it would be automatically 'optimized' to be an identity operation, since the U and U † cancel. To prevent such an unwanted optimization, we insert 'barriers' into the design of the quantum circuit to force the IBM system to actually implement the processes we require.
Since the IBM quantum experience only allows at most one measurement operation on any given qubit, we have to perform a CNOT gate on each measured qubit and an ancilla qubit. Here, the ancilla and measured qubits are respectively the target and control qubits [see Fig. 2 (b) and Ref. [59]]. The measurement results on the ancilla qubit refer to the outcomes i and leave behind the corresponding eigenstates |i . After the measurement at time t 1 , we apply the U † on the post-measurement state. Finally, the second measurement with outcome j at time t 2 can be implemented, without the need for ancillas. From this, the IBM quantum experience can return the result

A. Noise simulation
Every quantum computer suffers noise from the environment or other imperfections (e.g., incorrect gates). The IBM quantum experience provides decoherence rates in terms of an energy relation time T 1 and dephasing time T 2 . The T 1 time is determined by performing many identity gates in a quantum circuit with the initial state being |1 until the state decays towards the |0 state. The T 2 time is determined by a Ramsey experiment or an echo experiment [59]. The quantum circuit implementation of these two experiments are publicly available on the IBM quantum experience [1]. Note that all these parameters fluctuate on long-time scales, and thus experiments performed at different instances (e.g., from one day to the next) may give slightly different results [12]. Obviously, the total circuit time is determined by the number of quantum gates. In the IBM quantum experience, each quantum gate is constructed by a combination of frame change, Gaussian derivative, and Gaussian flattop pulses. The frame change is identical to performing a virtual Z gate in a classical computer taking zero gate time. The Gaussian derivative and flattop pulses are respectively described with amplitude and angle parameters. The gate times of the Gaussian derivative and flattop pulses are publicly given on the IBM processor information website. We note that the gate times between the two qubit processors (the IBM Q 5 Tenerife, and 14 Melbourne) are different.
In the following, to include the influence of decoherence and gate infidelities in a simulation of the quantum circuit we consider a simple strategy where we assume that each gate is performed perfectly, and instantaneously, and then (following each gate) allow for a period of noisy evolution. This encapsulates to some degree the noise, gate infidelity, and finite gate time. During such periods the dynamics of the system undergoing decoherence can be described by the following Lindblad master equation [60,61]: where σ i + , σ i − , σ i z represent the creation, annihilation, and Pauli-Z operators of the ith qubit, respectively, with coefficients γ T1 = 1/T 1 and γ T2 = 1 2 ( 1 T2 − 1 2T1 ), where T 1 = 46 µs and T 2 = 13.5 µs.
Although, in general, the values of the quantum witness only decrease under the two types of decoherence, we will show later that the experimental result with θ = 0 is not close to 0 [see Fig. 3]. The non-zero value is due to imperfect gate operations (particularly the ancilla CNOT in the direct-measure case and state preparation in the prepare-and-measure scenario). The effect of these imperfections excites the state |0 ⊗n into other states in a way which is not equivalent when the measurement at time t 1 is performed, and when it is not. This is precisely a classical "clumsy" measurement leading to a loophole violation. We can naively simulate the effects of such errors by the following extra Lindblad terms: where γ Errors is the coefficient to simulate the gate errors.
For the direct-measure scenario, we determine this value (γ Errors = 8.5 × 10 −2 µs −1 ) such that it approximately fits the CNOT gate infidelity error rate. To exclude the influence of these type of errors in the witness itself we consider a generic disturbance condition in the next section.

B. Disturbance
To address the non-invasive effect of the measurement at time t 1 , the disturbance parameter of a qubit system τ can be defined as [42] where p(M t2 0 ||τ ) and p(M t2 0 ||τ , O 1 ) are the probability distributions without and with applying the operation O 1 at time t 1 . With this parameter one can define a revised bound on the witness of W ≥ max(d |τ ). However, finding the maximum is difficult to implement for many qubits n because there is a total of 2 n circuits to be built for preparing all possible states |τ .
In this work, instead of preparing all possible states |τ , we only consider the states |0 ⊗n and |1 ⊗n for qubit number n > 2: The system is classified as non-macrorealistic when the quantum witness is greater than the maximal disturbance parameters: This bound relies on some experimentally determined assumption on the values of d |τ , and while a violation is is less strict than (10), it is still highly suggestive of nonmacrorealistic behavior.
Max 0.077 ± 0.008 0.004 ± 0.001 0.019 ± 0.004 0.011 ± 0.003 Ave 0.068 ± 0.006 0.003 ± 0.001 0.010 ± 0.006 0.005 ± 0.005  We test the disturbance parameters 25 times with 8192 runs. Although we do not explicitly test the quantum witness for a single-qubit, we do test the disturbance parameter for this case, to check the trend we describe below. For a two-qubit system, we prepare all possible quantum states to test the disturbance parameter. Specifically, the states |0 ⊗2 , |1 ⊗2 , |10 Q0 ⊗ |0 Q1 , and |0 Q0 ⊗|1 Q1 , where |i Qj is the jth qubit with eigenvalue i in the computational basis (see Fig. 2), are prepared. The average and maximum values of the disturbance parameters of single, two, four, and six qubits are shown in Table I. We note that the results of single, and two-qubit systems are obtained from the IBM Q 5 Tenerife, while the results for four, and six-qubit systems are obtained from the IBM Q 14 Melbourne.
From the experimental results of the IBM quantum experience, one can observe that the disturbance parameters approximately satisfy the following trend, which we use to justify the reduced bound in (12): (1) d |0 ⊗n ≥ d |τ ∀|τ , (2) d |0 ⊗n ≤ d |0 ⊗n ′ , n < n ′ , and (3) d |1 ⊗n ≥ d |1 ⊗n ′ , n < n ′ . This is because the contribution d |τ can only occur because of very particular errors [see also Sec. IV A], where an error in the measurement process takes a contribution from |τ to the state |0 ⊗n . In principle, when the number of qubits involved in the experiment increases, d |0 ⊗n becomes even more dominant, as a process which takes any particular state |τ to the collective state |0 ⊗n becomes statistically rarer. Although the above properties are consistently observed in the experiments, we retain d |1 ⊗n in the definition of disturbance because it is useful to consider how the contribution from the excited state |1 ⊗n at time t 1 can sometimes contribute to a false witness, as we will discuss later.

V. PREPARE-AND-MEASURE SCENARIO
An alternative approach (which can in principle allow for a larger number of measured qubits since no ancilla qubits are needed) relies on trading the measurement at time t 1 with ideal state preparation.
In this new scenario, the first circuit is performed with a unitary transformation U before the measurements at time t 1 . The IBM quantum experience returns the probability distribution p i (t 1 ) with outcomes i [see Fig. 2 (c)]. According to the probability distribution p i (t 1 ), we then prepare a new circuit with an initial state in the eigenstates |i . The U † operation is then performed before the measurements at time t 2 on the system. The results from the IBM quantum experience represent the conditional probability distributions p(j|i) (t1→t2) . Here, only the outcome j = 0 is used to analyse the quantum witness in Eq. (2).
Note that the prepare-and-measure scenario is not efficient as the number of qubits increases because the number of quantum circuits correspondingly increases with the number of outcomes i. We prepare all possible eigenstates |i for n = 2 and 4 qubits systems. For the 6 qubit case, we only prepare the eigenstates |i if p i (t 1 ) ≥ 10 −3 , which is chosen to be much smaller than the ideal outcome of, e.g., p 0 (t 1 ) = 0.5 (note that the error induced in the witness due to omission of these small terms can in principle be of the same order as the uncertainty in the experimental data we show later; but given that the observed violation is already lower than the disturbance condition, this error does not cause a false witness). Finally, we note that there are at most (i + 1) quantum circuits in this scenario. However, there are only two experimental circuits with the corresponding statistical data i p i (t 1 )p(j|i) (t1→t2) and p j (t 2 ) in the direct-measure scenario.
As with the direct-measure scenario, which suffers from a "clumsiness loophole" arising from the noninvasive measurement assumption, the prepare-and-measure scenario can similarly suffer from a clumsiness loophole related to non-ideal state preparation (and, in principle, non-Markovian effects [49,56]). Figure 3 shows experimental data for the small cat states (n = 2 with θ ∈ {0, π/8, 2π/8, 3π/8, 4π/8}). We also show the theoretical predictions both with and without noise simulation as well as the modified witness bound based on the disturbance parameters of Sec. IV B. From this figure we observe the maximum value of the quantum witness occurs when entanglement parameter θ = π 2 , which is the maximally entangled state. The different outcome probabilities for small cat states are listed for completeness in Appendix. At θ = 0 we find no evidence of the system being anything other than macrorealistic, because the value of the quantum witness is lower than the experimental disturbance parameter from Sec. IV B. Interestingly, there is a residual small violation of the witness even though such is not predicted by the simple 'pure states' expression in (6). This 'disturbance' represents either a classically invasive measurement (in the direct scenario) or an error in the state preparation (or non-Markovianity) in the prepare-and-measure scenario.

VI. EXPERIMENTAL RESULTS
For example, in our simulation of the direct scenario plotted in Fig. 3, we observe that the θ = 0 non-zero witness value arises directly from γ Errors in Eq. (9) (i.e., if we set γ Errors = 0 the witness value in the simulation falls to zero). This 'clumsiness' can be closely associated to the d |0 ⊗n disturbance parameter. In addition, for θ = π/2, increasing γ T1 in Eq. (8) actually increases the value of the witness. This is because the overall circuit time increases substantially when the ancilla measurement CNOT gates are applied, which increases the overall 'clumsiness' of disturbance of the measurement in the form increased relaxation from |1 ⊗n to |0 ⊗n . In this case this influence is encapsulated in the disturbance parameter d |1 ⊗n .
When we consider the results of the large cat states in Table. II, the quantum witness 'violation' dramatically decreases regardless of using the direct-measure or prepare-and-measured scenarios. Moreover, the value of the quantum witness for six qubits is even lower than the disturbance parameter in our experimental test. Thus, we can say the six-qubit system admits a macrorealistic description. One should note that, although the value of quantum witness for the four-qubit system is larger than the disturbance parameter d |0 ⊗4 , whether the fourqubit system is behaving in a "quantum way" is ambiguous because the disturbance parameters with n ≥ 2 are only tested for preparing state |0 ⊗n and |1 ⊗n . Although in our discussion in Sec. IV B the maximal disturbance parameter is contributed from state |0 ⊗n , for all states |τ are not implemented. This result shows that the IBM quantum experience tends to a macroscopic realistic behaviour as the number of experimental qubits increases. From our analysis in Sec. IV A, the superconducting qubits primarily suffer significant errors and take longer circuit time to the subsequently stronger influence of decoherence as the circuit complexity, or "depth" [15], We consider two different scenarios, referred to as direct-measure (red diamond) and prepare-and-measure (blue circle) to obtain the two-time correlations. The theoretical results with and without noise simulation are shown by pink dash and black solid, respectively. Obviously, the quantum witness increases with the parameter θ, but also shows a residual violation due to classically invasive measurement backaction and gate error at θ = 0. We simulate the influence of decoherence and gate infidelities by a Lindblad-form master equation (8). The coefficients of relaxation time T1 = 46 µs, dephasing time T2 = 13.5 µs, and gate-error coefficient γErrors = 8.5 × 10 −2 µs −1 are determined by approximately fitting the experimental results. The gray area is the regime of macrorealistic determined by the disturbance parameter (12). Note that the disturbance parameter including the standard deviation does not depend on θ.
increases. In general, the prepare-and-measure scenario can also test for qubit number n > 6. However, we do not do this cumbersome procedure because the directmeasure results shows that for the n = 6 case the system is already classified as macrorealistic.
Interestingly, the witness values from the prepare-andmeasure scenario are all slightly higher than the directmeasure ones. From the circuit-implementation point of view, the prepare-and-measure scenario significantly reduces the numbers of CNOT gates, which take almost four times longer than the U 3 gates. Therefore, the prepare-and-measure scenario effectively reduces the overall effect of noise on the witness.
Finally, we note that experimental uncertainties are derived from the multinomial distribution, which uses the formula δp = (1 − p)p/N , with N = 8192 being experimental runs, and standard error propagations [2,12].

VII. DIMENSION WITNESS
Finally, we utilize the quantum witness to test the dimension of the system in the IBM quantum experience. The dimension witness is based on the maximum value of the quantum witness derived by Schild and Emary [55], namely where D Ideal = 2 n is the ideal dimension of the system with qubit number n. We note that, recently, a different approach to dimension witnessing was also applied on the IBM quantum experience [62]. In order to reach the maximum value of the quantum witness, the maximally entangled states we used previously are now replaced by a product of superposition states at time t 1 [55], In order to obtain a product of superposition states at time t 1 in the quantum circuit, instead of the U operation in Sec. IV, we perform Hadamard gates on each qubit individually. At time t 2 , the Hadamard gates, which are its own inverse transformation, are performed to again obtain a state |0 ⊗n . We note that, the 'barriers' must be inserted between two Hadamard gates to avoid automatically combining there to an identity operation. The values we observe of the quantum witness are shown in Table III by using the direct-measure scenario. Although the observed quantum witness of each n cannot reach precisely the corresponding maximum values due to the inevitable decoherence, we do see that, as expected, quantum witness can experimentally function as a dimensionality witness.

VIII. CONCLUSIONS
In this work, we experimentally tested a quantum witness for n = 2, 4, and 6 qubits with the IBM quantum experience. While previous examples proposed by Huffman et. al., [48] showed a violation of a related Leggett-Garg inequality with a single qubit, here we design a circuit which should generate a highly entangled n-body state, such that it maximises the "disconnectivity" definition of macroscopicity. For n = 2, we observed a violation of the quantum witness for θ = π/8, 2π/8, 3π/8, and π/2. Thus, we claim the state is non-macrorealistic. For θ = 0, although the quantum witness is observed, it does not violate the disturbance condition. On the other hand, we found that six qubits, prepared in a GHZ state, did not violate the witness beyond a so-called disturbance condition, and are thus macrorealistic. The four-qubit case did violate such a condition, but strictly speaking can only be classified as macrorealistic under the assumption that certain states contribute negligibly to the disturbance condition. This latter is only justified by the observed experimental data.
A previous work [48] shows that the single-qubit system in the IBM quantum experience cannot be described by macrorealism by violating the LGI accompanied with a "clumsiness" test. Compared with their work, our experimental results effectively explore a higher dimensional system as well as maximizing the disconnectivity, making our test arguably more 'macroscopic'. Obviously, for a two qubit example, the experimental results are compatible with theoretical predictions, as seen in other works [12,48]. For larger numbers of qubits, it is of course far from the ideal theoretical predictions.
Overall, our results suggest that the IBM experience tends towards macrorealistic behavior for more than four qubits and for the resulting circuit depth [15] (i.e., overall run-time) on which the witness can be tested. A significant contribution to the circuit depth arises from the ancilla-based measurements. However even the prepareand-measure scenario, which has a much lower circuit depth, does not produce a violation for six qubits. Inter-estingly, since a CNOT gate is its own inverse, one can reinterpret the combination of the quantum witness, and our choice of circuit, as a test of a fundamental circuit identity under the conditions of macrorealism. In other words, we test whether CNOT 2 = 1 1 still holds under the condition of a projection onto a classical basis between the two CNOT gates.
Finally, instead of preparing entangled states, we also tested a product of superposition states to demonstrate the dimension witness capability of the quantum witness for n = 2, 3, 4, and 6 qubits. In our results, we found that as expected, the maximal violation increases with the number of qubits. In addition, the influence of noise on these results is substantially less than the GHZ-state based test. This is because single-qubit coherence tends to be less susceptible to noise than GHZ states, and because of the lower circuit depth.

Appendix
Here, we show the probability distributions obtained from IBM quantum experience to implement the quantum witness for small-cat state with prepare-andmeasure scenario and direct-measure one. Note that, the quantum witness is only justified by the observed experimental data.