Witnessing non-objectivity in the framework of strong quantum Darwinism

Quantum Darwinism is a compelling theory that describes the emergence of objectivity of quantum systems. Spectrum broadcast structure and strong quantum Darwinism are two extensions of quantum Darwinism with emphasis on state structure and information respectively. The complete characterisation of these three frameworks, however, requires quantum state tomography over both the system and accessible environments, thus limiting the feasibility and scalability of experimental tests. Here, we introduce a subspace-dependent objectivity operation and construct a witness that detects non-objectivity by comparing the dynamics of the system-environment state with and without the objectivity operation. We then propose a photonic experimental simulation that implements the witnesssing scheme.


I. INTRODUCTION
Everyday macroscopic systems are objective in the sense that certain information about their states are, in principle, knowable by multiple independent observers. In contrast, quantum systems are subjective in the sense that they can be measured by in multiple different bases, causing different observers to obtain different information about the system. Quantum Darwinism proposed by Zurek [1] describes how objectivity can emerge from microscopic quantum behaviour: as systems decohere via interactions with their environment, specific information about their state can be duplicated into multiple parts of the environment. Information is "Darwinistic" as certain classical information tends to proliferate to the detriment of other types of information. Spectrum broadcast structure [2] and strong quantum Darwinism [3] are two analogous theories which have stronger requirements for predicting objectivity than Zurek's quantum Darwinism [1]. They focus on different aspects of the quantum Darwinism process: spectrum broadcast structure describes a particular objective state structure, and strong quantum Darwinism imposes conditions on the type of information shared between system and environments. Strong quantum Darwinism establishes the minimum conditions in which objectivity emerges, and hence is the focus of our paper [3].
The importance of (non) objectivity for a physical system can be appreciated by considering incoherent systems. We typically understand incoherent systems to be classical, and indeed, this emergence is explained by decoherence theory [4]. However, decoherence theory does not resolve the quantum-to-classical transition, and the lack of quantum coherence is not sufficient for objectivity. For example, a multipartite maximally mixed state is considered nonobjective, as there is no information-neither quantum nor classical-shared between the systems. This leads to the concept of classical non-objectivity, which is particularly important for systems operating at the quantum-classical boundary. Those systems can be incoherent yet non-objective, and this classical non-objectivity can subsequently lead to advantages for a system operating in such a regime.
There are various general theoretical results about quantum Darwinism [1,[5][6][7], and there has been analytical and numerical exploration in many specific models (for example, Refs. [2,). In contrast, experimental exploration of quantum Darwinism is scarce, in part hindered by the fact that complete studies need quantum state tomography of the system and the accessible environments. Previously, there were a number of open quantum dots experiments whose behaviour was linked indirectly with the concept of quantum Darwinism [30][31][32][33][34][35], but quantitative information related to quantum Darwinism were never extracted. Meanwhile, Ref. [36] considered an experimental proposal linking the which-way information in the double-slit experiment and the subsequent interference patterns to quantum Darwinism, but it lacked a clear notion of "multiple observers" or multiple environments that is a key component of quantum Darwinism. Recently, there have been experimental exploration of quantum Darwinism in photonic cluster states [37], photonic quantum simulators [38] and nitrogen vacancy centers [39]. In the two photonic experiments [37,38], full quantum state tomography of the simulated system and environments is used to determine the state and hence characterise their (non-)objectivity. In the nitrogen vacancy experiment [39], however, only the classical Holevo information was determined. This is technically not sufficient for any of Zurek's quantum Darwinism [1], strong quantum Darwinism [3], nor spectrum broadcast structure [2] which require extra information such as the quantum discord.
Quantum state tomography is required to fully characterise quantum Darwinism. This hinders the scale and scope of experiments possible, especially of those with larger and more realistic environments. This problem is closely linked to the difficulties in characterising quantum entanglement and other quantum correlations [40,41]. One solution has been entanglement witnesses: operators and schemes which detect nonclassical correlations much more simply, albeit at the price of not being able to detect all forms of entanglement [42]. The most famous witness are the Bell inequalities [43]-mathematical inequalities that are satisfied by a classical theory but can be broken under some forms of quantum entanglement [44].
Here, we introduce a non-objectivity witness in the strong quantum Darwinism framework. We consider a preferred subspace in which objectivity could occur, analogous to the preferred basis of quantum coherence. Our scheme detects non-objectivity by comparing the evolution of the system-environment state with and without objectivity-enforcing operations. To motivate experimental tests of our scheme, we present an experimental quantum photonic simulation that follows our witnessing scheme. By comparing the number of measurements needed for state tomography versus those needed for the proposed witness, we show that, with sufficiently good components, our scheme provides a significant advantage.
The paper is organised as follows: in Sec. II we review the various mathematical frameworks of quantum Darwinism. In Sec. III we describe the non-objective witness scheme. In Sec. IV we propose a quantum photonic experiment and provide numerical simulation results. We end with a discussion in Sec. V.

II. FRAMEWORKS OF QUANTUM DARWINISM
Suppose we have a system S, numerous environments {E k } k , and hypothetical observers each with access to separate environments. Quantum Darwinism describes how the spread of information leads to system objectivity: S becomes objective when the environments E k contain full information about the system state, i.e., the state appears identical regardless of observation and observer. There are three mathematical frameworks that characterise the objectivity condition precisely: Zurek's quantum Darwinism [1], strong quantum Darwinism [3] and spectrum broadcast structure [2]. In this section, we will give these conditions, before focusing on strong quantum Darwinism.
Firstly, in Zurek's quantum Darwinism (QD) [1], objectivity occurs when the quantum mutual information between system and environment is equal to the information contained in the system: where I(S : is the quantum mutual information between system and environment E k , and H(S) = − tr ρ S log ρ S is the von Neumann entropy of the system with reduced state ρ S . In strong quantum Darwinism (SQD) [3], objectivity occurs when the quantum mutual information is equal to the information contained in the system and is entirely classical in nature: where χ(S : E k ) is the classical accessible information given by the Holevo quantity. The quantum discord is as follows: where ρ E k |i is the conditional state on subenvironment E k after measurement result i on S, using the positive-operatorvalued measure (POVM) {Π S } [45,46]. In SQD, the quantum discord-genuinely quantum correlations-must be zero. The Holevo information can then be defined as the difference with the quantum mutual information [47]: In spectrum broadcast structure (SBS) [2], objectivity occurs when the system-environment state has the following structure: where the conditional environment states ρ E k |i are perfectly distinguishable-thus allowing any observer with access to E k to construct a measurement that will perfectly determine the index i. Spectrum broadcast structure differs from strong quantum Darwinism with the condition of strong independence, where, conditioned on the system, the subenvironments share no (extra) correlations amongst each other [3]. These three frameworks correspond to different levels of objectivity: Zurek's quantum Darwinism describes apparent objectivity, whilst spectrum broadcast structure describes system objectivity with partial environment objectivity. In contrast, Strong quantum Darwinism describes the precise minimal conditions for objectivity of the system. Thus, in this paper, we will focus on strong quantum Darwinism.

A. Subspace-dependent Strong Quantum Darwinism
Here, we introduce the notion of subspace-dependent strong quantum Darwinism, upon which we will build our witness in Sec. III. Environments and systems tend to have a limited set of bases in which we routinely measure. Furthermore, subspace-dependent strong quantum Darwinism finds good company with basis-dependent discord [48,49], and with the basis-dependency in quantum coherence [50].
Strong quantum Darwinism is equivalent to bipartite spectrum broadcast structure [3]. That is, if the state on SF = SE 1 · · · E F has strong quantum Darwinism, then the system S with the full fragment F has bipartite spectrum broadcast structure: and the system S with the individual components also has bipartite spectrum broadcast structure: As the different conditional fragment states ρ F |i i are orthogonal, we can define disjoint subspaces H F |i i in which they lie. Similarly, the different conditional sub-environment states ρ E k |i i are orthogonal and we can also define the disjoint subspaces H E k |i i . Due to the state structure above, the conditional disjoint subspace of F is the tensor product of the conditional disjoint subspaces in E k : Let Π E k |i be the projector into the subspace H E k |i . The projector onto the tensor product space H F |i is simply The action of this projector Π F |i ρ F Π F |i preserves some correlations between the environment states in general as the projectors Π E k |i can have rank greater than one. This is allowed within the framework of strong quantum Darwinism. In contrast, these correlations are not allowed in spectrum broadcast structure due to strong independence [2]. In subspace-dependent strong quantum Darwinism, we define the preferred basis {|i S } i on the system, and we define the preferred objective subspace partitioning for the environments, which we encode in the projectors Π F |i from Eq. (9). The following objectivity operation projects any input state into an incoherent state satisfying strong quantum Darwinism, in the fixed subspaces as defined: This is comparable to the discord-breaking measurement in Ref. [51], or the decoherence operation in Ref. [52]. From this, we can define a measure of subspace-dependent non-objectivity using the trace norm distance: The maximum value of the measure is max ρ SF (t) M SQD (ρ SF (t)) = 1 and occurs when the system-fragment is completely nonobjective. The typical trace-norm distance ρ − σ 1 upper-bound (of two) occurs when ρ and σ have orthogonal support. However, due to the nature of the objectivity operation, orthogonality between the two terms only occurs when Γ SQD SF (ρ SF (t)) = 0. In the following section, we will employ the objectivity operation to create a witness that lower bounds the value of the measure in Eq. (11).

III. WITNESSING NON-OBJECTIVITY
We will construct a witness to detect non-objectivity between a system and some collection of subenvironments. A non-zero witness implies non-objectivity relative to the pre-defined basis and subspaces. Our scheme is illustrated in Fig. 1: by measuring the difference between two alternative system-environment evolutions, we can determine the  (13)] is applied on the "un-accessed" environment E\F to ensure that the witness does not detect extraneous correlations. We can either leave the system-fragment SF untouched (identity channel ISF ) or apply the objectivity operation Γ SQD SF [Eq. (10)]. The system-environment then undergoes some unitary evolution U (τ ), and is measured at the end of the protocol. The witness comes from the comparison between the final state with or without the objectivity operation.
amount of non-objectivity present. This method in the spirit of previous schemes for witnessing quantum discord [51] and quantum coherence [52]. We could alternatively use a combination of a discord witness [51,[53][54][55][56] combined with a measurement of the classical information. However, unlike the scheme we will introduce below, this will not produce a single witness value for non-objectivity, as non-objectivity scales as D − χ [3] i.e. with opposing dependence on quantum and classical information whilst experimental witnesses tend to lower-bound the information. Another possibility is to use a traditional witness operator W leading to values tr[W ρ]. However, we expect this to require multiple joint copies of the state at the same time, e.g. ρ ⊗4 SE in the discord witness of Ref. [53], which would make the experimental scheme more cumbersome given the already large dimensions of the environment.
The system-environment evolution proceeds as follows: At time t = 0, the system and environment starts out in joint initial state state, ρ SE (0). The full system-environment then evolves under the action of a unitary U (t) such that the state at time t is Our goal is to witness non-objectivity of the system and fragment ρ SF (t) = tr E\F [ρ SE (t)].
To do so, we first must ensure that the witness does not pick up on extraneous correlations between the observed system-fragment SF and the rest of the environment E\F. We use a point channel on the remainder environment, which discards the E F +1 . . . E N states and prepares a new (uncorrelated) state.
Note that the point channel is crucial to isolate the correlations we want to test. For example, the authors of Ref. [52] profess an ambiguity in one of their witnesses-as to whether it is detecting system coherences or system-environment correlations. This ambiguity would be removed with the addition of a correlation breaking channel, as we have done here.
If the point channel is the only operation that we enact at time t, then the system-environment subsequently evolves under some unitary evolution U (τ ): Finally, we conduct a measurement M SE , giving us the probability: One possible measurement operator M SE is simply the projector onto zero: M e.g. SE = |0 0| ⊗ |0 0| ⊗ · · · ⊗ |0 0|. In an alternative evolution, we could also apply the objectivity operation from Eq. (10) on the system-fragment at time t in conjunction with the point channel such that the subsequent final state is: leading to the alternative probability of measurement M SE at time t + τ : The absolute difference between these probabilities is our witness for non-objectivity: The witness lower bounds the non-objectivity measure from Eq. (11): This can be shown as follows: if we maximised over measurement operators in the witness, then where where the trace norm of the measure can be written as supremum over Hermitian operators B where B ≤ 1 The witness depends on suitable choices of the unitary operation U τ and final measurement M SE to ensure that the two alternative probabilities [Eqs. (15) and (17)] are different. Like coherence and its witnesses, the witness we have presented is system-basis and environment-subspace dependent. Quantum Darwinism, in its fullest, requires optimisation over all bases. However, we wish for a scheme less intensive than full quantum state tomography. Furthermore, realistically, we are constrained in what we can measure, especially when it comes to environments: often, there are only a limited number of degrees of freedom with encoded information that we can access, and/or only a limited number of degrees of freedom that can possibly encode information about the system of interest. As such, there should be a naturally preferred basis and subspace.

IV. QUANTUM PHOTONIC SIMULATION EXPERIMENTAL PROPOSAL
In this section, we propose an experiment to apply the witness onto a system and environment composed of photons with information encoded in the polarisation degrees of freedom. Our scheme is particularly suited to photonic qubits over spin qubits and other related systems using magnetic resonance as the objectivity operation from Eq. (10) relies on projective measurement. We will first present the setup, then numerical simulations, and followed by a comparison between the non-objectivity witness and quantum state tomography.

A. Overall Setup
The system is comprised on one photonic polarisation qubit, with fixed basis |0 S and |1 S (corresponding to horizontal and vertical polarisation respectively). In general, the dimension of each environment can be larger than the dimension of the system. Here, we consider each environment E k as being composed of two photons, E The parity of the environment state corresponds to the two disjoint subspaces that signal strong quantum Darwinism, which we depict in Fig. 2. If the system in state |0 S then the environment should be in the space spanned by {|00 , |11 }; and if the system is in state |1 S then the environment should be in the space spanned by {|01 , |10 }. The environment is comprised of two subenvironments E 1 and E 2 . This allows us to probe non-objectivity for fragments F = E 1 and F = E 1 E 2 .
The proposed experimental circuit consists of five overall steps (the case of F = E 1 is shown in Fig. 3): Our initial system-environment state is which has strong Quantum Darwinism when one environment is traced out, and is entangled over the full environment. From this base initial state, we will consider two different operations that reduce the objectivity in the reduced environment state: either mixing the initial state with the maximally noisy state 1 SE1E2 /d S d E1 d E2 , such that the state or applying local depolarisation on all photons: 1 , E 1 , E 2 , E where E (a) 1 , a = 1, 2 denotes the two sub photons in environment E 1 etc. The objective operation involves measuring the system and applying a non-destructive parity measurements on each environment in the fragment. If the results match then the system-fragment is objective in our pre-determined subspace. The nondestructive parity measurement on the environment means that we only need to reconstruct the system state after its measurement.
In the unitary phase of the witness, we apply Hadamards H on alternating photons: The final measurements are all in computational basis, i.e. in the horizontal/vertical polarisation basis. In the following subsection IV B we include some basic experimental detail on a hypothetical experiment. The numerical simulation of the circuit is given in subsection IV C.

B. Experimental Components
There are a number of major components in that circuit, which we will go in to with further detail: the large number of controlled-NOT (CNOT) operations which are required for the state preparation and the objectivity operation, the procedure to generate four-photon GHZ states required for the initial state preparation, and the objectivity operation itself.  26)] on all photons. The non-fragment component E\F, here E2, undergoes a point channel, where the photons are discarded and uncorrelated photons are produced. Meanwhile, the system S is measured and E1 undergoes a nondestructive parity check involving an ancillary photon. If the parity measurement matches the system measurement result, then the measured system state is recreated, and the system-environment has been projected into the objective subspace. The system-environment undergoes unitary evolution, here under Hadamards H (half wave plates). All measurements are in the computational basis of horizontal/vertical polarisation. If the objective operation results in a null state, then all measurement outcomes are zero.

Controlled-NOT Operation
The experimental procedure uses a number of nondestructive CNOT gates: in the preparation of the initial state in Eq. (24) and for the parity measurement for the objective operation, both seen in Fig. 3. This is also the main limiting factor of the witnessing scheme, and the scalability of the witnessing scheme depends heavily on the success probability and fidelity of the CNOT gates (see subsection IV D on this scaling).
If we use only linear optical elements, then CNOTs gates are necessary probabilistic with realistic success probabilities of 1/4 to 1/16 [57]. In contrast, if we employ nonlinearities, then we could make deterministic or near-deterministic CNOTs [58,59] with greater success probabilities. One explicit example is given in Fig. 4(a), which uses cross-Kerr nonlinearity to boost success probability up to 1/2.

Initial state preparation
The system state can be created with a Hadamard H produced with a half wave plate (HWP) at θ = π/2: . This is shown in the first left-hand box in Fig. 3. We then need to create a four-photon Greenberger-Horne-Zeilinger (GHZ) state on the environment. This procedure is depicted in Fig. 5, and proceeds as follows: we would first create Bell states (|00 + |11 )/ √ 2 on each environment via spontaneous parametric downconversion (SPDC). One photon from each pair then goes into a beam splitter with path lengths such that they arrive at the same time. Coincident detection in the outputs implies that each both photons are H-polarised or V -polarised, corresponding to projecting the four photons into the subspace spanned by {|00, 00 , |11, 11 }. After renormalising the state, the four-photon GHZ state is made. The probability of the four-photon GHZ state being made is 1/4 [60].
Finally, we would apply CNOT operations, with the system as the control, onto the first photons of each of the two environments of the following system-environment state, to produce the initial state given in Eq. (24). The second component of state preparation is degrading the objectivity in the system-environment. Mixing with the maximally noisy state [Eq. (25)] can be done simply by creating the initial state with probability 1 − p, and  [59]. This uses a cross-Kerr nonlinearity in the controlled-phase gates ±θ to boost gate success probability to 1/2. PBS denotes polarisation beam splitter, and BS denotes beam splitter. An ancillary coherent state |α is used. |X X| is a X homodyne measurement, and depending on its results, a further σx gate and phase φ gate may be applied to the target photon. (b) Nondestructive parity measurement from Ref. [58]. Aside from polarising beam splitters (PBS), it uses cross-Kerr nonlinearities that apply a shift of +θ, and −θ on the coherent state |α if there is photon in the corresponding control modes, followed by a X homodyne measurement |X X| in order to determine the parity {|00 , |11 } or {|01 , |10 } of the input photons. the maximally noisy state with probability p. Meanwhile, for the depolarisation channel, one possibility is to use the circuit described in Ref. [61]. An alternative procedure is to mix the local state of each photon with the local unpolarised state, i.e. keeping a photon with 1 − p or replacing it with a completely unpolarised state with probability p (e.g. by passing it through a multimode fibre, or by simply generating a new unpolarised state). By averaging over sufficiently many runs, this would give an effective depolarisation channel.

Objectivity Operation
The objectivity operation from Eq. (10) can be implemented with a system polarisation measurement and nondestructive parity checks on the environment. One possible parity check scheme is shown in Fig. 3, which uses an ancillary photon and two CNOT operations. Alternatively, a direct parity check scheme could be used, such as the one from [58], reproduced in Fig. 4(b). Either methods employ nonlinear cross-Kerr nonlinearities, which is crucial for increasing the success probability of the operation.

C. Numerical Simulations
In this section, we numerically simulate the circuit and scheme we proposed. The results are given in Figs. 6 and 7. Fig. 6 shows the presumes a perfect circuit, while Fig. 7 considers if the state preparation involved controlled-NOT operations with fidelities of approximately 0.79. We find that our witness is able to detect non-objectivity in the cases we considered.

Measurement Operators in the Witness
The final measurements of the system and fragment are always in the computational basis. This reflects our motivation that only particular measurements are possible, or preferable, in a realistic system. For a particular projective rank-1 measurement Π SF i in the computational basis, the non-objectivity witness W SQD Π SF i has a very small value in general, and this can be seen near the bottom of the plots in Figs. 6 and 7. However, we can construct a larger witness without any additional measurements by considering a collection of the computational measurement operators Π SF i : As the term within the absolute magnitude, tr Π SF i {ρ SE (t + τ ) − ρ SE (t + τ )} , can be either positive or negative, this leads to: The value of max {M SF } W SQD , as can be seen in Fig. 6 can provide a good, sometimes tight, lower bound to the true value of the measure M SQD . The bound is less tight when the fragment consists of the full environment, F = E 1 E 2 : these situations correspond to greater quantum correlations between the system and fragment. This is expected, as the unitary evolution is local, and the measurements are also local, and so are unable to capture the full quantumness component of the non-objectivity. Thus, one possible improvement would be to introduce an entangling or global unitary evolution. Despite this, the witness is successful in detecting non-objectivity.

Simulation with operation fidelities and noise
In a realistic experiment, gate operations are not perfect. In particular, we have employed various CNOT operations which can have fidelity F < 1. To simulate such results, we consider the output of a CNOT operation to bẽ C N OT (ρ) = f CN OT (ρ) + (1 − f )1/d, where f = 0.733, and 1/d refers to the maximally mixed state with correct dimension. That is, we assume depolarisation noise on the states after the application of the CNOT. The subsequent average gate fidelity isF ≈ 0.79 [62], since for a unitary basis {U j = σ α ⊗ σ β } where σ = 1, σ x , σ y , σ z and d = 4, = (63f + 17)/80.
The simulation results in Fig. 7 also assume that the initial state was imperfectly prepared and takes into account the probabilistic nature of the noisy mixing and depolarisation channels. The results are based on 6000 successful runs in total (i.e. runs where the state is successfully prepared and the objectivity operation is successful), where each run has one measurement outcome. The multiple runs builds up the measurement statistics, which we use to estimate the probabilities for the witness. The standard deviation across the runs is given as the error bars. We can see in Fig. 7 that the simulated experimental witness can overshoot the true measure values, especially when there is less non-objectivity. Therefore, experimentally, the witness should pass nonzero lower bound before non-objectivity can be declared.

D. Comparison with quantum state tomography
Let us make a back-of-the-envelope comparison between quantum state tomography versus our witness based on the number of trials required in either scheme to produce statistics with similar confidence. We will not count the preparation component for the state at time ρ SE (t) as this is required regardless of technique. Rather, we want to Let us suppose there are 1 + 2M photons in SF (one system photon, and M environments with 2 photons in each environment): For the state tomography, the photons in SF need to be measured in three different bases, in all the combinations. With 1 + 2M photons to measure, there are 3 1+2M basis combinations. Let C be the number of counts in one basis set required for sufficient statistics. Then quantum state tomography would naively require C · 3 1+2M runs in total.
Meanwhile, for the witness, we assume that the point channel and unitary evolutions are deterministic (such as the Hadamards we have chosen). And since we fix the final end basis, the first set requires C runs. The second set of runs requires the objectivity operation. Let p CN OT be the probability of the CNOT operation succeeding. Two CNOT operations are required per environment, therefore the probability of a successful parity check on one environment has probability p 2 CN OT . There are M environments, therefore the probability of all the parity checks successfully occurring is p 2 CN OT M . In order for there to be a total of C successful runs, there must have been at least C(1/p CN OT ) 2M copies of the state. Therefore, the witness scheme would require C + C(1/p CN OT ) 2M runs in total.
If the CNOT operation can be implemented with success probability of p CN OT 1/3, then the witness scheme outperforms quantum state tomography. If we introduce in the fidelities for the CNOT operation, then could roughly replace p CN OT → p CN OT · F CN OT where F CN OT is the fidelity. If F CN OT = 0.79, then a minimum success probability of p CN OT ≥ 1/3F CN OT ≈ 0.42 is required. In previous literature, for example Ref. [59], a theoretical p CN OT = 1/2 is possible, using nonlinear elements. This shows that with sufficiently good controlled-NOT operations with reasonable success probability, the witnessing scheme present here can scale better as more environments are added.

V. DISCUSSION
We presented a subspace-dependent witness of non-objectivity in the framework of strong quantum Darwinism. We defined an objectivity operation that projects the system-environment into a preferred objective subspace. The witness detects non-objectivity by comparing the evolution of the system-environment with and without the objectivity operation. We used a point channel to ensure that there is no correlations between the system-fragment and the rest of the environment that might accidentally trigger the witness. We showed that this witness scheme has the potential to scale better than quantum state tomography.
Our witness scheme relies on projective measurements. Photonic systems are ideal, and are our preferred system in our experimental proposal. In contrast, spin and magnetic resonance systems naturally support weak measurements. Recovering projective measurement results would require measuring multiple relevant observables.
When all the dimensions of the environments are equal to the dimension of the system, then strong quantum Darwinism and spectrum broadcast structure collapse to invariant spectrum structure (ISBS) (see Appendix A). Such states have the following form: for some local diagonal basis |i E k i for the various environments. In Appendix B, we consider the basis-dependent witness of non-objectivity in this framework. In this case, controlled-NOT operations are not required for the objectivity operation, thus greatly simplifying a hypothetical experimental setup. As for witnessing spectrum broadcast structure: Casting back to the discussion in subsection II A, our scheme would require us to chose a preferred subspace such that each conditional state ρ E1|i ⊗ · · · ⊗ ρ E N |i ∈ H E1···E N |i of the broadcast state, lies in disjoint subspaces H E1···E N |i = H E1|i ⊗H E2|i ⊗· · ·⊗H E F |i . We can define the projectors Π E k |i into H E k |i . In strong quantum Darwinism, the environment projector onto each conditional state is simply Π F |i = Π E1|i ⊗ · · · ⊗ Π E F |i , which preserves some correlations between the different environments. However, that is not allowed in spectrum broadcast structure. A spectrum broadcast structure witness in this style would require environment projectors that also break correlations conditional on i, which in turn requires discarding the current state and preparing a new one. The local conditional environment states Π E k |i ρ E k Π E k |i need to be known, and re-prepared exactly to ensure there are no extraneous correlations. This hypothetical SBS objectivity operation is a much more intensive unwieldy procedure, making spectrum broadcast structure unsuitable for this particular scheme. Non-objectivity in a system within the framework of strong quantum Darwinism can arise from two sources: the existence of quantum correlations, or the lack of perfect classical correlations. Our scheme does not distinguish between the two, instead giving a single measure that captures non-objectivity in its own right. If the source is required, we suggest using an extra discord witness [51]. This leads to an alternative two-part witness of non-objectivity: a discord witness, followed by some kind of characterisation of the classical information. If only a binary result is required (i.e. whether or not there is non-objectivity), then the two-part protocol can terminate the moment the discord is witnessed.
The resolution to the quantum-to-classical transition remains elusive. By choosing a naturally preferred basis in which objectivity may arise, we have presented the first witness of non-objectivity, analogous to the witnesses of established non-classical effects like quantum coherence, discord, and entanglement. With gate operations of sufficiently high fidelity and probability of success, the witness scheme scales better than full state tomography. Our work opens up further experimental development and testing of quantum Darwinism in larger and increasingly realistic scenarios, and thus, composes a step towards understanding the nature of the quantum-to-classical transition.
FIG. 8. Circuit for one particular run within the scheme to witness non-objectivity in the invariant spectrum broadcast structure framework. The system and each environment consists of one a photon polarisation qubit. The system-environment are first prepared in a five-photon GHZ state (starting from the four-photon GHZ from Fig. 5). In this particular run, we have some noise channel that acts on the system-environment state (Eq. (25)). The non-fragment component E\F, here environments E3 and E4, undergo a point channel. Meanwhile, the system and environments E1 and E2 are measured. If all measurements results match, corresponding to a successful objective projection, then that state is recreated and sent down the rest of the circuit. The system-environment undergoes unitary evolution, here under Hadamard gates H. All measurements are in the computational basis of horizontal/vertical polarisation.

Quantum Photonic Simulation Proposal
We take the system as one photon, and consider four environments each comprised of one photon each. The main departure from the scheme with strong quantum Darwinism is the initial state, and the objectivity operation, shown in Fig. 8 (compare with Fig. 3).
Here, we have initial five-photon GHZ state (|00000 + |11111 )/ √ 2 on the system and environments. This can be created from the extension of the four-photon procedure [60]: after creating the four-photon GHZ state for the environment, one prepares the system state in (|0 + |1 )/ √ 2 . Then, arrange the path-lengths such that the system photon and the first environment photon arrive at a polarising beam splitter the same time. Similarly, coincident detection in the outputs implies that each both photons are H-polarised or V -polarised, and after renormalising the state, the five-photon GHZ state is made [60]. This is shown on the left-hand-side in Fig. 8.
In invariant spectrum broadcast structure, the objectivity operation is now very simple: correlated measurement in the horizontal/vertical polarisation basis in the system and fragment photons. If all measurements on the system and fragment photons give the same outcome, then the system-fragment state is objective, and that state can be recreated and sent down to the rest of the circuit.
Our exact numerical results are shown in Fig. 9. The final unitary before measurement consists of Hadamards on all system-environment photons. Such a unitary gives a better lower bound to the witness than the Hadamard arrangement used in Fig. 3 for strong quantum Darwinism. We see that for the mixed, reduced system-fragment states, the witness is tight with the basis-dependent measure. It is not tight for the full system-environment state, which contains global quantum correlations, but it nonetheless successfully witnesses non-objectivity in the state.
This witness scheme scales extremely well due to the lack of CNOT operations, requiring only C + C total runs versus the C · 3 1+F runs for F photon environments in the case of quantum state tomography (cf. with subsec. IV D). Here, one could potentially afford an entangling unitary in order to witness the quantum correlations in the full system-environment state.