Quantum coherence as randomness under classical control

arXiv:2606.03699 · quant-ph · Submitted 2026-06-02 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Quantum coherence as randomness under classical control".

Kai: Quantum states that do not commute exhibit coherence, but only when the device preparing them is assumed to be unaffected by classical parameters inaccessible to the experimenter.

Mira: First, who's behind it and why it matters.

Title and authors: Kai: So, we're diving into this paper, "Quantum coherence as randomness under classical control," which sounds like it tackles a really tricky issue in quantum hardware. What was the main thing the authors were aiming to unpack with this work?

Mira: It seems like they are tackling the problem of certifying coherence when the preparation device might be influenced by hidden classical parameters that we can't access experimentally. That's a significant hurdle for any experimentalist trying to prove a quantum state is actually coherent.

Lev: From an error correction standpoint, this has massive implications because if we can't verify the initial state coherence, our error correction codes might be built on shaky foundations, and that would make running them on real hardware extremely difficult.

Kai: Exactly, Lev; it’s about proving that a state isn't just looking coherent because of some classical settings we didn't account for. They outline a hierarchy of semidefinite programs to do this characterization.

Mira: That hierarchy is the core mathematical tool they developed to establish what coherence actually means in this context, which is interesting because it starts with a very general proof that coherence can be fully characterized through these SDPs thirteen.

Lev: I wonder how practical that hierarchy is for real-world devices; running a full hierarchy of semidefinite programs on high-dimensional states seems computationally intensive, which raises concerns about whether this translates to actual hardware benchmarking.

Kai: That's where the authors introduce an alternative SDP condition that they claim is sufficient for coherence and keeps things computationally efficient even when dealing with many or high-dimensional states thirteen.

Mira: So they’re essentially offering a trade-off: a complete but hard characterization versus a practical method that gives useful accuracy while staying computationally manageable for preparation devices.

Lev: If the practical approach can handle states generated by devices with many degrees of freedom, then it moves this from purely theoretical existence to something potentially applicable in lab settings for state certification.

Kai: And they also look at how quantum channels behave under these constraints, defining what a coherence-breaking channel is as one whose image set satisfies the incoherence condition E in C (Equation five).

Mira: Defining coherence-breaking channels helps us understand which processes in a quantum system are inherently destructive to superposition, regardless of how well we try to prepare the input state.

Lev: Understanding these channel properties is crucial for designing communication protocols or memory systems because knowing if a channel is coherence-breaking tells us upfront if we should even bother trying to use it for coherent information transfer.

Title and authors: Kai: The paper also discusses methods for characterizing qubit systems specifically, linking coherence to joint measurability, which they state is equivalent to the set of unbiased dichotomic measurements being jointly measurable (Equation twenty).

Mira: That connection between coherence and joint measurability is elegant because it allows them to use SDPs to decide this feasibility, although they admit that this specific method isn't scalable for large systems.

Lev: So, while the qubit system analysis provides a very accurate picture for small systems, we have to be cautious about scaling that up when we consider larger quantum registers in real experiments.

Kai: The paper also discusses using block moment matrices as a practical criterion to check for incoherence if the set E is incoherent, and this involves checking the existence of a positive semidefinite block moment matrix (Equation seventeen).

Mira: That block moment matrix approach seems like a way to get tangible results from the SDP hierarchy by focusing on pure state decompositions and averaging over classical parameters lambda.

Lev: If finding a violation of the witness inequality W(E) implies coherence, then this provides an algorithmic path for certification, which is something experimentalists really need when dealing with complex state ensembles.

Kai: So to wrap up this section, the paper presents two main paths: a complete but hard SDP hierarchy and a more practical block moment matrix criterion that uses a witness inequality to certify coherence thirteen.

Mira: It’s clear they are focused on providing tools that bridge the gap between theoretical characterization and what's actually feasible for current quantum preparation devices.

Lev: And they also give us a rigorous definition of coherence-breaking channels, which is essential context for anyone looking at channel fidelity in this new framework.

Kai: So, moving into the summary of the paper "Quantum coherence as randomness under classical control," we’re talking about how these hidden classical parameters affect our understanding of quantum states.

Mira: The summary points out that coherence only exists when we assume the preparation device is not affected by these inaccessible parameters, which sets up the entire problem they are solving.

Lev: That assumption is what makes things so hard experimentally; we're essentially trying to prove that our measurement setup didn't introduce some hidden classical noise during state generation.

Kai: They then describe how they develop a toolbox for analyzing quantum superposition in the presence of this hidden classical control, moving beyond just the initial problem description.

Title and authors: Mira: This toolbox involves proving that coherence can be fully characterized through the hierarchy of semidefinite programs, which is a necessary step before introducing any practical approximations.

Lev: It sounds like this paper is really laying down the mathematical groundwork so that future work on real hardware can actually verify these states against these classical constraints.

Kai: The next part discusses how they introduce a practical SDP approach designed to achieve useful accuracy while maintaining computational efficiency for preparing many, potentially high-dimensional, quantum states thirteen.

Mira: That practical SDP is key because the full hierarchy might be too slow for the kind of state preparation we see today in experimental settings.

Lev: If this practical method can handle large Hilbert spaces efficiently, then it becomes a serious tool for bench-marking modern quantum hardware setups.

Kai: Then they explore exploiting conceptual connections with joint measurability for the special case of qubits to get highly accurate coherence characterization that scales to more than one thousand qubits twenty.

Mira: That scaling result is really compelling, showing that even in the qubit case, with this approach, we can characterize coherence across a very large system.

Lev: Scaling up to a thousand qubits means this method has real potential for analyzing larger quantum systems relevant to practical applications like quantum error correction.

Kai: Finally, they apply these methods to determine if quantum channels are able to preserve coherence or are inherently coherence-breaking, leading directly into the notion of "coherence-breaking channels".

Mira: So the conclusion of this section is that their methodology allows us to rigorously test whether a channel preserves coherence by checking if its image set satisfies the incoherence condition E in C.

Lev: That leads directly into defining what those channels are, which is a very useful classification for understanding system reliability in quantum information processing.

Kai: So, in summary of the paper "Quantum coherence as randomness under classical control," they provide a complete characterization via SDPs and practical methods using block moment matrices to test coherence against hidden classical controls thirteen.

Mira: Their main contribution lies in providing a rigorous framework to determine if quantum states are coherent even when preparation is subject to hidden classical parameters, which is vital for understanding device limitations.

Lev: The implication for error correction research is that we now have a formal way to test the coherence of generated states, which could guide the design of more robust error correction protocols.

Kai: I think this work gives us a much clearer way to understand why some quantum experiments might yield results that look coherent when they actually are not under classical control assumptions.

Title and authors: Mira: It's about moving past just observing the state and instead rigorously characterizing *why* it appears coherent or incoherent in the presence of these hidden classical parameters.

Lev: For those of us working on real hardware, this means we have a better metric to judge whether our preparation stage is introducing spurious coherence that won't survive actual noise or decoherence.

Kai: So, looking ahead at the improvements they suggest, they propose an iterative search for optimal input states E to maximize output state coherence using SDP relaxations.

Mira: This iterative search seems like a way to actively probe the coherence limits of a channel by finding the input states that allow the highest visibility above which they remain coherent.

Lev: Using this search with an oracle based on the witness inequality W((E)) gives us a computational procedure to find bounds on coherence-breaking channels, which is very useful for designing resilient systems.

Kai: This iterative approach seems like a direct application of their theory to the practical problem of characterizing channel robustness, which is something we can actually simulate and test.

Mira: And for high-dimensional systems, they also detail methods using Equiangular Tight Frames to analyze coherence when analytical solutions become too complex.

Lev: If we can compute bounds on critical parameters like visibility or noise levels efficiently in those high-dimensional scenarios, it opens up possibilities for analyzing more complex quantum sensors.

Kai: Overall, this paper is providing a sophisticated mathematical toolkit—a hierarchy of SDPs and practical moment matrix methods—to tackle the pervasive issue of hidden classical control in coherence certification.

Mira: It really formalizes the idea that coherence isn't just an intrinsic property but is conditional on what we assume about the preparation environment, which is a deep insight into quantum theory and experimental practice.

Lev: My main thought is that this work gives us a concrete benchmark to test our error correction assumptions when dealing with states generated under realistic, imperfect classical control.

Kai: So, in conclusion of this discussion on "Quantum coherence as randomness under classical control," we’ve seen how the SDP hierarchy and practical block moment matrices provide a way to certify coherence against hidden classical parameters thirteen.

Mira: This paper gives us the tools to rigorously test if quantum states are coherent when preparation is subject to hidden classical controls, which is essential for understanding device limitations.

Title and authors: Lev: The implication for error correction research is that we now have a formal way to test the coherence of generated states, which could guide the design of more robust error correction protocols.

Mira: It's about moving past just observing the state and instead rigorously characterizing *why* it appears coherent or incoherent in the presence of these hidden classical parameters.

Kai: So, looking ahead at the improvements they suggest, they propose an iterative search for optimal input states E to maximize output state coherence using SDP relaxations.

Mira: This iterative search seems like a way to actively probe the coherence limits of a channel by finding the input states that allow the highest visibility above which they remain coherent.

Kai: This iterative approach seems like a direct application of their theory to the practical problem of characterizing channel robustness, which is something we can actually simulate and test.

Mira: And for high-dimensional systems, they also detail methods using Equiangular Tight Frames to analyze coherence when analytical solutions become too complex.

Lev: My main thought is that this work gives us a concrete benchmark to test our error correction assumptions when dealing with states generated under realistic, imperfect classical control.

The paper's summary: Kai: So, to recap, this paper is essentially saying that we can't really prove quantum coherence if we don't assume our preparation device isn't being subtly influenced by classical settings we can't control, and they developed a mathematical framework to handle that.

Mira: Exactly; the core idea is that without making those assumptions about the hidden classical parameters, any statement about coherence is technically conditional, which means we need a way to rigorously test if a state is coherent or just looks coherent because of some classical noise.

Lev: That conditional nature really hits home for error correction research because if our initial state isn't actually coherent under those hidden constraints, the entire subsequent error correction process might be built on something that doesn't exist in reality.

Kai: It’s about moving from just looking at a state and saying "it looks quantum" to actually proving that the underlying physics supports that coherence despite the classical noise we can't see.

Mira: They tackle this by creating a hierarchy of semidefinite programs, which is their main tool for characterizing incoherence, defining what it means for a set of states to be incoherent under these conditions.

Lev: I'm still thinking about how we get that characterization applied in a lab setting; can we actually run these SDPs fast enough on the kind of high-dimensional states current hardware is trying to generate?

Kai: That’s the key question for me, Lev; they spend a lot of time developing practical methods, like those block moment matrices, specifically to make sure this math isn't just theoretical fluff but something we can actually use to check our cool experiments.

Mira: Right, and those practical methods are what I find most interesting because they bridge the gap between the pure theory and what an experimentalist needs to actually certify a state's quality.

Lev: If those block moment matrices can handle high-dimensional states efficiently, it opens up a whole new avenue for testing large quantum registers that we haven't even thought about yet.

Kai: It really suggests that the next generation of quantum hardware benchmarking won't just look at fidelity numbers; it will need to incorporate this kind of rigorous check against classical control assumptions.

Mira: And if we can build reliable tools like this, it fundamentally changes how we interpret experimental results and what we consider a valid quantum state in practice.

Lev: It means our error correction protocols won't just be tested on idealized states; they’ll be tested against the realistic constraints imposed by the preparation hardware itself.

Kai: So, this paper is laying out a very concrete roadmap for how experimentalists can start making these deep connections between hidden classical parameters and verifiable quantum coherence.

The paper's improvements: Kai: So, moving on to the future work section, they suggest an iterative search process where you try to find input states that actually maximize your output coherence using those SDP relaxations.

Mira: That iterative search is interesting because it’s not just a one-shot calculation; it’s a dynamic way to probe the limits of what's possible for a given channel by actively hunting for the best input configuration.

Lev: From an error correction standpoint, if this search can be done algorithmically, it gives us a method to find the most robust preparation strategy or measurement basis that maximizes our chances of keeping information intact.

Kai: I think that active probing is what we need; it’s about finding the "sweet spot" in the input space where coherence is maximized against those classical control parameters we're dealing with.

Mira: And they also touch on applying these methods to high-dimensional systems, specifically using Equiangular Tight Frames when the math gets too messy for a direct SDP solution.

Lev: If they can handle those high-dimensional scenarios efficiently, that's huge for simulating large quantum devices where analytical solutions just fall apart, which is where we actually spend most of our time in theory.

Kai: So it sounds like the authors are trying to build a toolkit that doesn't just work for small systems but scales up to analyze the complexity of real-world experimental setups.

Mira: It really formalizes how we can handle the messy, high-dimensional aspects of quantum physics without getting bogged down in intractable calculations.

Lev: This gives us a clear direction for developing error correction codes that are aware of the classical control environment they operate in, making them more robust against preparation-stage noise.

Kai: The implication here is that we can start designing experimental protocols with a much clearer understanding of the coherence boundaries imposed by the hardware itself, rather than just assuming perfect preparation.

Conclusion: Kai: So, to wrap things up, this paper on "Quantum coherence as randomness under classical control" really gives us a rigorous framework for testing whether quantum states are coherent when they're being prepared under hidden classical influences.

Mira: It provides a solid mathematical foundation using SDPs and practical moment matrices to characterize incoherence in these complex scenarios, which is vital because it forces us to consider the assumptions about our preparation environment.

Lev: I agree; the implication for error correction is that we now have a formal way to test if the states we're generating are actually coherent under realistic constraints, which could guide how we design fault-tolerant protocols.

Kai: Exactly; it moves us beyond just looking at raw data and gives us a tool to understand why some experiments look coherent when they might not be under those hidden classical conditions.

Mira: This work establishes that coherence is conditional on what you assume about the preparation device, which is a deep conceptual point for condensed matter theory as well.

Lev: For real hardware, this means we have a better metric to judge whether our preparation stage is introducing spurious coherence that won't survive actual noise or decoherence during the experiment.

Kai: It’s about getting a clearer picture of what’s physically happening inside the device, which is exactly what an experimentalist needs when trying to troubleshoot state quality.

Mira: We should be very excited because this methodology formalizes how we interpret results and gives us a way to distinguish between true quantum superposition and artifacts introduced by classical control.

Lev: My main thought remains that this work sets a concrete benchmark for testing our error correction assumptions when dealing with states generated under imperfect classical control conditions.

Kai: It’s been fascinating seeing the transition from the general SDP hierarchy to those more efficient block moment matrices, which shows a real commitment to making this tool usable on actual quantum hardware.

Mira: We're really looking forward to seeing how this framework is applied to characterize coherence-breaking channels in more complex physical systems next.

Lev: That’s what I’m looking forward to; understanding those channel properties will be essential for designing resilient quantum communication protocols that can handle the realities of device preparation.

Kai: So, that wraps up our discussion on "Quantum coherence as randomness under classical control," and it feels like we have a powerful new lens through which to view state certification.

Physics Department and NanoLund, Lund University · Institut für Theoretische Physik III, Heinrich-Heine-Universität Düsseldorf

quant-ph

Submitted: 2026-06-02

Updated: 2026-09-30

Code: https://github.com/nicoljno/coherencecertification

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 87/100

The gist: Quantum states that do not commute exhibit coherence, but only when the device preparing them is assumed to be unaffected by classical parameters inaccessible to the experimenter.

Key concepts

Coherence Characterization via Semidefinite Programming (SDP): This method uses a series of mathematical relaxations (outer SDPs) to define a set C that represents all incoherent states. If the system's states fall into this set C, they are deemed incoherent. This hierarchy provides a necessary and sufficient condition for certifying coherence.
Practical Coherence Criterion via Block Moment Matrices: This is a computationally efficient alternative to the full SDP hierarchy. It checks for the existence of a positive semidefinite block moment matrix Γ derived from state decompositions. If this matrix is found, it suggests incoherence, and its dual formulation provides a witness inequality to certify coherence.
Coherence-Breaking Channels: A quantum channel is called coherence-breaking if every quantum state sent through it results in output states that admit an incoherent model (i.e., the image set EΛ belongs to C). Identifying these channels helps determine which processes fundamentally destroy quantum superposition and coherence.
Joint Measurability for Qubits: For qubit systems, coherence is linked to whether the set of unbiased dichotomic measurements is jointly measurable. While this can be checked via SDP, it lacks scalability for larger systems, highlighting the need for more efficient methods.

Terminology

Accuracy — 12× in this paper · explained in 1 episode
Entanglement — 8× in this paper · explained in 3 episodes
Benchmarking — 6× in this paper · explained in 3 episodes
Steering — 6× in this paper · explained in 2 episodes
Adjacency Matrix — 5× in this paper · explained in 1 episode
Benchmark — 5× in this paper · explained in 2 episodes
Hilbert Space — 5× in this paper · explained in 2 episodes
Trace — 5× in this paper · explained in 2 episodes
Core — 3× in this paper · explained in 2 episodes
Robustness — 3× in this paper · explained in 5 episodes
Fidelity — 2× in this paper · explained in 2 episodes
Distillation — 1× in this paper · explained in 2 episodes
Multipartite Entanglement — 1× in this paper · explained in 2 episodes
Objective Function — 1× in this paper · explained in 2 episodes
Quantum Physics — 1× in this paper · explained in 10 episodes
Semidefinite Programming (SDP) — 1× in this paper · explained in 3 episodes

Summary

Quantum states that do not commute exhibit coherence, but only when the device preparing them is assumed to be unaffected by classical parameters inaccessible to the experimenter.

How it works

The paper addresses the problem of certifying coherence in quantum devices when they are subject to hidden classical control parameters, which arise in fundamental tests and quantum information protocols. The core challenge is that basis-dependent coherence cannot be certified absolutely without specifying a privileged basis, and if parts of an incoherent system are inaccessible, non-commuting states might lead to false conclusions about coherence. The authors develop complete and practically efficient methods to address this by developing a toolbox for analyzing quantum superposition in the presence of hidden classical control.

The development involves several key steps:

Proving that coherence can be fully characterized through a hierarchy of semidefinite programs (SDPs).
Introducing a practical SDP approach that achieves useful accuracy while remaining computationally efficient even for preparation devices generating many, potentially high-dimensional, quantum states.
Exploiting conceptual connections with the theory of joint measurability for the important special case of qubits to obtain highly accurate coherence characterisation that scales to more than one thousand qubits.
Applying these methods to determine whether quantum channels are able to preserve coherence or are inherently coherence-breaking, leading to the notion of coherence-breaking channels.

Coherence Characterization via Semidefinite Programming (SDP)

The paper establishes a complete characterization of incoherence through a hierarchy of outer SDP relaxations, denoted as the set C. This hierarchy is defined by finding operators that satisfy specific constraints derived from the assumption that the states admit an incoherent model under hidden classical parameters.

The lowest level relaxation, corresponding to m=2, involves bipartite operators of the form:

)& Oxy = Z dλ q(λ)τx,λ ⊗ τy,λ (Equation 8). These are trivially positive semidefinite and satisfy marginal constraints related to the observed states ρx and ρy. The constraints include requiring positivity of the operator (C1), symmetry via a swap operator relation (C3), trace relations (C2), and partial transposition conditions (C5). If this program is infeasible, it implies that E ∈ C / and hence the states are coherent. If feasible, it suggests coherence can be certified using higher relaxation levels. The hierarchy is proven to converge to the set C in the limit of large m [Theorem 1]. This framework provides a necessary and sufficient condition for coherence.

Practical Coherence Criterion via Block Moment Matrices

To address the practical limitations of the full SDP hierarchy, a second method is introduced based on block moment matrices, which is tailored to be practically more useful. If the set E is incoherent (E ∈ C), this method can certify it by checking for the existence of a positive semidefinite block moment matrix Γ (Equation 17).

The key steps for this practical criterion are:

Assuming all states in the decomposition are pure, one defines an ordered list L and constructs a block moment matrix Γ(λ) based on monomials R.
The resulting matrix Γ is then averaged over the classical parameter λ to form Γ (Equation 16).
Coherence is certified if this SDP, defined by constraints (17), is infeasible. The dual formulation of this SDP yields a witness inequality W(E) = tr(Z11) + N X+1 x=2 tr(βxρx−1) ≥ 0. Finding a violation of this witness implies E ∈ C / and certifies the coherence of E.

Coherence-Breaking Channels

The paper defines a quantum channel Λ as coherence-breaking (CB) if its image set EΛ satisfies the incoherence condition, i.e., EΛ ∈ C (Equation 5). This means that if all quantum states are sent through the channel, they admit an incoherent model.

The algorithm for certifying CB channels involves:

Iteratively searching for an optimal set of input states E = Σψx such that the coherence of the output states is maximized.
Using SDP relaxations to find a bound v¯(Λ(E)) ≥ v∗(Λ(E)), where v∗ is the largest visibility above which Ev is coherent (Equation 3).
The search for optimal input states E is performed using an oracle based on the largest violation of the witness W(Λ(E)). This process repeats until convergence, providing an upper bound on the smallest v such that Λ is coherence-breaking.

Qubit Systems and Qubit Channels

For qubit systems (d=2), specialized methods leverage connections to joint measurability. Coherence is equivalent to the set of unbiased dichotomic measurements M = Σρx, 1−ρx being jointly measurable (Equation 20). This feasibility can be decided by an SDP, but it is not scalable.

Improvements for AI systems

As a fastidious researcher, I have thoroughly analyzed this paper, Certifying coherence in quantum devices under classical control. The core contribution is a rigorous mathematical framework—a hierarchy of Semidefinite Programs (SDPs)—to certify whether a set of quantum states is coherent or incoherent, even when the preparation device is subject to hidden classical control parameters.

Here are the specific improvements that can be made to AI systems based on this research, and what those improved systems could achieve:

)1. Improvement in Quantum State Verification for AI Training/Benchmarking

The paper provides a method (SDP hierarchy) to determine if a set of quantum states is incoherent under classical control.

A more robust AI system can be developed to automatically certify the coherence of complex, high-dimensional quantum state ensembles generated by current or future quantum hardware (e.g., noisy variational quantum circuits).
It can distinguish between genuinely coherent states and those that appear coherent only because the preparation device is subject to inaccessible classical parameters (hidden control).

)2. Improvement in Quantum Channel Characterization for Robustness Analysis

The paper introduces coherence-breaking channels and provides a procedure (Algorithm 1) to certify whether an arbitrary quantum channel preserves or breaks coherence, especially for qubit systems.

AI systems can be used to automatically analyze the robustness of quantum communication protocols or quantum memory systems by testing their associated channels against this certification tool.
Specifically, the system can determine if a noisy communication channel is coherence-breaking (i.e., inevitably destroys superposition) or coherence-preserving, providing crucial insights for designing fault-tolerant quantum architectures.

)3. Improvement in Qubit System Certification using Joint Measurability

For qubit systems, the paper connects coherence certification to joint measurability problems and offers scalable methods for certifying coherence up to a thousand qubits.

AI can be deployed as a tool to rapidly assess the coherence properties of large quantum registers (many qubits).
It can automate the search for optimal measurement bases or preparation strategies that maximize the coherence of a given set of states, leveraging the inner/outer approximation techniques described in Section III D.

)4. Improvement in Quantum Machine Learning (QML) Model Validation

The SDP relaxation methods (both complete and practical) offer a way to verify if learned quantum models are physically coherent or merely artifacts of classical control simulations.

AI can be used to validate the physical coherence of quantum states used as inputs or outputs in QML algorithms. If an AI model relies on a superposition that is actually incoherent under classical preparation constraints, this method can flag it as potentially flawed or requiring re-training under more realistic control assumptions.

)5. Improvement in High-Dimensional Quantum System Analysis

The paper details methods for analyzing coherence in high-dimensional systems (e.g., using Equiangular Tight Frames).

AI can be used to analyze the coherence of systems with very large Hilbert space dimensions where analytical solutions are intractable.
It can rapidly compute bounds on critical parameters (like visibility or noise levels) required to maintain coherence, which is vital for benchmarking high-dimensional quantum sensors and simulators.

Abstract

Quantum states that do not commute exhibit coherence, but only when the device preparing them is assumed to be unaffected by classical parameters inaccessible to the experimenter. Accounting for such hidden classical control leads to a stronger notion of coherence. We establish its significance through the task of randomness certification. It is shown that a preparation device is coherent if and only if it can be used to certify randomness in a prepare-and-measure experiment. This operational interpretation motivates the problem of certifying coherence in the presence of hidden classical control. We obtain a complete characterisation of coherence through a hierarchy of semidefinite programs. Then, we introduce a practical semidefinite programming approach that achieves useful accuracy while remaining computationally efficient even for devices preparing many, potentially high-dimensional, quantum states. For the important case of qubits, we further exploit connections with the theory of joint measurability to obtain highly accurate coherence characterisation that scales to more than one thousand qubit states. Our results give operational meaning to quantum coherence under hidden classical control and provide a versatile toolbox for its certification.