Certifying coherence in quantum devices under classical control

This paper presents a comprehensive and computationally efficient framework, utilizing semidefinite programming and connections to joint measurability, to certify quantum coherence in devices subject to hidden classical control, enabling the analysis of high-dimensional states and the characterization of coherence-preserving channels.

The Gist

Imagine you are a detective trying to prove that a magician is using real magic (quantum superposition) rather than just clever tricks (classical physics). Usually, to prove magic, you check if the magician's cards don't follow normal rules. But what if the magician has a hidden assistant?

This paper tackles a specific problem: How do we prove a quantum device is truly "magical" (coherent) when it might be secretly controlled by a hidden classical variable?

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Hidden Puppeteer"

In the quantum world, "coherence" is like a state of being in two places at once (superposition). Usually, we say a system is coherent if its parts don't line up perfectly (they don't "commute").

However, imagine a machine that prepares these quantum states. What if this machine is secretly being guided by a hidden switch (a classical variable, let's call it $\lambda$) that the experimenter can't see?

• The Trick: The hidden switch tells the machine to prepare a "boring" state when the switch is set to "1," and a different "boring" state when it's set to "2."
• The Illusion: When you average out the results (because you don't know which switch setting was used), the final mix looks like a complex, "magical" quantum state that shouldn't exist.
• The Danger: You might think you've discovered a new quantum phenomenon, but it's actually just a classical trick. The paper asks: How do we prove the machine is truly quantum and not just being puppeteered by a hidden classical variable?

2. The Solution: A "Mathematical Sieve"

The authors built a set of mathematical tools (called Semidefinite Programs or SDPs) to act as a sieve. These tools test whether a set of states could have been faked by a hidden classical switch.

They developed two main tools:

A. The "Perfect but Slow" Sieve (The Hierarchy)

• How it works: This is a step-by-step ladder of tests. The first step is a quick check. If it fails, you know it's fake. If it passes, you move to a harder, more detailed step.
• The Promise: If you keep climbing this ladder forever, you will eventually get a 100% perfect answer. It proves that coherence can be completely defined by math.
• The Catch: It's like trying to count every grain of sand on a beach to prove it's a beach. It's accurate, but it takes too long for real-world experiments with many states.

B. The "Fast and Smart" Sieve (The Practical Method)

• How it works: This is a shortcut. It doesn't climb the whole ladder; it just takes a very smart snapshot.
• The Benefit: It is incredibly fast. The authors showed it can handle hundreds of quantum states (even in high dimensions) in minutes on a standard computer.
• The Result: Even though it's a shortcut, it's surprisingly accurate. It can tell you with high confidence whether a device is truly coherent or just faking it.

3. The Special Case: The "Qubit" Super-Tool

For the most common type of quantum bit (the qubit, which is like a coin that can be heads, tails, or both), the authors found a clever shortcut.

• They connected the problem of "coherence" to a different known problem called "joint measurability" (asking if you can measure two things at once without disturbing them).
• By using this connection, they created a tool that can certify coherence for over 1,000 qubits at once. It's like having a super-fast scanner that can check a whole library of books in seconds.

4. Testing the "Pipe" (Quantum Channels)

Finally, they applied these tools to quantum channels (the "pipes" that send quantum information from one place to another).

• The Question: Does this pipe preserve the magic, or does it destroy it?
• The New Concept: They defined "Coherence-Breaking Channels." These are pipes that are so noisy or destructive that no matter what you send through them, the output will always look like a boring, classical mixture. It's like a pipe that turns gold into lead no matter what you put in.
• The Test: Their tools can now tell you exactly when a pipe is safe (preserves coherence) and when it is broken (destroys coherence).

Summary

The authors have built a toolbox for quantum scientists.

1. Theoretical Proof: They proved that you can mathematically define "true quantumness" even with hidden classical tricks.
2. Practical Tool: They created a fast, efficient method to test real devices with many states.
3. Scalability: For simple qubits, they made a tool that scales to massive numbers (1,000+).
4. Channel Testing: They gave a way to test if a communication channel destroys quantum magic or keeps it alive.

In short, they gave us the magnifying glass needed to spot real quantum magic, even when a hidden classical puppeteer is trying to pull the strings.

Technical Summary

Technical Summary: Certifying Coherence in Quantum Devices under Classical Control

Problem Statement
Quantum coherence, fundamentally linked to the non-commutativity of states, is typically defined relative to a specific basis. However, a basis-independent certification of coherence requires demonstrating that a set of states cannot be simultaneously diagonalized. A significant conceptual limitation arises when the preparation device is subject to "hidden classical control"—inaccessible classical parameters (hidden variables, $\lambda$) that stochastically influence the device's operation. In such scenarios, a device may generate a set of non-commuting states that appear coherent to an observer, yet admit a classical model where, conditioned on $\lambda$, the underlying states are commuting (incoherent). Existing methods to certify genuine coherence in the presence of such hidden variables are limited to simple cases. The central problem addressed is how to rigorously certify that a quantum preparation device is genuinely coherent when it may be influenced by inaccessible classical parameters.

Methodology
The authors develop a suite of methods based on Semidefinite Programming (SDP) to characterize the set $\mathcal{C}$ of state sets that admit an incoherent model (i.e., can be simulated by a classical mixture of commuting states).

1. Complete Hierarchy of SDP Relaxations:
   The paper first establishes a necessary and sufficient condition for coherence by constructing a converging hierarchy of outer SDP relaxations ($\mathcal{C}_2 \supset \mathcal{C}_3 \supset \dots \supset \mathcal{C}_\infty = \mathcal{C}$).

   • Level $m=2$: This involves defining bipartite operators $O_{xy} = \int d\lambda q(\lambda) \tau_{x,\lambda} \otimes \tau_{y,\lambda}$, where $\tau_{x,\lambda}$ are commuting states. Constraints include positive semidefiniteness, positive partial transpose (PPT), and marginal constraints recovering the observed states $\rho_x$.
   • Convergence: Higher levels ($m > 2$) utilize operators acting on $m$ systems. The hierarchy converges to the exact set $\mathcal{C}$ as $m \to \infty$, proven via a correspondence to de Finetti theorems for POVMs. While complete, this approach scales poorly ($N^2$ matrices of size $d^2$), making it impractical for large systems.

2. Practical SDP Criterion (Block Moment Matrices):
   To address scalability, the authors introduce a single, efficient SDP condition based on block moment matrices.

   • Formulation: By assuming the underlying states $\tau_{x,\lambda}$ are pure and commuting, they construct a block moment matrix $\Gamma$ containing the observed states $\rho_x$ and optimization variables $M_{ij}$ representing products of commuting states.
   • Constraints: The matrix $\Gamma$ must be positive semidefinite, with diagonal blocks corresponding to $\rho_i$ and off-diagonal blocks $M_{ij}$ satisfying $\rho_i \succeq M_{ij} \succeq 0$ and Hermiticity.
   • Efficiency: This method scales linearly with the number of states $N$ and dimension $d$ (matrix size $d(N+1)$), making it computationally feasible for large $N$ and high $d$.
   • Witnesses: The dual of this SDP yields coherence witnesses (inequalities) that are satisfied by all incoherent sets but violated by coherent ones.

3. Specialized Method for Qubits (Joint Measurability):
   For qubit systems ($d=2$), the authors exploit the equivalence between coherence certification and the joint measurability of associated dichotomic measurements.

   • Approximation: They utilize convex programming techniques to approximate the set of jointly measurable observables using inner and outer polyhedral approximations of the Bloch sphere.
   • Scalability: By using a polyhedron with 220 vertices (approximating the Bloch sphere with a shrinking factor $r \approx 0.9934$), they achieve high-accuracy certification for sets exceeding 1,000 qubit states.

4. Coherence over Channels:
   The methods are extended to quantum channels $\Lambda$. A channel is defined as Coherence-Breaking (CB) if its image (the set of all output states) admits an incoherent model.

   • Certification: The authors propose an iterative algorithm to find the set of input states that maximizes the coherence of the output, thereby tightening the bounds on whether a channel is CB.
   • Qubit Channels: For qubits, the polyhedral approximation method is used to determine if the image of the channel admits an incoherent model, effectively distinguishing between coherence-preserving and coherence-breaking regimes.

Key Results

• Theoretical Characterization: The paper proves that coherence under hidden classical control can be fully characterized by a hierarchy of SDPs.
• Scalability: The practical SDP criterion (Eq. 17) successfully certifies coherence for:
  • $N=100$ random qubit states in under 8 minutes.
  • $N=70$ random qutrit states in under 11 minutes.
  • High-dimensional systems ($d=150$) with just two states, certifying coherence for visibility $v \gtrsim 0.9246$.
• Accuracy Comparison: Benchmarks against the complete hierarchy show the practical SDP method is not only significantly faster but also provides tighter (more accurate) bounds on the critical noise threshold for coherence certification.
• Qubit Scaling: The joint measurability approach enables accurate coherence certification for over 1,000 qubit states, a regime inaccessible to the general SDP hierarchy.
• Channel Analysis:
  • For the depolarization channel in $d=20$, the iterative algorithm converges to known theoretical bounds on the coherence-breaking threshold.
  • For composed channels (depolarization + dephasing; depolarization + amplitude damping), the authors map the regions of parameters where the channels are coherence-breaking versus coherence-preserving.
  • For qubit depolarization channels, the method certifies the CB threshold at $v \lesssim 0.4999$ and the preservation threshold at $v \gtrsim 0.5029$, nearly matching the exact theoretical value of $0.5$.

Significance and Claims
The authors claim to provide a "powerful toolbox" for analyzing quantum superposition in realistic scenarios where classical control is imperfect or hidden.

• Foundational Impact: The work resolves the conceptual limitation that non-commuting states might be classically simulable, providing rigorous criteria to distinguish genuine quantum coherence from classical mixtures.
• Practical Utility: The developed methods are designed for "state-of-the-art quantum preparation devices," offering computationally efficient tools that remain effective even for high-dimensional systems and large numbers of states.
• New Concepts: The introduction of coherence-breaking channels parallels the concept of entanglement-breaking channels, offering a new framework to classify quantum channels based on their ability to preserve or destroy superposition in the presence of hidden classical variables.
• Benchmarking: The tools are presented as essential for benchmarking the noise and loss advantages of high-dimensional superposition in experimental settings, particularly where analytical solutions are unavailable.

The paper does not propose new experimental setups but rather provides the theoretical and computational framework necessary to interpret experimental data where classical control assumptions are relaxed.