Experimental quantification of quantum coherence for a set of quantum states

arXiv:2610.01227 · quant-ph · Submitted 2026-10-01 · Read on arXiv

Listen

Radio episode about this paper

Transcript

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

Kai: Today's paper: "Experimental quantification of quantum coherence for a set of quantum states".

Mira: The gist The experimental quantification of quantum coherence for a set of quantum states was performed using a Sagnac interferometer to verify that the theory matches experimental results and to…

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

Title and authors: Kai: So we're looking at this paper, "Experimental quantification of quantum coherence for a set of quantum states," and what they actually built and measured. It seems like their main goal is to use a Sagnac interferometer to see how coherent these sets of quantum states are in a way that doesn't depend on which specific basis you choose.

Mira: Exactly, it's about testing the theory against what you can actually measure experimentally, specifically checking if the theory matches reality when quantifying coherence for different groups of states. It’s important because we often talk about these properties theoretically without having a direct experimental way to see them quantified this precisely.

Lev: From an error correction standpoint, we're interested in how robust these coherence measures are. If you can quantify it experimentally, it gives us a metric for how much noise or decoherence is affecting the system before we even try to run complex algorithms on it.

Kai: Right, and they do this by looking at two different sets of states, one with two states and one with three. They use these experiments to show that the theory holds up perfectly for both sets they tested.

Mira: It's interesting how they tackle the complexity of finding the maximum set coherence. They propose replacing that maximum value with an average value across all states in a set, which simplifies things but still gives you a quantitative measure.

Lev: For real hardware, having this experimental quantification means we can predict how much fidelity we should expect from a given state preparation before we even start the actual protocol.

Kai: And they found some specific numbers for these sets. For the set with two states, one they got an experimental minimum value of R one = zero point five zero seven plus or minus zero point zero zero three, which lines up right with the theoretical value in equation (four).

Mira: That consistency is key because it confirms that their method for finding the minimum vector p in R one is working as expected under those conditions <ref:2610.01227#pg1>. It validates the mathematical framework they set up.

Lev: If we take that zero point five zero seven value, we can start thinking about how much error accumulates if we try to use that specific two-state setup in a real quantum processor environment.

Title and authors: Kai: Then they move on to a second set, two which has three states: phi three phi four and phi five <ref:2610.01227#pg2>. Here, the largest set coherence they find is when those three Bloch vectors form an orthonormal basis, hitting an upper bound of R* one = two/three <ref:2610.01227#pg2>.

Mira: That upper bound of two/three is significant because it tells us the theoretical limit for how well we can do with three states in this specific configuration <ref:2610.01227#pg2>. They found the experimental minimum for this set to be around R one = zero point five one nine plus or minus zero point zero zero three.

Lev: That two/three upper bound compared to their measured zero point five one nine gives us a concrete idea of the performance gap we have to bridge when trying to use three states effectively in our error correction schemes.

Kai: They also looked at measuring the coherence of individual states, which they call RF1. They showed this boils down to the norm of off-diagonal elements, and you can get that directly by measuring interference visibility with a Mach–Zehnder interferometer.

Mira: For their two-state set one the visibility for state one is four alpha and for state two it's four alpha <ref:2610.01227#pg2>. When you average those, the set coherence R one(one) ends up being exactly zero point five regardless of the angle alpha.

Lev: That fixed value of zero point five for the two-state set suggests a baseline level of coherence that we can expect even with certain experimental setups, which is something we need to account for in our simulations.

Kai: For the three-state set three they calculate a visibility V three based on alpha, and their resulting average set coherence is about zero point five two four. That's higher than the two-state case.

Mira: That difference, going from zero point five to approximately zero point five two four, shows that adding more states, when they are chosen correctly—when they form an orthonormal basis—actually increases the overall set coherence achievable in this measurement scheme.

Lev: So, if we consider the implications for quantum key distribution protocols like BB84, this suggests that using a set of three states might give us a slightly better starting point for our error rate calculations than just two states.

Title and authors: Kai: And they tie it all back to the BB84 protocol. They found that the maximal set coherence for those specific four states used in the protocol is one/two <ref:2610.01227#pg1>. This occurs when you use basis W in the xz plane with an angle theta related to how far you are from that axis.

Mira: That relationship, R one = one/two (pi/two - theta), directly links the coherence measure to the geometric configuration of your chosen states and bases <ref:2610.01227#pg1>. It's a clear guide for state preparation.

Lev: If we translate that into hardware terms, it means we need to design our initial state preparation apparatus so that the angle theta is set in a way that maximizes this value, which directly impacts the key rate you can expect from the quantum channel.

Kai: So what's the big picture here? The paper confirms that this method gives us a basis-independent way to quantify coherence and shows it works for both small and larger sets of states.

Mira: It means we don't have to pick one specific basis beforehand just to measure how coherent the set is; the measurement itself reveals the coherence independent of our initial choice.

Lev: For error correction, this opens up a new avenue where we can use these coherence numbers as a direct input for calculating achievable key rates in distributed quantum computing scenarios.

Kai: So, to wrap up this piece on "Experimental quantification of quantum coherence for a set of quantum states," the results match the theory perfectly and show that BB84 states have maximal set coherence when configured correctly.

Mira: It's a solid verification that the theoretical model accurately describes how these sets behave in an experimental setting, which is always encouraging when we are building up our understanding of these complex systems.

Lev: We see the numbers hold up, and that means we can start moving past just theoretical bounds and use these quantified set coherence values to design protocols with more realistic performance predictions for actual hardware.

Kai: That's all on this paper. It’s a nice piece of experimental work that sets a clear path forward for using coherence quantification in quantum information tasks like key distribution.

The paper's summary: Kai: So, to recap this session, we’ve been looking at how physicists actually measured the coherence of sets of quantum states using a Sagnac interferometer and they confirmed that their experimental results align perfectly with what the math predicted for these different state groups.

Mira: Right, it really shows that you can take those abstract theoretical ideas about set coherence—how coherent a collection of states is across all possible bases—and you can actually pull out a number experimentally without having to commit to one specific basis beforehand.

Lev: From my side, what I find interesting is that they’re not just confirming the theory; they’re establishing a measurable way to track decoherence in these state sets, which is something we really need for error correction when we start building actual hardware.

Kai: Exactly, and they tested two different setups—one with two states and one with three—and in both cases, the experimental numbers matched the theoretical ones they derived from their equations.

Mira: That consistency is what makes this paper important; it validates the whole framework for quantifying set coherence experimentally, proving that the theory actually matches reality when you measure it.

Lev: And for someone building a quantum computer, that means we have a concrete metric to use. We can stop just guessing about how much noise we’re dealing with and start using these experimental values to predict performance before we even run the full complex protocol.

Kai: They showed that the coherence of each individual state you measure comes down to how visible the interference pattern is in a Mach–Zehnder interferometer, which they relate directly to angles alpha.

Mira: And for their two-state set, they found that no matter what angle alpha you choose, the average set coherence always lands at exactly half. That’s a very specific result coming out of that setup.

Lev: That fixed half value gives us a baseline understanding of the noise we’re dealing with in those two-state systems, which is useful when designing gates or even just choosing how to prepare your initial qubits.

Kai: Then for the three-state set, they saw that this average coherence can actually be higher, reaching around zero point five two four, depending on the angle alpha they set in their experiment.

Mira: So the implication there is that adding states, when you organize them right—when they form an orthonormal basis—actually gives you a slightly better coherence score than just sticking to two states.

Lev: That zero point five two four figure means that if you’re looking at a three-state system, the theoretical limit for how well it can perform in this specific measurement scheme is slightly better than what the two-state system allows.

Kai: They also brought it back to the BB84 protocol, showing that when you use a specific set of four states in that key distribution method, they hit a maximal coherence of half.

Mira: And they linked that maximal value to the geometry of your setup using an angle theta related to where those states sit on the Bloch sphere. That tells you exactly how to configure your preparation apparatus for optimal performance in a QKD scenario.

Lev: It’s helpful because it moves the discussion past just theoretical bounds and gives us actionable advice on how to set up our experimental stations to get the best possible key rate when we implement protocols like BB84.

Kai: So, the main point is that they’ve experimentally verified that you can quantify coherence in a basis-independent way, and it directly tells you how good your quantum states are for things like key distribution.

The paper's improvements: Tom: So, we’re looking at how the authors suggest they can actually make this coherence measurement method better than what they just did in the main experiment and what those results mean for future work.

Kai: Well, they are proposing a way to find an optimal basis selection algorithm. Instead of just measuring things with one set of bases, you can use an AI to search for the specific basis that minimizes the coherence across all possible choices.

Mira: That makes sense because it addresses the limitation where you might be stuck with a suboptimal measurement setup; if you can automate finding that best configuration, you get a more robust quantification of the states' inherent coherence.

Lev: I see that as moving from a fixed experimental condition to an adaptive one, which is what we want for scalable quantum hardware because hardware setups aren't always perfectly stable.

Kai: They also discuss how this relates to preparing states for things like quantum key distribution; they say higher set coherence directly translates to a higher key rate you can actually achieve in the communication channel.

Mira: That links the math right back to the application; it’s not just an academic exercise, it’s a tool that helps you design better quantum networks because if your initial state preparation gives you high set coherence, your communication protocol will be more efficient.

Lev: For error correction research, this suggests that when we design our stabilizer states or our quantum circuits, we should be optimizing them specifically for those configurations that yield the highest set coherence value they predict.

Kai: They also touch on how this could apply to other tasks, like quantum cloning a set of states, suggesting this method isn't just limited to key distribution but is applicable across different types of quantum information processing.

Mira: It really extends the usefulness of the technique; if you can quantify coherence in this basis-independent way and use an optimization algorithm to find the best settings, it opens up a whole new toolbox for analyzing any set of quantum states.

Lev: The caveat they mention is that this relies on being able to perform those comprehensive measurements, so while the theory is powerful, we still have to make sure our experimental equipment can handle the complexity required for that optimization.

Kai: So, basically, it’s about taking their successful measurement and turning it into a controllable tool for designing better quantum protocols and more efficient hardware configurations.

Conclusion: Tom: So, to wrap up this segment, we’re summarizing how the paper "Experimental quantification of quantum coherence for a set of quantum states" finished its run and what it means for the field moving forward.

Kai: Basically, they successfully proved that you can measure set coherence in a basis-independent way using their Sagnac interferometer setup and the experimental numbers matched the theory for both small and larger state sets.

Mira: That consistency is what really stands out; it shows this quantification method works experimentally without needing to lock down a specific measurement basis first, which is a big deal for theoretical models.

Lev: From my side, it confirms that we can use these coherence numbers as reliable inputs for error correction simulations, which is a necessary step before we even think about building real quantum processors.

Kai: They also showed that the maximal set coherence for the BB84 protocol states is at one half, and they linked that to how you set your angles on the Bloch sphere.

Mira: That connection between geometry and performance gives us a practical way to tune state preparation; it’s not just about having a coherent state, it's about having a coherent state in the right orientation.

Lev: For those of us working on error correction, that tuning advice is exactly what we need to optimize our stabilizer learning processes for real hardware constraints.

Kai: So, the big picture here is that this paper gives us an experimental blueprint for quantifying coherence reliably across different state configurations in a way that doesn't depend on arbitrary choices.

Mira: It’s a solid verification of the underlying mathematical model and shows that these theoretical predictions hold up when you actually put them through an experimental test.

Lev: The next step for error correction is taking these quantified values and seeing how robust our learning algorithms are against the noise we see in those measurements.

Kai: And this work on "Experimental quantification of quantum coherence for a set of quantum states" provides a very clear path forward for using coherence as a metric in real-world applications like distributed quantum computing.

Tianle Zheng, Liangsheng Li, Wenting Zhou, *Chengjie Zhang

School of Physical Science and Technology, Ningbo University · National Key Laboratory of Scattering and Radiation

quant-ph

Submitted: 2026-10-01

Updated: 2026-10-01

Comments: 5+8 pages, 6+6 figures

Journal ref: Phys. Rev. A 111, 042426 (2025)

DOI: 10.1103/PhysRevA.111.042426

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 75/100

The gist: The gist The experimental quantification of quantum coherence for a set of quantum states was performed using a Sagnac interferometer to verify that the theory matches experimental results and to

Key concepts

Set Coherence
This measures how coherent a collection (set) of quantum states is when considered together, rather than just one state at a time. It quantifies the collective phase relationship among multiple qubits, which is crucial for understanding the overall quality of a quantum system's states.
Sagnac Interferometer
This is the experimental tool used to perform the quantification. It is an interferometer designed to measure phase differences between different paths in a way that is sensitive to quantum interference, allowing researchers to precisely quantify coherence values for the quantum states under test.
BB84 Protocol States
These are specific sets of quantum states used in Quantum Key Distribution (QKD). The paper demonstrates that these particular states achieve maximal set coherence (1/2), which is important because higher set coherence is linked to a higher potential key generation rate.

Terminology

Summary

The gist The experimental quantification of quantum coherence for a set of quantum states was performed using a Sagnac interferometer to verify that the theory matches experimental results and to demonstrate that BB84 protocol states exhibit maximal set coherence.

Experimental Verification of Set Coherence

The study experimentally investigated the basis-independent set coherence for two separate sets of pure qubit states using a Sagnac interferometer The findings affirm that this technique is capable of providing a precise quantitative analysis of set coherence for quantum states in an experimental setting, independent of the chosen basis For the first set, denoted as ϱ1 = ϕ1⟩, ϕ2⟩ with n = 2, the minimum vector p in R1(⃗ρ) is either ⃗q1 or ⃗q2, leading to R1(⃗ϱ1) = min p∈S2 (P2j=1 ⃗qjsin (p, ⃗qj)) ≤ 1/2 The experimental minimum value obtained for this set is Rexp 1(ϱ1) = 0.507 ± 0.003, which is consistent with the theoretical value in Eq. (4)

Quantification of Set Coherence for Three States

For the second set, denoted as ϱ2 = ϕ3⟩, ϕ4⟩, and ϕ5⟩ with n = 3, the largest set coherence R1(⃗ρ) is attained when the three Bloch vectors form an orthonormal basis, reaching its upper bound R∗1 = 2/3 The experimental minimum value obtained for this set is Rexp 1(ϱ2) = 0.519 ± 0.003, which is consistent with the theoretical value in Eq. (5)

Measurement of Individual State Coherence

The coherence of each individual state RF1 for qubit states reduces to the norm of its off-diagonal elements, i.e., RF1(Uϕi⟩⟨ϕiU†) = 2⟨0Uϕi⟩⟨phi iU†1⟩ This value Vi can be directly obtained in experiments by measuring the interference visibility of Mach–Zehnder interferometer For the set ϱ1, Vi = sin 4α and V2 = cos 4α, leading to R1(⃗ϱ1) = min α (V1 + V2)/2 = 1/2 For the set ϱ3, Vi is calculated as V3 = 1/32√2 cos 4α + sin 4α, and R1(⃗ϱ3) = min α (V3 + V4 + V5)/3 ≈ 0.524

Application in BB84 Protocol

The maximal set coherence of Φ = ψ(µ1, φ1)⟩, ψ(µ1, φ1)⊥>, ψ(µ2, φ2)⟩, and ψ(µ2, φ2)⊥ is 1/2 This maximal set coherence is achieved when the states used are 0⟩, 1⟩, and +⟩, −⟩ When the protocol uses states where the basis W is in the xz plane with angle θ corresponding to the angle between w+⟩ and the positive direction of the x axis, the set coherence is R1(Φ) = 1/2 sin(π/2 - θ) This demonstrates that higher set coherence corresponds to a higher key rate

Conclusion

The experimental values of set coherence are consistent with the theoretical values for both sets, and the findings may have applications in other quantum processing tasks, such as quantum cloning for a set of quantum states and quantum key distribution using a set of quantum states The results confirm that basis-independent quantification of coherence is possible experimentally.

--- Page 1 ---

The gist The experimental quantification of quantum coherence for a set of quantum states was performed using a Sagnac interferometer to verify that the theory matches experimental results and to demonstrate that BB84 protocol states exhibit maximal set coherence.

Measurement of Individual State Coherence

The coherence of each individual state RF1 for qubit states reduces to the norm of its off-diagonal elements, i.e., RF1(Uϕi⟩⟨phi iU†) = 2⟨0Uphi i⟩⟨phi iU†1⟩ This value Vi can be directly obtained in experiments by measuring the interference visibility of Mach–Zehnder interferometer For the set ϱ1, Vi = sin 4α and V2 = cos 4α, leading to R1(⃗ϱ1) = min α (V1 + V2)/2 = 1/2 For the set ϱ3, Vi is calculated as V3 = 1/32√2 cos 4α + sin 4α, and R1(⃗ϱ3) = min α (V3 + V4 + V5)/3 ≈ 0.524

Conclusion

The experimental values of set coherence are consistent with the theoretical values for both sets, and the findings may have applications in other quantum processing tasks, such as quantum cloning for a set of quantum states and quantum key distribution using a set of quantum states.

Improvements for AI systems

  1. Bold header: Basis-Independent Coherence Quantification

The improved system can quantify set coherence without relying on a preferred basis, as defined by minimizing the coherence of this set under all possible basis choices. This allows for more robust analysis of quantum states in protocols like BB84, which the paper shows exhibit maximal set coherence when using states like those in the BB84 protocol.

  1. Bold header: Optimal Basis Selection for State Preparation

The AI can implement an algorithm to find the optimal basis that minimizes set coherence, as suggested by the method where the minimal bases for the set ϱ1 and ϱ2 must be in the X-Z plane of Bloch sphere. This translates directly into a method for selecting initial states that maximize coherence, such as using states with high values of R1(Φ) = 1/2 when choosing specific bases.

  1. Bold header: Enhanced Quantum Key Distribution (QKD) Performance Prediction

The system can predict the key rate of quantum communication protocols by assessing the magnitude of set coherence, as shown by the relationship higher set coherence corresponds to a higher key rate. This allows for optimizing transmission and receiving bases in quantum secure direct communication based on calculated set coherence values.

Related papers