Complete Relational Description of Spin in a Quantum Background

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

Kai: We're looking at the paper: "Complete Relational Description of Spin in a Quantum Background".

Mira: Here's the gist of the paper. The gist The standard description of a spin in quantum mechanics presupposes externally fixed directions—a classical background, and this work shows that when the quantum reference system is augmented to two large spins, the standard quantum mechanical description of a spin is recovered in the limit of large quantum numbers for the reference system.

Kai: How it works The paper investigates how a single fundamental spin can be fully described in relation to other quantum mechanical systems, moving beyond descriptions that presuppose fixed classical frames The standard quantization of a spin makes reference to a fixed external classical frame, which defines directions along which the spin may be measured If all physical systems obey quantum mechanics, including spacetime, the general description of a quantum system—of a spin in particular—should be in relation to other quantum systems The description of a system relative to a quantum frame of reference will not depend only on properties of the system of interest, as the algebra of relative observables will depend on the frame The authors address two difficulties in achieving this relational description: first, it is not enough to relationally encode that the joint state is a (mostly) aligned or a (mostly) anti-aligned state, and second, it is necessary to encode complementarity through a composite reference system with two angular momenta that do not commute The calculation builds on an idea by Poulin from 2007 where they considered the joint state of a spin-1/2 system with a second, larger spin, removing the fiducial information of the coordinate axes by averaging the density matrix over rotations This yields a mixed state for the spin, no different than the state of a classical probabilistic bit Augmenting the quantum reference frame with a second large spin and following a similar procedure yields a relational state that includes information on the coherence of the spin-1/2 system The encoding becomes approximately exact and the state of the spin-1/2 becomes approximately pure when quantum numbers for the reference are taken large Spin in Relation to One Reference The initial strategy discussed by Poulin involves starting from a joint state of a spin-1/2 system and a reference spin, removing dependence on classical externally defined directions by incoherently group averaging the joint state over rotations The resulting state after this procedure is approximately corresponding to a mixture, see Section II When the quantum number of the reference spin G is large, the state approaches a probabilistic mixture of a classical spin with probability α2 of being in the spin-up state and probability β2 of being in the spin-down state The apparent loss of coherence is not due to averaging incoherently over total system rotations, but rather because only the magnitude of the spin amplitudes can be encoded, not their relative phase Spin in Relation to Two Reference Spins Augmenting the reference system with a second spin system H leads to a relational state that approaches an exact encoding of the qubit S as the quantum numbers of the reference grow, see Section III The calculation involves changing basis to the eigenbasis of angular momentum jSG and magnetic moment mSG, and then changing basis again to the eigenbasis that has as eigenvalues the angular momentum jSGH of the joint system of S, G, and H The incoherent group average applied to the density matrix yields a state where the total angular momentum J is in a probabilistic mixture, but this relational description has encoded information on the coherence of S Large Reference Spins Limit When taking G = H ≫ 1, the matrix elements corresponding to terms like S(4), S(5), and S(6) become negligible due to pre-factors falling as 1/√G, 1/√G, and 1/G respectively The state E˜C (ρ SGH) becomes approximately E˜C (ρ SGH) = ρjSGH ⊗ ψjSG⟩⟨ψjSG where ψjSG⟩ = α G + 1/2⟩ + β G − 1/2⟩ This state is the probabilistic mixture of a classical spin with probability α2 of being in the spin-up state and probability β2 of being in the spin-down state Coherent vs Incoherent Average The incoherent group averaging destroys coherence between the total angular momentum sectors of the entire system GHS, treating the total angular momentum as classical The coherent group average projects a vector ψ⟩ on its invariant component, which can be interpreted as treating the total angular momentum of the system as fiducial, a gauge effect that must be removed for a completely background independent relational description For a completely background independent relational description, the coherent average should be used Relation to Reference Frame Quantization The construction is compared to reference frame quantization, where the encoding maps are defined by tracing over the reference system's coherent states The general moral of the calculation carries through to target systems of many degrees of freedom: a reference quantum system with two non-commuting operators satisfying the algebra of angular momenta increasingly resolves the quantum state of a target system that transforms under SU(2) as the quantum numbers of the reference grow larger, and does so exactly in the limit The result is that a pair of gyroscopes G, H can fully resolve a single spin-1/2 state in the classical limit G, H → ∞ A quantum reference system with two operators that do not commute and satisfy the algebra of angular momenta will yield an exact encoding of the target system in the limit of large quantum numbers for the reference The incoherent group averaging does not result in loss of coherence of the target system—here the spin-1/2 It is the coherence between sectors of different total angular momentum that is lost; the coherence between the angular momenta of subsystems is encoded For a completely background independent relational description, the coherent average should be used The result for the coherent average procedure in the limit of large quantum numbers for the reference can be extracted from the relational state resulting from the incoherent average In this limit, only one difference between the two procedures is that the coherent average forces a zero total angular momentum while the incoherent average fixes it to the classically expected value The general conclusion of our case study for a spin-1/2 target system, carries through for arbitrarily large target systems that transform under SU(2): a quantum reference system with two operators that do not commute and satisfy the algebra of angular momenta, will yield an exact encoding of the target system in the limit of large quantum numbers for the reference The spin state also approaches the aligned/anti-aligned pure state (22), defined relationally with respect to the jSG = G±1/2 eigenstates The final result is that a pair of gyroscopes G, H can fully resolve a single spin-1/2 state in the classical limit G, H → ∞ Page 9 --- The gist The standard description of a spin in quantum mechanics presupposes externally fixed directions—a classical background, and this work shows that when the quantum reference system is augmented to two large spins, the standard quantum mechanical description of a spin is recovered in the limit of large quantum numbers for the reference system.

Title and authors: Kai: So we're starting by looking at the title and authors of this paper, "Complete Relational Description of Spin in a Quantum Background." It’s Hannah Troger, Ofek Bengyat, Thomas D. Galley, and Marios Christodoulou from the Institute for Quantum Optics and Quantum Information in Vienna.

Mira: They are clearly tackling a fundamental issue here: moving the description away from presupposing fixed external directions for a spin. The title itself signals that they are aiming for a fully relational description of spin within quantum mechanics.

Lev: For hardware, that means we're thinking about how to build systems where the spin state is defined by its relationships with other parts of the quantum system, not by an external magnetic field or coordinate system.

Kai: They suggest that when you augment your reference system with two large spins and take the limit of large quantum numbers for those reference systems, you can recover the standard quantum mechanical description of a spin.

Mira: That's a very bold claim because it implies that this relational description is not just an approximation, but it can reproduce what we already know in the classical limit for the reference frame.

Lev: If you’re designing an experiment, knowing that large quantum numbers give you back to the standard model when you take that limit is useful for understanding how much fidelity we need to push those reference systems.

Kai: The core idea is moving away from fixed frames and into a framework where the description depends on other quantum systems.

Mira: It really sets up a challenge for theorists because it requires encoding complementarity through two non-commuting reference angular momenta, which isn't trivial stuff.

Lev: I'm thinking about the physical realization of that composite reference system—how you actually get those two large spins to interact in the way they need to be for this limit to work.

Kai: That’s a key practical hurdle we have as experimentalists, figuring out how to realize that interaction on our superconducting qubits.

The paper's summary: Mira: So the paper summarizes how they go about this, and it starts by addressing the standard description which relies on classical background directions. They say you can remove that dependence by incoherently group averaging the joint state over rotations.

Kai: They then show that when you take the limit of large angular momentum for your reference spin, say G, the resulting state approaches a probabilistic mixture of a classical spin with probability alpha squared of being in the spin-up state and probability beta squared of being in the spin-down state.

Mira: That's where I see it. The paper is showing that this averaging over rotations doesn't just lead to a simple classical bit, but it actually encodes information on the coherence of your spin-one/two system <ref:2606.15873#pg1>.

Lev: So that means we aren't losing all quantum information just because we averaged incoherently; the structure of the state is still there, and it’s just encoded differently.

Kai: The key distinction here is that the incoherent averaging destroys coherence between different total angular momentum sectors of the entire system GHS, treating it as a classical thing.

Mira: But if you use a coherent group average instead, it projects onto an invariant component which can be interpreted as treating the total angular momentum of the whole system as fiducial.

Lev: That suggests that we need to choose our averaging method carefully to get what we actually want for our physical description of the system.

Kai: The authors then show that augmenting the reference frame with a second spin H and following a similar procedure yields a relational state that includes information on the coherence of the spin-one/two system <ref:2606.15873#pg1>.

Mira: And when you take G equals H greater than one, this relational state approaches an exact encoding of your qubit S as the quantum numbers of the reference grow.

Lev: So they are showing that this entanglement between systems can be used to keep a pure state for the spin-one/two system as long as those reference numbers are large <ref:2606.15873#pg1>. That's important for maintaining coherence during computation.

Kai: The result is that a pair of gyroscopes G and H can fully resolve a single spin-one/two state in the classical limit where G and H both go to infinity <ref:2606.15873#pg1>.

The paper's improvements: Mira: So the improvements they suggest are pretty clear. They say you can switch from incoherent averaging to coherent averaging for a background independent description because it should be used instead, as that's what we need for a completely background independent view.

Lev: From an error correction point, this is significant because it means we can use this relational state even when dealing with the limitations of the incoherent approach.

Kai: They also show that the incoherent group averaging does not result in loss of coherence for the spin-one/two system, which is a key result here <ref:2606.15873#pg1>.

Mira: So while it doesn't lose coherence between sectors of different total angular momentum, it encodes information on the coherence between subsystem angular momenta.

Lev: That means we can use this encoded information to potentially reconstruct the state if we know how to map it back properly.

Kai: The paper also points out that a pair of gyroscopes G and H can fully resolve a single spin-one/two state in the classical limit where G and H both go to infinity <ref:2606.15873#pg1>. That's what they call exact encoding in the classical limit for the reference.

Mira: And we can extend this general moral to target systems with many degrees of freedom, meaning a reference quantum system with two non-commuting operators satisfying the algebra of angular momenta will yield an exact encoding for qubits and qudits in the limit.

Lev: So if we think about building this, it tells us that you don't need infinite spins to get the full benefit; you just need large enough ones.

Kai: The practical implication is that a pair of gyroscopes G and H can fully resolve a single spin-one/two state in the classical limit where G and H both go to infinity <ref:2606.15873#pg1>.

Conclusion: Mira: So to wrap up, the paper "Complete Relational Description of Spin in a Quantum Background," it shows that by using two large, non-commuting reference spins, you can get an exact encoding of a target spin when those reference numbers are large.

Kai: The main thing is that the coherent average gives us the right path for a background independent description because it should be used instead of the incoherent one.

Lev: Lev thinks that this paper provides a clear framework for defining states without relying on external backgrounds, which is essential for designing error correcting codes and other quantum protocols.

Mira: We’re done with this paper, but we need to get ready for what’s next in the research and how this relates to other papers.

Lev: Lev thinks that the paper's main contribution is showing that you can define states without relying on external backgrounds, which is essential for designing error correcting codes and other quantum protocols.

Mira: We're done with this paper, but we need to get ready for what’s next in the research and how this relates to other papers.