Quantum Mathematics Learning Guide 72: Quantum Reference Frames, Superselection Rules, Asymmetry and Relational Quantum States

A quantum state is never described without a reference convention. Phase is defined relative to an oscillator, direction relative to an orientation, time relative to a clock, and particle-number coherence relative to a phase standard. When two observers do not share that reference, mathematically available superpositions can become operationally inaccessible.

Quantum reference-frame theory turns this into group theory. A symmetry group G acts through a unitary representation U(g). If the reference parameter g is completely unknown, the observer must average—or twirl—over that group. Coherences between different symmetry sectors disappear from the observer’s effective state, producing a superselection-like restriction.

This guide develops group twirling, U(1) phase references, particle-number superselection, relational encodings, SU(2) orientation frames, the resource theory of asymmetry, relative entropy of asymmetry, finite reference tokens, quantum clocks and the Wigner–Araki–Yanase measurement constraint. Guide 49 owns gauge geometry; this guide owns operational reference systems and symmetry asymmetry.

Identify symmetry group → specify shared or missing reference → apply group action → twirl unknown frame → separate global asymmetry from relational information → quantify and consume reference resources.

1. Reference frames are physical resources

A coordinate label by itself is not operational.

To implement “rotate about laboratory X”, the apparatus must define X. To prepare (|0⟩+|1⟩)/√2 when |0⟩ and |1⟩ differ in particle number, the laboratory needs a phase reference connecting those sectors.

A perfect classical reference is an idealisation. Real reference frames are finite physical systems and can become noisy, disturbed or misaligned.

2. Group action

Let symmetry/reference transformations be labelled by g∈G.

The system transforms as

ρ→𝒰_g(ρ)=U(g)ρU(g)†.

Examples:

• G=U(1) for phase/number reference;
• G=SO(3) or SU(2) for spatial orientation/spin direction;
• time translations generated by H for clock references;
• permutations for particle-label conventions in selected relational problems.

The relevant group is determined by what reference information is missing.

3. Twirling over an unknown reference

If g is completely unknown with invariant Haar prior, the observer assigns

𝒢(ρ)=∫_G dg U(g)ρU(g)†.

For finite groups the integral becomes an average

(1/|G|)Σ_g.

The twirling map is completely positive, trace preserving, idempotent and projects states onto the G-invariant operator algebra.

4. U(1) phase reference

Let N be a number operator.

Changing phase reference by φ acts as

U(φ)=e^{-iφN}.

Without a shared phase reference, twirl over φ:

𝒢(ρ)=(1/2π)∫_0^{2π}dφ e^{-iφN}ρe^{iφN}.

This removes coherence between different eigenvalues of N while retaining coherence inside degenerate fixed-N sectors.

5. Worked U(1) twirl of |+⟩

Let |0⟩ and |1⟩ carry number 0 and 1, and prepare

|+⟩=(|0⟩+|1⟩)/√2.

The density matrix is

ρ=(1/2)(|0⟩⟨0|+|0⟩⟨1|+|1⟩⟨0|+|1⟩⟨1|).

Under phase shift,

|0⟩⟨1|→e^{iφ}|0⟩⟨1|.

Averaging e^iφ over 0…2π gives zero.

Therefore

𝒢(ρ)=(|0⟩⟨0|+|1⟩⟨1|)/2.

The observer without the phase reference cannot operationally access the number-sector coherence.

6. Effective superselection rule

A superselection rule (SSR) forbids coherent operations/observations between designated sectors.

Missing-reference descriptions can produce an effective SSR: if all accessible operations are U(1)-covariant and no phase token is available, coherence between different N sectors has no observable operational effect.

This should be distinguished from a proposed fundamental superselection law. Bartlett, Rudolph and Spekkens emphasised that many SSR restrictions can be understood as lack of a shared reference frame. [1]

7. Covariant operations

An operation 𝓔 is G-covariant when

𝓔(U_gρU_g†)=U_g𝓔(ρ)U_g†

for every g.

If an agent lacks a reference frame and their laboratory obeys the symmetry, G-covariant operations are the natural free operations.

They cannot create asymmetry from a G-invariant input.

8. Asymmetry as a resource

A state ρ is symmetric/free when

U_gρU_g†=ρ

for all g.

An asymmetric state changes under the group action and can serve as a token of the missing reference.

Examples:

• optical coherent phase relative to particle-number U(1);
• a spin polarised along a direction relative to SU(2);
• energy coherence used as a clock under time translations.

Resource theory asks which asymmetric states can be converted into which others using only covariant operations.

9. Relative entropy of asymmetry

For a compact-group twirl, a standard asymmetry measure is

A_G(ρ)=S(𝒢(ρ))−S(ρ),

equivalent to a relative-entropy distance from the symmetric set under the usual conditions.

It is zero for invariant states and nonincreasing under G-covariant operations.

For pure |+⟩ across number sectors, S(ρ)=0 while the U(1)-twirled state is maximally mixed on two sectors, giving one bit of asymmetry when logarithms are base two.

10. Relational information can survive a global twirl

Consider two modes A and B with total particle number one:

|ψ_rel⟩=(|10⟩+e^{iφ}|01⟩)/√2.

A global U(1) transformation generated by N_A+N_B multiplies both terms by the same phase e^−iθ.

The density matrix is therefore invariant under global phase twirling.

Yet the relative phase φ between the modes remains measurable by interference between A and B.

Global phase asymmetry is absent; relational coherence inside the fixed-number sector remains.

11. Worked relational-phase measurement

Define one-excitation basis

|0_L⟩=|10⟩, |1_L⟩=|01⟩.

Then |ψ_rel⟩ is an ordinary logical qubit

(|0_L⟩+e^{iφ}|1_L⟩)/√2.

Measure logical

X_L=|10⟩⟨01|+|01⟩⟨10|.

Then

⟨X_L⟩=cosφ.

Likewise the logical Y observable yields sinφ.

Both observables conserve total particle number and are therefore compatible with the global U(1) SSR.

12. Decoherence-free relational encoding

If noise applies the same unknown phase e^−iθN to both modes, every state in a fixed total-number sector acquires only a global phase.

The logical qubit encoded across |10⟩ and |01⟩ is immune to collective phase noise.

This is the same mathematical reason decoherence-free subspaces can protect information against collective symmetry noise.

Reference-frame independence and noise protection can therefore coincide.

13. Shared versus unshared phase references

Alice may describe a state as |+⟩ relative to her oscillator.

Bob, with no knowledge of Alice’s oscillator phase, assigns the twirled mixture.

If Alice sends Bob a sufficiently good phase-reference token, Bob can partially align his frame and recover access to the coherence.

The difference between their state assignments reflects available relational information, not contradictory physical predictions.

14. Finite reference tokens

A finite reference system cannot encode a continuous group element perfectly.

Examples:

• a finite-amplitude coherent state as a phase token;
• a spin-j coherent state as a directional token;
• a finite-energy clock wavepacket as a time token.

Measurement of the token estimates the relative group element with finite precision.

Larger asymmetry resources generally support better alignment, but repeated use can degrade a quantum reference if operations disturb it.

15. SU(2) orientation references

If Alice’s Cartesian axes are unknown to Bob, spin states transform under SU(2).

Bob’s frame-free description is

𝒢_{SU(2)}(ρ)=∫dR U(R)ρU(R)†.

For a single spin-1/2 pure state, full rotational twirling gives

I/2.

Without an orientation reference, “spin points along +z” has no invariant meaning.

16. Multiple spins retain rotational invariants

Two spins can encode orientation-independent information through total angular momentum.

The singlet

|ψ^-⟩=(|01⟩−|10⟩)/√2

is invariant under collective U⊗U rotations.

Singlet/triplet labels are relational rotational invariants even when no external spatial direction is shared.

Reference-free communication can encode information into such multiplicity/invariant sectors.

17. Representation decomposition under twirling

A compact-group representation decomposes Hilbert space schematically as

ℋ=⊕_λ (𝓜_λ⊗𝓝_λ)

where 𝓜_λ carries an irreducible representation and 𝓝_λ is a multiplicity space.

Group twirling depolarises representation degrees of freedom inside irreps while preserving information in multiplicity spaces.

These multiplicity spaces are natural places to encode information independent of the missing reference.

18. Reference-frame-free subsystems

When a collective symmetry acts as

U_g=⊕_λ U_λ(g)⊗I_{𝓝_λ},

the multiplicity subsystem 𝓝_λ is untouched by the unknown group action.

Quantum information stored there is both reference-frame independent and protected against collective symmetry noise.

This is closely related to noiseless subsystems and decoherence-free subspaces from Guide 52.

19. Coherence versus asymmetry

“Coherence” depends on a chosen basis. “Asymmetry” depends on a chosen group representation.

For U(1) generated by a nondegenerate N, coherence between N eigenstates is asymmetry.

With degeneracies, coherence inside one N sector is not U(1) asymmetry because the group acts identically there.

Resource theories of coherence and asymmetry overlap but are not universally identical.

20. Modes of asymmetry

For U(1), density-matrix elements |n⟩⟨m| transform as

|n⟩⟨m|→e^{-iφ(n−m)}|n⟩⟨m|.

The difference k=n−m labels a Fourier mode of asymmetry.

Covariant operations cannot create arbitrary new mode structure. This refines the simple statement “coherence cannot increase” by tracking which symmetry frequencies are available.

21. Quantum clocks

Time translations act as

U(t)=e^{-iHt}.

A state diagonal in energy is invariant and cannot function as a clock indicating time under isolated evolution.

Coherence between distinct energies creates time-translation asymmetry and allows observables to change with time.

Quantum metrology of time/frequency is therefore a resource theory of asymmetry relative to time translations.

22. Relational time

In relational approaches, one subsystem acts as a clock and another as the system.

A globally stationary joint state can contain correlations such that conditioning on a clock reading yields an evolving conditional system state.

Page and Wootters developed a canonical version of this idea. [4]

This does not mean laboratory time disappears; it shows that change can be encoded in correlations between quantum subsystems.

23. Wigner–Araki–Yanase constraint

If an additive conserved quantity L=L_S+L_A must be exactly conserved during measurement, exact repeatable measurement of a system observable O that does not commute with L_S is constrained.

The Wigner–Araki–Yanase theorem and later quantitative forms show that a large apparatus/reference uncertainty in the conserved quantity can enable increasingly accurate approximate measurements.

A reference frame is therefore not merely coordinate bookkeeping; it can supply the asymmetry resource needed to implement otherwise symmetry-forbidden operations.

24. Phase reference activates forbidden operations

Under a strict number-conserving operation set, one cannot create |0⟩+|1⟩ coherence from a number eigenstate.

Coupling to a large coherent reference mode allows a globally number-conserving unitary to transfer relative phase information into the small system.

From the reduced-system viewpoint it appears that particle-number superselection has been “violated”; globally, conservation remains exact because the reference participates.

25. Entanglement under a superselection rule

A state can be mathematically entangled yet have less accessible entanglement when local operations are restricted by an SSR and parties lack a shared reference.

Providing a shared phase reference can activate operational tasks that were impossible under the restricted operation set.

Resource accounting must therefore specify both the quantum state and the available reference frames.

26. Reference frames and cryptography/communication

Communication protocols often assume aligned polarisation axes, phases or clocks.

If those references are missing, one can:

• send alignment tokens;
• estimate the relative transformation;
• encode into invariant/decoherence-free subsystems;
• design reference-frame-independent protocols.

Reference alignment consumes bandwidth and may leak information; invariant encodings trade rate for robustness.

27. Reference changes are not gauge transformations in every sense

Both gauge theory and reference-frame theory involve group transformations and invariant quantities, but their physical interpretations differ.

Gauge redundancy can be a local redundancy in field variables constrained by Gauss law (Guide 64). A reference-frame change may instead relate observers with different physical orientations or clocks.

The mathematical tools overlap; the ontology and operational scenario must be stated.

28. Common misconception: twirling physically destroys the original state

Twirling can represent an observer’s effective description after averaging over an unknown reference. It need not mean a physical decoherence channel was applied. Operational predictions coincide for observers restricted to reference-independent measurements.

29. Common misconception: a superselection rule forbids every useful superposition

It forbids/access-limits coherence between specified sectors under the restricted operation set. Coherence within a degenerate fixed-charge sector and relational information can remain fully usable.

30. Common misconception: relational states contain no phase information

The absolute global phase reference may be absent while relative phases between subsystems remain measurable by invariant interference observables.

31. Worked synthesis problem

Compare a single-mode phase state and a two-mode relational phase state under global U(1).

Step 1: Single mode. |+⟩=(|0⟩+|1⟩)/√2.

Step 2: Twirl. Unknown phase multiplies |0⟩⟨1| by e^iφ; integration sets it to zero. Result is I/2 in the number basis.

Step 3: Two-mode relational state. |ψ_rel⟩=(|10⟩+e^{iϕ}|01⟩)/√2.

Step 4: Global U(1). Both basis states have total number one, so they acquire the same global phase. The density matrix is unchanged by twirling.

Step 5: Measure relative phase. ⟨X_L⟩=cosϕ and ⟨Y_L⟩=sinϕ.

Step 6: Interpretation. Absolute phase asymmetry has been removed, but a relational qubit survives inside a fixed-charge sector.

32. Practice set

1. How does a group G act on a quantum state?
2. Define the group-twirling map.
3. What does U(1) number twirling remove?
4. Why can lack of a reference induce an effective superselection rule?
5. Define a G-covariant operation.
6. What is a free state in the resource theory of asymmetry?
7. State the relative entropy of asymmetry formula used here.
8. Why does the state (|10⟩+e^iφ|01⟩)/√2 survive global U(1) twirling?
9. What information survives SU(2) twirling of multiple spins?
10. Why can energy coherence act as a clock resource?
11. What does the WAY theorem constrain?
12. Why is a finite quantum reference frame a consumable/imperfect resource?

Answers

1. ρ→U(g)ρU(g)†.
2. 𝒢(ρ)=∫dg U(g)ρU(g)† with Haar-normalised measure.
3. Coherence between different eigenvalues of the number generator, while preserving within-sector structure.
4. Without the reference, operationally allowed preparations/measurements are restricted to symmetry-covariant/invariant ones, making cross-sector coherence inaccessible.
5. An operation commuting with the group action: 𝓔∘𝒰_g=𝒰_g∘𝓔.
6. A state invariant under every group transformation.
7. A_G(ρ)=S(𝒢ρ)−S(ρ) for the compact-group setting used here.
8. Both components have the same total number and acquire only the same global phase.
9. Invariant irrep labels and multiplicity/relational subsystems, such as singlet/triplet structure.
10. A state with coherence between distinct energies changes under time translation and can encode elapsed phase/time.
11. Exact measurements of observables not commuting with an exactly conserved additive quantity are restricted; large reference/apparatus resources allow better approximation.
12. A finite token encodes a continuous group element with finite precision and can be disturbed/degraded by use.

Sources and further study

[1] Stephen D. Bartlett, Terry Rudolph and Robert W. Spekkens, Reference frames, superselection rules, and quantum information, Reviews of Modern Physics 79, 555–609 (2007). The standard reference connecting missing frames, twirling and effective superselection.

[2] Gilad Gour and Robert W. Spekkens, The resource theory of quantum reference frames: manipulations and monotones, New Journal of Physics 10, 033023 (2008). Develops asymmetry/frameness as a resource theory.

[3] Iman Marvian and Robert W. Spekkens, Extending Noether’s theorem by quantifying the asymmetry of quantum states, Nature Communications 5, 3821 (2014). Connects symmetry, asymmetry measures and conservation-law resource constraints.

[4] Don N. Page and William K. Wootters, Evolution without evolution: Dynamics described by stationary observables, Physical Review D 27, 2885 (1983). A canonical relational-time construction.

[5] Eugene P. Wigner, Die Messung quantenmechanischer Operatoren, Zeitschrift für Physik 133, 101–108 (1952), with later Araki–Yanase developments. Foundational conservation-law limitations on quantum measurements.

Batch 18 series navigation

• Guide 69: Quantum State Preparation, Amplitude Encoding, QROM, Data Loading and State-Synthesis Algorithms
• Guide 70: Quantum Hamiltonian Learning, System Identification, Bayesian Estimation and Spectroscopy
• Guide 71: Quantum Graph Algorithms, Connectivity, Triangle Finding, Shortest Paths and Quantum-Walk Search
• Guide 72: Quantum Reference Frames, Superselection Rules, Asymmetry and Relational Quantum States
• Return to the BTT Mathematics Learning Hub

Educational note: a superselection rule can be fundamental, symmetry-imposed, or an effective restriction caused by unavailable reference information. Claims should state which operational scenario is intended.