Symmetry-Constrained Operations in QFT

Symmetry constrains information processing through the operations and observables available to an agent. A channel may transform covariantly, a unitary may conserve charge, an instrument may transform its outcomes, and a QFT intervention may be localized in spacetime. These conditions answer different questions. A resource claim is meaningful only after the operation class, causal access, admissible ancillas, and reference frames have been stated.

Required background. Factorization failure and local algebras fixes the continuum subsystem; symmetry of a QFT supplies the global group action.

Helpful background. Local operations, separability, and distillability supplies the operational bipartite language.

Chapter map. See the chapter-wide path from a task specification to an information resource, claim-validity table, and three gates before an operational claim.

Five operation classes to keep distinct

Let $U_A(g)$ and $U_B(g)$ represent a group $G$ on the input and output systems. A channel $\mathcal E:A\to B$ is $G$-covariant when

$$\mathcal E\!\left(U_A(g)\rho U_A(g)^\dagger\right) =U_B(g)\mathcal E(\rho)U_B(g)^\dagger$$

for every $g\in G$. Covariance means that transforming the input and then applying the channel is equivalent to applying the channel and transforming the output. It does not say that a particular laboratory can implement the channel locally.

Several nearby definitions must not be collapsed into this one:

1. An invariant observable satisfies $U(g)OU(g)^\dagger=O$. For $U(1)$ with $U(\theta)=e^{i\theta Q}$, this is $[O,Q]=0$.

2. A symmetric unitary satisfies $[V,U(g)]=0$, or $[V,Q]=0$ for a connected $U(1)$ representation.

3. A covariant channel obeys the equation above; its system charge need not be conserved because charge may flow to an environment.

4. A covariant instrument includes the group action on its classical outcome. With the convention that $g\cdot x$ is the relabeled outcome,

   $$\mathcal E_x\!\left(U_A(g)\rho U_A(g)^\dagger\right) =U_B(g)\mathcal E_{g^{-1}\!\cdot x}(\rho)U_B(g)^\dagger.$$

5. A local or causally implementable operation must additionally respect the laboratories’ spacetime access. Global covariance alone supplies no such guarantee.

An invariant outcome label is sometimes exactly right. A charge measurement has effects $\Pi_q$ with $[\Pi_q,Q]=0$, so each charge outcome is invariant. By contrast, a phase or orientation measurement has outcomes that are relabeled by the group. Requiring all such pointer values to be invariant would exclude the intended covariant measurement.

Nor must every Kraus operator of a covariant channel be invariant. For a two-level charge system with $Q=|1\rangle\langle1|$, amplitude damping has

$$\begin{aligned} K_0&=|0\rangle\langle0| +\sqrt{1-\gamma}\,|1\rangle\langle1|,\\ K_1&=\sqrt{\gamma}\,|0\rangle\langle1|. \end{aligned}$$

Here $U(\theta)K_1U(\theta)^\dagger=e^{-i\theta}K_1$. The jump Kraus operator carries a definite charge mode, but its phase cancels between $K_1$ and $K_1^\dagger$, so the channel is covariant. The channel-mode decomposition and its use for states, measurements, and transformations are developed in Marvian and Spekkens 2014, §§ II–IV.

Symmetric dilations and hidden reference systems

A sufficient microscopic construction of a covariant channel uses an invariant environment state $\sigma_E$ and a symmetric joint unitary:

$$\mathcal E(\rho) =\operatorname{Tr}_E\!\left[V(\rho\otimes\sigma_E)V^\dagger\right], \qquad [V,U_S(g)\otimes U_E(g)]=0.$$

A charge eigenstate is invariant up to a phase and is therefore a valid $U(1)$ ancilla state. An ancilla coherent across charges is instead asymmetric. It can make the reduced system dynamics look symmetry breaking, but the ancilla’s asymmetry is then an input resource.

The hidden-resource test is exact in two charge qubits. Let $Q_S=Q_R=|1\rangle\langle1|$, let $W$ swap $S$ and $R$, and define

$$|+\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2}.$$

The swap obeys $[W,Q_S+Q_R]=0$, yet

$$W\bigl(|0\rangle_S|+\rangle_R\bigr) =|+\rangle_S|0\rangle_R.$$

After $R$ is hidden, the induced system process has mapped the invariant vacuum to an asymmetric state and is therefore not covariant. The relative entropy of $U(1)$ asymmetry, $A_{U(1)}$, has moved from the reference to the system: $A_{U(1)}(|+\rangle)=\ln2$, whereas $A_{U(1)}(|0\rangle)=0$. Replacing the reference by a charge eigenstate makes the apparent symmetry breaking impossible. This adversarial replacement reveals the resource that the reduced description concealed.

For a connected Lie symmetry such as $U(1)$, a finite-dimensional asymmetric reference cannot broadcast asymmetry while remaining locally unchanged, even if correlations with the target are permitted. Weaker indefinite repeatability requires an infinite-dimensional reference. These are precise no-go statements, not a license to assume that every finite reference marginal degrades monotonically after every possible interaction; the joint correlations must also be tracked Lostaglio and Müller 2019, Theorems 1–3.

Charged wavepacket benchmark

Let $a^\dagger$ and $b^\dagger$ create orthogonal unit-charge wavepackets in laboratories $A$ and $B$, and set

$$Q=a^\dagger a+b^\dagger b, \qquad U_{ab}(\vartheta) =\exp\!\left[\vartheta(a^\dagger b-b^\dagger a)\right].$$

The generator moves a particle between the modes but leaves their total number unchanged, so

$$[U_{ab}(\vartheta),Q]=0.$$

For the displayed convention,

$$\begin{aligned} U_{ab}(\vartheta)^\dagger a^\dagger U_{ab}(\vartheta) &=a^\dagger\cos\vartheta+b^\dagger\sin\vartheta,\\ U_{ab}(\vartheta)^\dagger b^\dagger U_{ab}(\vartheta) &=b^\dagger\cos\vartheta-a^\dagger\sin\vartheta. \end{aligned}$$

Thus $\vartheta=\pi/4$ coherently mixes the two wavepackets inside the fixed one-particle sector without a charge reference. This does not make the operation local: if $A$ and $B$ are separated laboratories, the coupling $a^\dagger b-b^\dagger a$ is bilocal. The modes must be brought together, coupled through an allowed mediator, or connected by an allowed quantum communication channel.

By contrast, the local target operation $|0\rangle_A\mapsto|+\rangle_A$ changes the local charge coherence. Without a reference it is operationally replaced by the normalized $U(1)$ twirl

$$\mathcal G_A(\rho) =\frac{1}{2\pi}\int_0^{2\pi}d\theta\, e^{i\theta Q_A}\rho e^{-i\theta Q_A},$$

which removes matrix elements between unequal local charges. The beam splitter and the hidden-reference swap therefore make two independent checks: charge conservation does not imply locality, and a symmetric global dilation does not make an asymmetric ancilla free.

The interpretation of twirling as a missing-reference restriction, and the recovery of relational coherence when a reference is supplied, are developed in Bartlett, Rudolph, and Spekkens 2007, §§ II.B–III.

Local operations in QFT

In the Heisenberg picture, a QFT instrument is described by normal completely positive maps on the selected observable algebra, with the nonselective sum unital. Algebraic admissibility is still weaker than physical localization. In the Fewster–Verch measurement framework, a system field couples to a probe field in a compact spacetime region; probe effects induce system observables in the causal hull, and causal factorization controls the composition of separated instruments Fewster and Verch 2020, §§ 3.2–3.5. A formally covariant map with no such local implementation can therefore lie outside the task’s operation class.

Gauge transformations require another distinction. Transformations acting trivially on the relevant boundary data are redundancies, whereas transformations with nontrivial boundary or asymptotic action may generate physical charges. Fix the boundary conditions, dressings, and charge algebra before assigning a resource theory; physical operations must still act on the selected gauge-invariant observables and respect the Gauss constraints.

Common pitfalls

Equating covariance with charge conservation. A system channel can be covariant while exchanging charge with an invariant environment. State whether the conserved charge belongs to the system or to a dilation.

Equating symmetry with locality. The wavepacket beam splitter is symmetric but bilocal for separated modes. Add the spacetime implementation constraint independently.

Treating a reference as a free catalyst. A coherent ancilla supplies asymmetry. Track its final marginal and its correlations with every target.

Exercises

1. Covariant damping with a charged Kraus operator

Show explicitly that the amplitude-damping channel above is $U(1)$-covariant even though $K_1$ does not commute with $Q$.

Solution

The two Kraus operators transform as

$$U(\theta)K_0U(\theta)^\dagger=K_0, \qquad U(\theta)K_1U(\theta)^\dagger=e^{-i\theta}K_1.$$

For either Kraus term, the phase multiplying $K_j$ is canceled by its complex conjugate multiplying $K_j^\dagger$. Hence

$$\begin{aligned} U(\theta)\mathcal E(\rho)U(\theta)^\dagger &=\sum_j U(\theta)K_j\rho K_j^\dagger U(\theta)^\dagger\\ &=\sum_jK_jU(\theta)\rho U(\theta)^\dagger K_j^\dagger\\ &=\mathcal E\!\left(U(\theta)\rho U(\theta)^\dagger\right). \end{aligned}$$

Covariance is a property of the complete channel, not of one chosen Kraus representation.

2. What the charge twirl removes

Let $Q=\sum_q q\Pi_q$. Derive the block form of $\mathcal G(\rho)$.

Solution

Insert $I=\sum_q\Pi_q$ on both sides of $\rho$:

$$\rho=\sum_{q,q'}\Pi_q\rho\Pi_{q'}.$$

Conjugation gives a phase $e^{i\theta(q-q')}$ to the $(q,q')$ block. Therefore

$$\mathcal G(\rho) =\sum_{q,q'} \left[\frac{1}{2\pi}\int_0^{2\pi} d\theta\,e^{i\theta(q-q')}\right] \Pi_q\rho\Pi_{q'} =\sum_q\Pi_q\rho\Pi_q.$$

Only coherence between unequal charges is removed; coherence inside a degenerate charge sector remains.

3. Symmetry, locality, and the hidden ancilla

Verify $[a^\dagger b-b^\dagger a,Q]=0$ and explain why this does not make $U_{ab}$ local. Then show that the swap construction cannot create $|+\rangle_S$ when the reference starts in $|0\rangle_R$.

Solution

Using $[a^\dagger a,a^\dagger]=a^\dagger$ and $[b^\dagger b,b]=-b$ gives

$$[Q,a^\dagger b]=a^\dagger b-a^\dagger b=0,$$

and similarly $[Q,b^\dagger a]=0$. The unitary is therefore symmetric. Its generator nevertheless contains operators from both laboratories, so a separated pair cannot implement it using local operations alone.

For the reference test,

$$W|0\rangle_S|0\rangle_R=|0\rangle_S|0\rangle_R.$$

Tracing out $R$ leaves the vacuum, not $|+\rangle$. The missing off-diagonal charge coherence was supplied entirely by the asymmetric state $|+\rangle_R$ in the earlier protocol.

References

• Bartlett, Stephen D., Terry Rudolph, and Robert W. Spekkens. “Reference Frames, Superselection Rules, and Quantum Information.” Reviews of Modern Physics 79 (2007): 555–609. DOI.
• Fewster, Christopher J., and Rainer Verch. “Quantum Fields and Local Measurements.” Communications in Mathematical Physics 378 (2020): 851–889. DOI.
• Lostaglio, Matteo, and Markus P. Müller. “Coherence and Asymmetry Cannot Be Broadcast.” Physical Review Letters 123 (2019): 020403. DOI.
• Marvian, Iman, and Robert W. Spekkens. “Modes of Asymmetry: The Application of Harmonic Analysis to Symmetric Quantum Dynamics and Quantum Reference Frames.” Physical Review A 90 (2014): 062110. DOI.