Quantum information · Theory
Measurement incompatibility and noiseless quantum codes
arXiv:2610.37347v1 ↗PDF ↗Submitted 27 October 2026
- Question
- A quantum channel can destroy the incompatibility of measurements but never create it. Which channels preserve the incompatibility of every finite family of measurements? Chitambar and Zhang conjectured that, when the input is larger than the output, such a channel must carry a noiseless copy of its output space, and Uola and coauthors asked whether every dimension has a measurement family that is incompatible on the full space but compatible on every proper subspace.
- Result
- Both questions are settled. A channel from dimension \(D\) to \(d\) preserves all incompatibility if and only if an isometry \(W\) encodes every output state noiselessly, \(\mathcal N(W\rho W^\dagger)=\rho\). One explicit family of \(d^2\) binary measurements is a complete test: its compatibility threshold is \(s=1\), and any channel that keeps that threshold has such a code. The same family is incompatible on \(\mathbb C^d\) and compatible on every proper subspace, for every \(d\ge2\). If the threshold shifts by a small \(\delta\), an approximate encoding has diamond-norm error at most \(2c^2\delta^2\).
The paper in detail
A family of measurements is compatible if all of them arise from one parent measurement by classical post-processing. For a channel \(\mathcal N\) with Heisenberg dual \(\Lambda\), measuring after the channel equals measuring \(\Lambda(M_{a|j})\) before it; \(\Lambda\) maps parents to parents, so compatibility is always preserved, while incompatibility can be lost. Through the one-to-one correspondence between joint measurability and steering, preserving all incompatibility is equivalent to every incompatible family demonstrating steering on one fixed bipartite state associated with the channel.
The key object is a rank-one minimal informationally complete measurement. Fix \(\ket x\), set \(P_0=\ket x\bra x\) and \(Q=2P_0-\mathbb I\), and for unit \(\ket u\perp\ket x\) take \(P_u=(\ket x+\ket u)(\bra x+\bra u)\); \(d^2-1\) independent \(P_u\) together with \(P_0\) span all Hermitian operators. With \(T=\sum_iP_i\), \[G_i=T^{-1/2}P_iT^{-1/2},\qquad F_0=T^{1/2}QT^{1/2},\] the \(G_i\) are rank-one effects summing to \(\mathbb I\), \(\operatorname{Tr}(F_0G_i)=\delta_{0i}\), and \(F_0\) is invertible. The test family is \(\{G_i,\mathbb I-G_i\}\) for \(i\ge1\) plus \(\{B_s,\mathbb I-B_s\}\) with \(B_s=(\mathbb I-sG_0)/2\), which is compatible exactly for \(s\le1\).
For the converse, take a Stinespring isometry \(V\) and expand \(\mathbb I-VV^\dagger=\sum_iF_0G_i\otimes A_i\). The coefficient of \(G_0\) is \(C_0=\operatorname{Tr}_{\rm sys}(\mathbb I-VV^\dagger)\ge0\). If \(C_0\) is positive definite, the expansion lets one adjust the splitting of the image parent so that the image family stays compatible for some \(s>1\), contradicting preservation. So \(C_0\) has a kernel vector \(\ket e\), and \(W\ket\psi=V^\dagger(\ket\psi\otimes\ket e)\) is the noiseless encoding.
Taking \(V\) to be the inclusion of a subspace with projector \(\Pi\) gives \(C_0(\Pi)=\operatorname{Tr}(\mathbb I-\Pi)\ge1\), and a compactness argument picks one \(t>0\) that makes the family compatible after compression to every proper subspace while it stays incompatible on the full space. For approximate preservation, with threshold \(1+\delta\) and \(\varepsilon=\lambda_{\min}(C_0)\), the paper shows \(\varepsilon\le c^2\delta^2\) and builds \(W=LS^{-1/2}\) from the corresponding eigenvector, with \[\|\mathcal N\circ\mathcal W-\mathrm{id}_d\|_\diamond\le2\varepsilon\le2c^2\delta^2.\]
Transferable methods
Dual-frame expansion of a dilation defect
Expanding the defect \(\mathbb I-VV^\dagger\) of a Stinespring dilation in an informationally complete frame, with environment operators as coefficients, isolates one positive operator \(C_0\) on the environment. Whether \(C_0\) has a kernel decides whether a noiseless code exists, and its kernel vector gives the code.
With \(\operatorname{Tr}(F_0G_i)=\delta_{0i}\) and \(F_0\) invertible, write \[\mathbb I-VV^\dagger=\sum_iF_0G_i\otimes A_i.\] Multiplying by \(F_0^{-1}\otimes\mathbb I\) and compressing with \(V\) gives \(\sum_iV^\dagger(G_i\otimes A_i)V=0\), because \((\mathbb I-VV^\dagger)V=0\); the system trace gives \(A_0=C_0=\operatorname{Tr}_{\rm sys}(\mathbb I-VV^\dagger)\ge0\). If \(C_0\ket e=0\), the positive operator \(\mathbb I-VV^\dagger\) vanishes on every \(\ket\psi\otimes\ket e\), so \(W\ket\psi=V^\dagger(\ket\psi\otimes\ket e)\) satisfies \(VW\ket\psi=\ket\psi\otimes\ket e\) and the channel returns \(\rho\) unchanged.
Editorial check: for amplitude damping with decay \(\gamma\), \(C_0\) has eigenvalues \(\gamma\) and \(2-\gamma\), so there is no noiseless code for \(\gamma>0\) and \(\varepsilon=\gamma\). For a channel from \(\mathbb C^3\) to \(\mathbb C^2\) that acts unitarily on a two-dimensional subspace and sends the third level to the maximally mixed state, \(C_0\) has a zero eigenvalue, and the \(W\) built from its kernel is an isometry that returns every test state exactly.
How to reuse it
The kernel of \(\operatorname{Tr}_{\rm sys}(\mathbb I-VV^\dagger)\) is a direct computational test for a noiseless code of full output dimension, and its smallest eigenvalue measures how far a channel is from having one. More generally, when a channel property must hold for all inputs of some kind, look for a finite probe whose threshold is one coefficient in an informationally complete frame; expanding the dilation defect in the dual frame then turns failure of the property into an explicit perturbation.
Rigid parents of rank-one measurements
When the fixed measurements are rank-one elements of an informationally complete measurement, their parent is forced, and compatibility of an added effect reduces to checking that its coefficients in that measurement’s basis lie in \([0,1]\). This gives a family with a sharp, analytically known threshold.
A joint effect for two first outcomes lies below two rank-one effects with different supports and so vanishes; the marginals then fix the parent as \(\{G_i\}\). Splitting a rank-one \(G_i\) into two positive parts forces both to be multiples of \(G_i\), so \(B=\sum_ib_iG_i\) is jointly measurable with the family if and only if every \(b_i\in[0,1]\). For \(B_s=(\mathbb I-sG_0)/2\) the coefficients are \((1-s)/2\) for \(i=0\) and \(1/2\) otherwise, so the threshold is \(s=1\). \(G_0\) is not a projection, since otherwise all effects would be block diagonal, so \(s_{\max}=\lambda_{\max}(G_0)^{-1}>1\) and the incompatible range is nonempty.
Editorial check: constructing the measurement numerically for \(d=2,3,4\) reproduces \(\sum_iG_i=\mathbb I\), rank-one effects and the dual relation, with \(s_{\max}=2.90\), \(6.32\) and \(13.5\) for one random choice of the vectors \(\ket u\). The coefficients of \(B_{1.3}\) are \(-0.15\) and \(0.5\), so the family is incompatible there, as the criterion predicts.
How to reuse it
Rank-one informationally complete measurements make joint-measurability questions linear: a candidate effect is compatible with them exactly when its frame coefficients are valid probabilities. This gives cheap certificates of compatibility or incompatibility without semidefinite programming, and a way to design measurement families with a sharp threshold.
Context & limitations
Why it matters. It turns a property quantified over all measurement families into a finite test and links joint measurability, steering and error correction through one threshold. By the steering correspondence, whether every incompatible family can steer the state associated with a channel decides whether the channel has a noiseless code, and the proof builds the code explicitly from a Stinespring dilation.
Limitations. The test family has \(d^2\) measurements, and whether fewer suffice is open. The stability constant \(c\) depends on the chosen measurement and is not optimized, and the quadratic bound holds only for small shifts. The results are characterization theorems; the paper contains no numerical study or experiment.