Measurement incompatibility and noiseless quantum codes

Technical Summary: Measurement Incompatibility and Noiseless Quantum Codes

Problem Statement
This paper addresses the fundamental relationship between the preservation of measurement incompatibility under quantum channels and the existence of noiseless quantum error-correcting codes. While it is established that quantum channels preserve measurement compatibility (joint measurability), the conditions under which they universally preserve incompatibility (i.e., no finite family of incompatible measurements becomes compatible after passing through the channel) were not fully characterized for arbitrary input and output dimensions. Specifically, the work targets two open questions:

1. The Chitambar-Zhang Conjecture: Whether a channel universally preserves incompatibility if and only if its input space contains a noiseless code subspace of the full output dimension.
2. The Uola et al. Existence Question: Whether, for every dimension $d \geq 2$, there exists a family of measurements that is incompatible on the full space but becomes compatible when compressed to any proper subspace.

Methodology
The authors employ a constructive approach centered on the design of a specific family of measurements and the analysis of the channel's Heisenberg dual map ($\Lambda = \mathcal{N}^*$).

1. Construction of a Rank-One MIC: The authors construct a rank-one minimal informationally complete measurement (MIC), denoted $\{G_i\}$, with an invertible dual operator $F_0$. This MIC spans the space of Hermitian operators.
2. Binary Measurement Family: They define a family of binary measurements $\{ \{G_i, \mathbb{I}-G_i\} \}_{i \geq 1}$ and a parameterized binary measurement $\{B_s, \mathbb{I}-B_s\}$ where $B_s = \mathbb{I} - sG_0/2$. The parameter $s$ controls the transition from compatibility ($s \leq 1$) to incompatibility ($s > 1$).
3. Stinespring Dilation and Expansion: Using the Stinespring isometry $V$ for the channel $\mathcal{N}$, the authors analyze the image of the parent measurement under the dual map $\Lambda$. They expand the operator $I - VV^\dagger$ (the projection onto the environment) in the basis of the MIC effects, utilizing the invertibility of $F_0$ to isolate coefficients $A_i$.
4. Threshold Analysis: The proof hinges on determining whether the "splitting" of the parent measurement effects required to make the image family compatible can be achieved without altering the total effect. This is linked to the existence of a vector in the kernel of a specific operator $C_0$ derived from the channel's structure.

Key Contributions and Results

• Proof of the Chitambar-Zhang Conjecture: The paper proves that for arbitrary finite input dimension $D$ and output dimension $d$, a channel $\mathcal{N}$ universally preserves measurement incompatibility if and only if there exists an isometry $W: \mathbb{C}^d \to \mathbb{C}^D$ such that $\mathcal{N}(W\rho W^\dagger) = \rho$ for all states $\rho$. This establishes a direct equivalence between universal incompatibility preservation and the existence of a noiseless code subspace of dimension $d$.
• Resolution of the Uola et al. Question: The authors demonstrate that for every $d \geq 2$, there exists a family of $d^2$ binary measurements that is incompatible on the full Hilbert space $\mathbb{C}^d$ but becomes jointly measurable (compatible) when compressed to any proper subspace. This is achieved by showing that the compatibility threshold of the constructed family is preserved only if the compression is trivial (i.e., the full space).
• Stability and Approximate Encoding: The paper provides an explicit construction for approximate encodings when the compatibility threshold is shifted by a small amount $\delta$. It proves that if the threshold shift is small, the transmission error (measured by the diamond norm) is bounded by a constant times the square of the shift ($\|\mathcal{N} \circ W - \text{id}_d\|_\diamond \leq 2c^2\delta^2$). This quantifies the stability of the noiseless code condition.
• Explicit Encoding Construction: An exact encoding is explicitly defined using the kernel of the operator $C_0$. If $|e\rangle$ is a unit vector in the kernel of $C_0$, the isometry is given by $W|\psi\rangle = V^\dagger(|\psi\rangle \otimes |e\rangle)$.

Significance and Claims
The paper claims to resolve the conjecture regarding the structural requirements for channels that universally preserve incompatibility, firmly linking this property to quantum error correction. By constructing a specific family of binary measurements, the authors show that the "compatibility threshold" serves as a precise diagnostic tool: if a channel preserves this threshold, it must admit a noiseless code.

Furthermore, the work establishes a tripartite connection between measurement compatibility, quantum steering, and quantum error correction. The authors argue that universal preservation of incompatibility is equivalent to the requirement that every finite incompatible measurement family demonstrates steering on the specific bipartite state associated with the channel. The results suggest that the existence of a noiseless code can be determined by examining the steering threshold of a finite measurement family on a shared state.

The paper concludes by noting that while the construction uses $d^2$ binary measurements (due to the MIC spanning the Hermitian space), it remains an open question whether fewer measurements suffice for this criterion. Additionally, the authors suggest that the derived stability bounds may be improvable, pointing toward future quantitative investigations into the relationship between measurement information and quantum transmission.