Characterizing quantum channels from local-unitary invariants

Abstract: We develop systematic frameworks for characterizing the entanglement properties of two-qubit channels beyond unitary settings. We introduce averaged local-unitary invariants, referred to as moments, obtained from Haar integrals over input states or unitaries. These moments provide computable descriptions of how a quantum channel can create, preserve, or destroy bipartite entanglement. We first show that second-order moments yield criteria for non-entangling and entanglement-breaking channels, which allow us to detect entanglement-creating and entanglement-preserving channels. We then demonstrate that higher-order moments can capture additional information and distinguish channels beyond second-order moments alone. Finally, we show that combinations of moments associated with different channel families improve the discrimination of locally inequivalent two-qubit unitaries.

• The paper introduces a moment-based framework employing Haar-averaged LU invariants to reliably distinguish entanglement-creating, preserving, and breaking quantum channels.
• It utilizes second and fourth order moments to resolve channels with indistinguishable entangling powers under conventional measures, providing finer discrimination.
• The approach offers practical, numerically tractable criteria applicable to both noiseless and noisy two-qubit channels, with potential extensions to multipartite scenarios.

Essay: Moment-Based Characterization of Quantum Channels via Local-Unitary Invariants

Introduction and Motivation

The manipulation and assessment of quantum entanglement under the influence of quantum channels are fundamental to quantum information processing. Traditional approaches address the creation (entangling power), preservation, and destruction (entanglement-breaking) of entanglement mainly in the idealized setting of unitary channels, often leading to limited applicability in realistic, noisy environments. The lack of injectivity of entangling power and the qualitative nature of existing entanglement-breaking criteria preclude a comprehensive, fine-grained analysis of general quantum channels, especially beyond the unitary regime.

This work introduces a moment-based framework centered on local-unitary (LU) averaged invariants for the systematic characterization of two-qubit quantum channels' entangling behavior. By operationalizing Haar-averaged invariants over product and entangled input states, the paper establishes practical and computable criteria for distinguishing entanglement-creating, preserving, and breaking action of channels that transcend previous, unitary-limited approaches.

LU Invariants and Moment Construction

For two-qubit states, LU invariants such as $I_2(\rho_{AB})$ and $I_4(\rho_{AB})$—quantities constructed as polynomials in the state's Pauli correlation tensor, invariant under local unitaries—form the backbone of entanglement detection. These invariants admit extremal values on pure product and Bell states ($I_t = 1, 3$ for $t=2,4$ respectively). The convexity of $I_t$ and their monotonicity under local operations constitute an analytic link to separability.

To address channel characterization, the output state under a channel $\Lambda$ is considered for (random) product or entangled inputs, and the LU invariants are averaged over Haar-distributed input ensembles, yielding channel moments $\mathcal{C}^{(t)}(\Lambda)$ (for entanglement-creation from products) and $\mathcal{P}^{(t)}(\Gamma)$ (for entanglement-preservation/breaking on one qubit of an entangled pair).

Criteria for Entanglement Creation, Breaking, and Preservation

Numerical thresholds:

• Any two-qubit channel $\Lambda$ with $\mathcal{C}^{(t)}(\Lambda) > 1$ ($I_4(\rho_{AB})$0) is entanglement-creating; $I_4(\rho_{AB})$1 is necessary for non-entangling action.
• Similarly, any single-qubit channel $I_4(\rho_{AB})$2 with $I_4(\rho_{AB})$3 is entanglement-preserving on at least some entangled input.

The invariance of the channel moments under local twirling and their tractable expressions via the channel’s Kraus representation facilitate direct numerical or analytic evaluation in practice.

Higher-order discrimination:

The main technical advance is the demonstration that higher-order moments (specifically, the fourth) further resolve channels not discerned by the second moment alone. For instance, the channels induced by the CNOT and B gates share $I_4(\rho_{AB})$4 but differ in $I_4(\rho_{AB})$5:

• $I_4(\rho_{AB})$6
• $I_4(\rho_{AB})$7, $I_4(\rho_{AB})$8

Combined-moment domains:

By examining models that interpolate unitary and depolarizing channels, e.g.,

$I_4(\rho_{AB})$9

and projecting the admissible region in $I_t = 1, 3$0 space, the authors obtain a finer boundary for entanglement-creation versus non-entangling behavior than using $I_t = 1, 3$1 alone. Similar constructions hold for entanglement-preserving characterization via $I_t = 1, 3$2 for unital (bistochastic) single-qubit channels. The explicit geometric boundaries in moment space are derived via optimization subject to PPT/separability constraints.

Distinguishing nonlocally inequivalent channels:

Although even the fourth moment is not injective on the set of two-qubit unitaries, the joint analysis of moments for channels related via local and nonlocal gates provides improved discrimination, especially when combining data from entanglement-creation and entanglement-breaking criteria.

Interrelation Between Two-Qubit Unitary and Single-Qubit Derived Channels

A further layer of analysis explores the connection between two-qubit unitary actions and the single-qubit channels they induce via coupling to a mixed environment. This relation is probed by plotting $I_t = 1, 3$3 against $I_t = 1, 3$4, where $I_t = 1, 3$5 is the channel induced on one qubit by $I_t = 1, 3$6 acting on $I_t = 1, 3$7.

This combined-moment approach allows for the differentiation of unitary gates that are otherwise indistinguishable by any one family of moments, demonstrating a nuanced hierarchy of distinguishability—for example, the $I_t = 1, 3$8SWAP and SWAP gates, or $I_t = 1, 3$9SWAP and CNOT.

Implications and Directions for Future Work

From a practical standpoint, this moment-based framework circumvents the need for explicit optimization over product inputs or entangled states, providing analytic and numerically tractable entanglement criteria whose application extends to both noiseless and noisy channels. The dual employment of different orders of moments is shown to strictly refine the operational detection of entanglement creation and preservation.

Theoretically, this approach bridges the gap between binary qualitative channel classifications and quantitative, hierarchy-sensitive channel discrimination. The invariant structure of the moments under local (and some global) transformations situates the method naturally within the broader paradigm of resource theories, motivating extensions to multipartite and higher-dimensional channels, as well as towards operationally relevant resource-breaking and resource-creating aspects beyond entanglement.

Potential future directions include the integration of these invariants with experimental protocols harnessing randomized measurements, generalization to multipartite scenarios via higher-order trace polynomials or sector lengths, and adaptation to non-entanglement resource regimes. Additionally, statistical characterization and error analysis with finite data merit pursuit for deployment in NISQ-era devices.

Conclusion

This paper presents a comprehensive, moment-based formalism for characterizing two-qubit quantum channels regarding their entanglement manipulation capabilities, employing Haar-averaged LU invariants. By leveraging higher-order moments and combined-moment domains, the framework achieves enhanced discrimination of entanglement-creating, preserving, and breaking channels—compared to traditional measures—across both unitary and mixed cases. The analytic structure and operational accessibility of these criteria underscore their potential for both foundational studies and applications in quantum information science (2606.03722).