Quantifying Coherence-to-Entanglement Conversion Efficiency under Noisy Operations

Abstract: We investigate the noise-limited conversion of local quantum coherence into bipartite entanglement in a minimal two-qubit protocol comprising a coherent single-qubit input, an incoherent ancilla, an ideal CNOT operation, and subsequent environmental noise. Employing the $l_1$-norm of coherence and the entanglement negativity as resource quantifiers, we establish an exact closed-form correspondence between local single-qubit input coherence and the two-qubit entanglement generated in the noiseless limit, showing that the output negativity is precisely one half of the initial $l_1$-coherence. We then derive analytic expressions for the surviving entanglement and the associated coherence-to-entanglement conversion efficiency under two representative noise mechanisms: independent phase damping and global two-qubit depolarizing noise. The two channels exhibit qualitatively distinct degradation behavior. Phase damping induces a universal multiplicative suppression of the generated entanglement, yielding a coherence-independent conversion efficiency and no finite-noise entanglement sudden death. In contrast, global depolarization introduces an isotropic mixing contribution that shifts the partial-transpose spectrum, producing coherence-dependent degradation and a finite sudden-death threshold. We show that maximally coherent inputs not only maximize the entanglement generated by the CNOT protocol but also optimize its robustness against depolarizing noise. Direct density-matrix simulations validate the analytic results to numerical precision. These findings provide a compact analytic benchmark for assessing how different noise mechanisms constrain coherence-to-entanglement conversion in elementary quantum-information protocols and near-term quantum devices.

• The paper presents a closed-form mapping between input l1-norm coherence and output entanglement negativity using a CNOT gate.
• It demonstrates that phase damping scales entanglement linearly with (1-p) while global depolarization induces sudden entanglement death above a critical noise threshold.
• Numerical simulations confirm the analytic predictions, providing practical benchmarks for optimizing quantum circuits in NISQ devices.

Quantifying Coherence-to-Entanglement Conversion Efficiency under Noisy Operations

Introduction and Motivation

This work presents an exact analytic treatment of noise-limited conversion of local quantum coherence into bipartite entanglement in an elementary two-qubit circuit, formalizing operational resource quantification in noisy intermediate-scale quantum (NISQ) devices. The protocol under analysis consists of a single-qubit coherent input, an incoherent ancilla, an ideal CNOT entangling unitary, and subsequent noise modeled as either independent phase damping (pure dephasing) or a global two-qubit depolarizing channel. The primary aim is to obtain closed-form mappings between input $l_1$-norm coherence and output entanglement negativity, quantifying the conversion efficiency and the degradation effect of distinct noise mechanisms.

Theoretical Framework and Protocol Description

The protocol initializes qubit $A$ in the pure state $\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$, and qubit $B$ (ancilla) in the incoherent state $\ket{0}$. The system then undergoes a CNOT gate (control: $A$, target: $B$), resulting in the output $\cos\theta\ket{00} + e^{i\phi}\sin\theta\ket{11}$. In the noiseless scenario, the bipartite entanglement generated solely depends on the input coherence. Entanglement negativity—specifically suited for two-qubit characterization due to the sufficiency of the PPT criterion—is used as the entanglement measure.

Figure 1: Ideal coherence–entanglement relation in the noiseless CNOT protocol: $C_{l_1}$ coherence of input is mapped to output negativity $\mathcal{N}_0 = \frac{1}{2}\sin(2\theta)$.

The protocol's density matrix before noise has an $A$0-state form with only the populations $A$1, $A$2 and the anti-diagonal coherence nonzero.

Analytic Results: Noiseless and Noisy Coherence-to-Entanglement Mappings

Ideal (Noiseless) Mapping

The analysis establishes that the output negativity after the CNOT operation, as a function of the input angle $A$3, is

$A$4

This proportionality is a direct, closed-form mapping: regardless of the input phase, the $A$5-coherence is deterministically converted into negativity entanglement with conversion factor $A$6.

Phase Damping

Phase damping is treated as independent decoherence on each qubit. The anti-diagonal element $A$7, responsible for entanglement, is suppressed by $A$8, while populations remain unaltered. The output state retains an $A$9-form, leading to a compact analytic expression for the negativity: $\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$0 The conversion efficiency, defined as the ratio of output to ideal negativity, is

$\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$1

which is independent of the input coherence angle $\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$2. There is no entanglement sudden death: entanglement persists for all $\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$3 for any nonzero input coherence.

Figure 2: Entanglement negativity $\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$4 for maximally coherent input under increasing noise for both phase damping and global depolarization.

Global Depolarizing Channel

Global depolarizing noise is modeled as a CPTP map replacing the output state with the maximally mixed state with probability $\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$5: $\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$6 Unlike individual dephasing, this channel isotropically mixes all components. The negativity after depolarization is given by

$\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$7

leading to entanglement sudden death at the threshold

$\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$8

For maximally coherent input ($\cos\theta\ket{0} + e^{i\phi}\sin\theta\ket{1}$9), $B$0, recovering the Werner-state separability point.

The conversion efficiency is coherence-dependent: $B$1 Here, input coherence also increases robustness to depolarization, in contrast to phase damping where robustness is uniform with respect to $B$2.

Figure 4: Entanglement sudden-death threshold $B$3 for global two-qubit depolarizing channel; maximal for maximal input coherence.

Figure 6: Two-dimensional negativity landscape $B$4 for both phase damping and global depolarization, showing the entangled and separable regions.

Figure 5: Coherence-to-entanglement conversion efficiency $B$5 as a function of depolarizing or phase-damping noise; global depolarization leads to a thresholded drop in efficiency.

Numerical Validation and Spectral Insights

The analytic predictions for the output negativity were corroborated through explicit density-matrix simulations, demonstrating agreement to within numerical roundoff. The partial transpose spectra under both noise channels confirm that only one eigenvalue can become negative and that its behavior under the two noise channels fundamentally diverges: smooth decay for phase damping and thresholded (sudden) death for depolarization.

Implications for Quantum Information Processing

These findings rigorously characterize two physically distinct noise-induced degradation mechanisms for coherence-to-entanglement conversion in quantum circuits:

• Phase damping preserves conversion efficiency proportional to remaining single-qubit coherence, suggesting entanglement distribution via dephased but coherent ancillae can remain robust in systems dominated by pure dephasing noise.
• Global depolarization enforces a coherence-dependent ceiling on both entanglement and conversion efficiency, reflecting universal mixing (“white noise”) processes prevalent in crosstalk- or reset-dominated quantum devices.

These results establish analytic reference points for benchmarking coherence-to-entanglement protocols in NISQ devices and for calibrating optimal input state preparation in the presence of realistic device noise.

Additionally, the exact threshold for entanglement sudden death under depolarization aligns with and generalizes the Werner state boundary, which is a widely used reference in open-system quantum information theory.

Future Directions

Potential extensions include analysis of local depolarizing noise, amplitude damping, noisy entangling gates, and generalizations to multi-qubit “cascaded” entanglement networks. The interplay between restricted $B$6-state structures and efficient entanglement negativity calculations may also be exploited in larger quantum circuits and for novel resource theories.

Conclusion

This work provides a comprehensive analytic characterization of how the conversion efficiency between quantum coherence and entanglement in a two-qubit CNOT protocol is fundamentally constrained by both phase damping and depolarizing noise. The derived closed-form mappings and efficiency landscapes offer a quantitative framework for analysis, benchmarking, and future engineering of elementary quantum-information technologies operating under nontrivial environmental noise (2606.16916).