Quantum coherence and state conversion: theory and experiment

The study addresses the core state-conversion problem within the resource theory of quantum coherence: given operational constraints (incoherent operations, IOs), what are the limits and optimal probabilities for converting one quantum state into another? Motivated by the broader context of resource theories (e.g., LOCC and entanglement) and the practical necessity to manipulate coherence under decoherence and physical restrictions, the work focuses on single-shot and asymptotic regimes for qubits. It seeks to fully characterize stochastic incoherent conversions between mixed qubit states, extend to assisted scenarios where correlations with a second party enhance capabilities, and demonstrate these conversions experimentally using photonic platforms, thereby quantifying the practical value of coherence in quantum technologies.

Resource theories provide a structured way to quantify and manipulate nonclassical properties such as entanglement and coherence. Prior work has defined coherence measures (e.g., Baumgratz et al.), characterized pure-state and some qubit mixed-state conversions under IO/SIO in single-shot settings, and explored asymptotic coherence distillation and cost. Stochastic conversions between pure states were studied previously, and robustness-based and rank-based constraints on coherence transformations have been developed. Assisted coherence distillation and distributed coherence manipulation (LQICC) frameworks were also introduced. However, a complete solution for stochastic interconversion probabilities between mixed qubit states, and optimality in assisted conversion for key state families, remained open. This work fills those gaps and refines bounds on asymptotic conversion rates by tying single-shot probabilities to asymptotic rates.

Theoretical framework:

• Coherence setting: Fix an incoherent basis (eigenbasis of σz). Incoherent states are diagonal in this basis. Free operations are incoherent operations (IO), which admit an incoherent Kraus decomposition; strictly incoherent operations (SIO) are a subset with both Kraus operators and their adjoints incoherent.
• Stochastic IO/SIO: Any stochastic map with (strictly) incoherent Kraus operators can be embedded in a deterministic IO/SIO (Proposition 1), legitimizing postselection-based probabilistic protocols.
• IO–SIO equivalence: For single-qubit states, existence and optimal probability of a stochastic IO achieving ρ→σ equals that under SIO (Theorem 2), allowing analysis to focus on SIO.
• Bloch-sphere parametrization: Represent qubit states via Bloch vectors r=(rx,ry,rz). Rotations about z are free, so only r⊥=sqrt(rx^2+ry^2) matters for coherence; denote r=√(rx^2+ry^2).

Optimal conversion without assistance:

• Reachable region characterization (Theorem 3): For fixed initial ρ (Bloch vector r) and success probability p, a target σ (Bloch vector s) is reachable by IO/SIO iff two inequalities hold, which define the intersection of an ellipsoid and a cylinder in the Bloch sphere. For p≤1−rz, the ellipsoid sits inside the cylinder, so lowering p below 1−rz does not enlarge the reachable set, implying a discontinuity in optimal probabilities and a nonzero-measure region unreachable even with arbitrarily small p.
• Maximal success probability (Corollary 4): Provides expressions giving Pmax(ρ→σ); zero whenever σ lies outside the ellipsoid constraint; otherwise a closed-form probability depending on r, rz, and s.
• Necessary conditions beyond qubits: Any coherence measure C that is monotone on average under selective IO yields upper bounds P(ρ→σ)≤C(ρ)/C(σ). A stronger necessary condition under SIO uses the robustness of coherence CAR: CAR(σ)≤CAR(ρ) (Theorem 5). Coherence rank cannot increase stochastically.

Assisted incoherent state conversion (one-way LQICC):

• Task: Alice and Bob share ρAB; Bob is restricted to IO; Alice can perform general local operations and send classical information, enabling Bob to implement conditional IO.
• Pure two-qubit inputs: For |ψ⟩AB with Bob’s local state having Bloch vector r, the optimal probability to prepare σ (Bloch vector s) on Bob’s side via one-way LQICC is given in closed form (Theorem 6), depending only on rz and s (explicit min-expression provided in the paper).
• Werner states: For ρAB=q|Φ+⟩⟨Φ+|+(1−q)I/4, an explicit Pmax(ρAB→σ) under one-way LQICC is derived (Theorem 7). Even when Bob’s local state is maximally mixed, correlations can enable creation of coherent σ.
• General statement: For any correlated, non–quantum-incoherent ρAB and qubit Bob, one-way LQICC strictly enlarges Bob’s accessible set compared to his reduced state alone (Theorem 8).

Asymptotic IO conversions:

• Define asymptotic rate R(ρ→σ) as the maximal rate of converting many copies of ρ to σ via IO in the trace-norm sense. Show single-shot success probability lower-bounds the rate: R≥P(ρ→σ).
• Bounds via distillable coherence and coherence cost: Ca(ρ)=S(Δρ)−S(ρ); Cc(ρ)=Cr(ρ) (coherence of formation). For qubits, Cc has a closed form via binary entropy of 2|ρ01|. Then: Ca(ρ)≤R(ρ→σ)≤min{Ca(ρ)/Ca(σ), Cc(ρ)/Cc(σ)}.
• Example family: Analyze ρ and σ being mixtures of |±⟩; compare bounds numerically, showing tightness of P(ρ→σ) as a lower bound in relevant parameter regimes.
• Special case with unit rate: Identify families of qubit states for which R(ρ→σ)=1 (Theorem 9).
• Irreversibility analysis: For nonfree states, R(ρ→σ)R(σ→ρ)≤1; quantify attainable pairs (Ca,Cc). Proposition 10 shows the |±⟩ mixture family minimizes Ca for a fixed Cc (and maximizes Cc for fixed Ca), delineating the boundary of the allowed region; pure states lie on Ca=Cc.

Experimental implementation (linear optics):

• Setup comprises three modules: (I) state preparation, (II) incoherent operations with/without assistance, (III) tomography. Photon pairs at 808 nm are generated via type-I phase-matched BBO crystals pumped at 404 nm (SPDC). Polarization optics (HWP, QWP), beam displacers, interferometers, and filters tailor states and implement SIO Kraus maps.
• Prepared inputs: (a) Single-qubit states for Bob: ρB=(I+rxσx+rzσz)/2 with tunable rx,rz via waveplate angles and dephasing crystals; (b) Pure two-qubit entangled states |Ψ⟩AB with controlled Schmidt coefficients; (c) Werner states via unbalanced Mach–Zehnder interferometer mixing a Bell state with white noise.
• SIO realization: Sequences of HWPs and beam displacers implement non-unitary SIOs; stochastic conversions realized by postselecting Kraus outcomes κ1 vs κ2; probabilities estimated from coincidence counts N1/(N1+N2) in appropriate measurement bases.
• Assistance: Alice performs optimal local projective measurements and communicates outcomes to condition Bob’s IOs.
• Tomography: Standard polarization tomography (HWPs, QWPs, PBSs, SPDs) reconstructs output states; fidelities reported near unity for prepared entangled states; count rates and noise conditions detailed.

• Theoretical equivalence: For qubit states, stochastic IO and SIO achieve exactly the same conversion possibilities and optimal success probabilities (Theorem 2).
• Reachable-region characterization: For any qubit ρ and success probability p, the set of achievable σ under IO/SIO is the intersection of a universal ellipsoid and a p-dependent cylinder in the Bloch sphere (Theorem 3). Lowering p below 1−rz does not enlarge this set, implying a discontinuity of Pmax and a finite region unreachable at any nonzero p.
• Optimal probability: Closed-form expressions determine the maximal success probability P(ρ→σ); P=0 outside the ellipsoid and takes an explicit value inside (Corollary 4). General necessary conditions include CAR(σ)≤CAR(ρ) (Theorem 5) and non-increase of coherence rank under stochastic IO.
• Assisted conversion (one-way LQICC):
  • For pure two-qubit shared states, an explicit optimal probability formula is derived depending only on rz of Bob’s reduced state and the target σ (Theorem 6).
  • For Werner states, optimal probabilities are given in closed form (Theorem 7). Assistance strictly enlarges Bob’s accessible set for any correlated, non–quantum-incoherent shared state (Theorem 8).
• Asymptotic rates: Single-shot success probability lower-bounds asymptotic conversion rate R; upper bounds via Ca and Cc are provided. For qubits, Ca and Cc have closed forms. Numerical analysis on mixtures of |±⟩ shows regions where P(ρ→σ) nearly saturates the best known upper bound. A family of states achieves R=1 (Theorem 9).
• Irreversibility boundary: Among single-qubit states, convex mixtures of |±⟩ minimize distillable coherence Ca for a fixed coherence cost Cc (Proposition 10), mapping the feasible (Ca,Cc) region; pure states saturate Ca=Cc.
• Experimental validation: Implemented non-unitary SIOs in a photonic platform. Measured deterministic and stochastic conversion boundaries on the Bloch sphere match theoretical predictions for both pure and mixed inputs. With assistance, Bob can reach the full Bloch sphere (stochastically). For noisy shared states (Werner), experimental deterministic boundaries align with theory; stochastic postselection does not extend beyond the predicted limits.

The work fully resolves the single-shot stochastic state-conversion problem for qubits under incoherent operations by characterizing the probability-achievable region and providing optimal success probabilities. The geometrical description clarifies how coherence (distance from the incoherent axis) and population imbalance constrain conversions, revealing nontrivial discontinuities in achievable probabilities when reducing success thresholds. Robustness-based and rank-based monotones furnish necessary conditions that extend beyond qubits. In distributed settings, the study shows that classical assistance from a correlated partner substantially enhances conversion capabilities, with exact optimality results for pure two-qubit inputs and Werner states. This formalizes the operational advantage of correlations for coherence manipulation under LQICC. Linking single-shot probabilities to asymptotic rates, the results yield practical lower bounds that can be tight, improving upon previous bounds in significant parameter regions. The identification of families with unit-rate conversions and of extremal states in the (Ca,Cc) plane advances understanding of coherence theory’s irreversibility and resource interconversion efficiency. Experimentally, the realization of optimal non-unitary SIOs with linear optics confirms the theoretical predictions quantitatively, demonstrating that current photonic technologies can implement optimal coherence transformations and assisted protocols. This validates resource-theoretic prescriptions and provides a benchmark platform for more complex, higher-dimensional transformations.

This paper provides a complete solution to stochastic incoherent state conversion for single-qubit states: IO and SIO are equivalent in power, the reachable set at fixed success probability is characterized geometrically, and closed-form optimal probabilities are derived. In distributed scenarios, the authors introduce and solve assisted incoherent conversion for pure two-qubit states and Werner states, proving that correlations strictly enhance Bob’s capabilities. They connect single-shot and asymptotic regimes by establishing tight bounds on conversion rates, identify classes with unit conversion rate, and chart the extremal trade-off between distillable coherence and coherence cost. An experimental linear-optical implementation realizes non-unitary SIOs and achieves optimal conversion probabilities, both with and without assistance, corroborating the theory. Future directions include: extending optimal assisted conversion results to general mixed shared states; exploring higher-dimensional systems and other operation classes; integrating these protocols in larger quantum networks; and developing more general experimental modules for multi-qubit or CV coherence transformations.

• Most exact optimality results are restricted to qubits; higher-dimensional systems are addressed only via necessary conditions (e.g., robustness and rank monotones), not full characterizations.
• Assisted conversion is solved optimally for pure two-qubit states and Werner states; general mixed shared states remain open.
• Experimental demonstrations, while high-fidelity, are subject to imperfections (e.g., non-ideal beam splitters, finite visibility, dephasing control), which may slightly reduce achievable fidelities and count rates.
• Theorems rely on IO/SIO as the operational class; results may differ under alternative free operation sets in coherence theory.