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Measuring Quantum Coherence with Entanglement
Alexander Streltsov, Uttam Singh, Himadri Shekhar Dhar, Manabendra Nath Bera, Gerardo Adesso
TL;DR
The paper addresses the limited quantitative characterization of quantum coherence as an operational resource by relating it to entanglement through incoherent operations. It defines coherence measures from generated bipartite entanglement and proves their monotonicity, convexity, and qubit formulas for the geometric measure.
Problem
Quantitative characterization of coherence as an operational resource remains limited, motivating a systematic connection between coherence and entanglement.
Method
The paper defines coherence through maximal entanglement generated with an incoherent ancilla, using generalized CNOT operations for geometric measures.
Results
The resulting quantities are coherence monotones, geometric coherence is convex, and arbitrary single-qubit geometric coherence has a closed-form expression.
Takeaways & Limitations
The results establish a quantitative operational connection between coherence and entanglement for distance-based quantifiers under contractive quantum operations.
Abstract
from arXiv · showhide
Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for quantum technologies.
Appendix: Supplemental Material · Measuring Quantum Coherence with Entanglement · 1. Proof of monotonicity (C3) in Theorem 3
The appendix proves property (C3): the coherence quantifier defined from any entanglement monotone E does not increase on average under selective incoherent operations. The proof uses incoherent ancillas and a contradiction argument to establish that CE is a coherence monotone for every entanglement monotone E.
- 1. Proof of monotonicity (C3) in Theorem 3: The proof considers selective incoherent operations with outcome probabilities p_i, conditional states σS_i, and incoherent Kraus operators K_i.The states satisfy σS_i = K_iρS K†_i/p_i, with p_i = Tr[K_iρS K†_i].
- 1. Proof of monotonicity (C3) in Theorem 3: By definition, entanglement generated between the system and an incoherent ancilla cannot exceed CE under any incoherent operation ΛSA.The same upper-bound statement remains valid after adding an incoherent particle B in state |0⟩⟨0|B and applying a tripartite incoherent operation ΛSAB.
- 1. Proof of monotonicity (C3) in Theorem 3: The argument proceeds by contradiction: violating the desired monotonicity inequality would imply a violation of the defining upper bound for CE.The contradiction construction assumes a suitable family of incoherent operations exists for sufficiently large ancilla dimension dA.
- 1. Proof of monotonicity (C3) in Theorem 3: An additional incoherent particle B is introduced, together with a general relation valid for every entanglement monotone E.The conditional states σS_i arise from ρS through incoherent Kraus operators, allowing the proof to apply the corresponding relation to the averaged outputs.
- 1. Proof of monotonicity (C3) in Theorem 3: The relevant state is represented as the output of a tripartite incoherent operation on ρS ⊗ |0⟩⟨0|A ⊗ |0⟩⟨0|B.The construction introduces Kraus operators Mij from incoherent operators Lij associated with the original operation ΛSA.
- 1. Proof of monotonicity (C3) in Theorem 3: The constructed Mij are incoherent Kraus operators, and the resulting ΛSAB satisfies the required relation Eq. (A.9).Substituting Eq. (A.9) into the preceding inequality yields the contradiction needed for the proof.
- 1. Proof of monotonicity (C3) in Theorem 3: The contradiction establishes property (C3) for CE, proving that CE is a coherence monotone for any entanglement monotone E.This is the appendix’s final conclusion for the monotonicity proof.
2. Proof of convexity (C4) in Theorem 3
The section proves that the coherence quantifier C_E is convex whenever E is a convex entanglement measure. The proof takes suprema over incoherent operations and the ancilla-dimension limit, then concludes convexity.
- Proof of convexity (C4) in Theorem 3: C_E is shown to be convex for any convex entanglement measure E.This establishes the target convexity property for the quantifier defined in Eq. (A.1).
- Proof of convexity (C4) in Theorem 3: Convexity of E supplies the inequality used to analyze the coherence quantifier.The argument begins from convexity of the entanglement quantifier E for the relevant probabilities and states.
- Proof of convexity (C4) in Theorem 3: The proof takes the supremum over all incoherent operations Λ_S A and the limit d_A →∞ on both sides of the inequality.These operations and the ancilla-dimension limit yield the stated result in Eq. (A.16).
- Proof of convexity (C4) in Theorem 3: Applying the supremum and limit to each term individually cannot decrease the right-hand side, completing the proof of convexity in Eq. (A.14).This final monotonicity step is combined with Eq. (A.16) to establish the result.
3. Geometric entanglement and coherence
This section shows that the geometric entanglement–coherence bound can be saturated when the ancilla dimension is at least the system dimension, with the generalized CNOT achieving equality. The argument also extends to distance-based quantifiers under contractive distances and maximally correlated states.
- Saturation of the bound: The geometric entanglement and geometric coherence measures saturate the bound from Theorem 1.The proof establishes an incoherent operation that achieves equality for these measures.
- Geometric measures: Geometric entanglement is defined through fidelity maximization over separable states, while geometric coherence uses the analogous maximization over incoherent states.The geometric entanglement expression also coincides with a convex-roof construction for mixed states.
- Saturation of the bound: The saturation holds when the ancilla dimension satisfies d_A ≥ d_S, and the optimal operation is the generalized CNOT.The generalized CNOT is the operation defined in Eq. (9) of the main text.
- Proof of equality: For the generalized-CNOT output, geometric coherence is bounded above by geometric entanglement, while Theorem 1 supplies the reverse inequality needed for equality.The proof uses the corresponding separable maximally correlated state and an incoherent state with matching coefficients.
- Generalization: The reasoning extends to distance-based entanglement and coherence quantifiers for contractive distances and maximally correlated states.The geometric measures are included when the distance is D(ρ, σ) = 1 − F(ρ, σ).
4. Geometric entanglement for maximally correlated states
For any maximally correlated state, the section constructs a separable maximally correlated state that attains the state’s geometric entanglement bound. The proof uses optimal pure-state decompositions, closest product states, and matching upper and lower fidelity bounds.
- Existence and optimality: Any maximally correlated state admits a separable maximally correlated state satisfying the desired optimality equality.The candidate state is constructed from closest product states associated with an optimal decomposition.
- Construction: An optimal decomposition of a maximally correlated state consists of pure states that are linear combinations of product states |i⟩⊗|i⟩.The closest product states can therefore be chosen as |lk⟩⊗|lk⟩, with lk determined by the corresponding coefficient magnitude.
- Bounding argument: The constructed separable state provides an upper bound on the geometric entanglement because it is separable.The proof then applies strong concavity of the square-root fidelity to derive the corresponding lower bound.
- Conclusion: Combining the upper and lower bounds proves that the constructed separable maximally correlated state is optimal.This establishes the claimed equality for geometric entanglement.
5. Geometric coherence for arbitrary single-qubit states
For arbitrary single-qubit states, geometric coherence is obtained by applying the generalized CNOT to an incoherent ancilla and evaluating the resulting two-qubit state's geometric entanglement. The construction uses concurrence for the maximally correlated output state and yields a closed expression for geometric coherence.
- Optimal incoherent operation: The generalized CNOT optimally attains the maximization defining geometric coherence when geometric entanglement E_g is used.This applies to a single-qubit system, for which the output with the ancilla is a two-qubit state.
- Output state: Applying the CNOT to an arbitrary single-qubit state and an initially incoherent ancilla produces a maximally correlated two-qubit state.The input is expressed in a reference basis {|i⟩}.
- Concurrence and entanglement: The concurrence of the maximally correlated output state is evaluated to obtain its geometric entanglement.Geometric entanglement for any bipartite two-qubit state is computable in closed form from concurrence.
- Final formula: The resulting geometric coherence for an arbitrary single-qubit state is given by a closed expression.The expression is reported in the main text.