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Trace-distance measure of coherence
Swapan Rana, Preeti Parashar, Maciej Lewenstein
TL;DR
The paper asks whether trace-distance coherence is a strong monotone and how it compares with other coherence measures. It analyzes qubit, X-state, pure-state, and arbitrary-state cases using analytic arguments and a semidefinite program, finding strong monotonicity for qubits and X-states while other lp- and Schatten-p-based measures fail for all p∈(1,∞).
Problem
The paper investigates whether trace-distance coherence satisfies strong monotonicity for all states and how it relates to established coherence measures.
Method
The paper derives analytic results for qubits and X-states, provides a semidefinite program for arbitrary states, and studies relations among norm- and relative-entropy-based measures.
Results
Trace-distance coherence is strongly monotone for qubits and X-states, while Clp and Cp violate strong monotonicity for all p∈(1,∞).
Takeaways & Limitations
The results support trace-distance coherence as a valid measure in the qubit and X-state settings and provide an SDP for evaluating it on arbitrary states.
Abstract
from arXiv · showhide
We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, $X$ states. An expression for the trace distance coherence for all pure states and a semi definite program for arbitrary states is provided. We also explore the relation between $l_1$-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both $l_p$-norm- and Schatten-$p$-norm-based measures violate the (strong) monotonicity for all $p\in(1,\infty)$.
I. INTRODUCTION
The paper studies coherence measures within a resource-theoretic framework, focusing on whether trace-distance coherence satisfies strong monotonicity and how coherence quantifiers relate. It establishes results for qubits and X-states, proposes an SDP for arbitrary states, and identifies failures of other norm-based measures.
- Coherence corresponds to off-diagonal density-matrix elements in a computational or measurement-selected basis.
- Strong monotonicity requires coherence not to increase on average under incoherent operations and has implications for additivity questions.
- Trace distance was proposed as a coherence measure, with the open question of whether Ctr(ρ) satisfies strong monotonicity for all states.
- Cl1 and Cr were known strong monotones for all states, but Cl1 lacked an exact entanglement analog or physical interpretation, motivating their interrelation.
- The paper studies trace-distance coherence for qubits and general states, relates Cl1 to Cr, and tests other norm-based candidates.
A. Qubit and X-states
For 2 × 2 matrices, the closest diagonal matrix in trace norm is the diagonal part, making trace-distance coherence equal to the l1 norm. The same conclusion extends to X-shaped matrices and establishes strong monotonicity in these classes.
- Qubits: For any 2 × 2 matrix A, the closest diagonal matrix in trace norm is diag(A), so Ctr(A) = Cl1(A).
- Qubits: The qubit result requires no normality or positivity restrictions on A or its closest diagonal matrix.
- X-states: The X-state result improves an earlier theorem by applying directly to direct sums of qubits.
- Qubits: Strong monotonicity holds for Ctr(ρ) for any 2 × 2 matrix ρ, regardless of the dimensions of the Kraus operators.
- X-states: For X-shaped matrices, whose nonzero entries lie only on the diagonal and anti-diagonal, the nearest diagonal matrix is diag(X), yielding Ctr(X) = Cl1(X).
B. Pure states
For pure states, the nearest incoherent state is straightforward for qubits but becomes difficult to characterize analytically for higher dimensions. The analysis reduces the problem to minimizing the largest root of a degree-d polynomial.
- Higher-dimensional difficulty: Beyond qubits, the intuitively expected diagonal state need not be closest; for a pure qutrit, diag{1/2, 1/2, 0} is closer.The example uses |ψ⟩ = 2/3|0⟩ + 2/3|1⟩ + 1/3|2⟩.
- Optimization structure: For a pure state, the trace distance is twice the sole positive eigenvalue of H = |ψ⟩⟨ψ| − δ, minimized over diagonal δ.Because H is traceless and has exactly one positive eigenvalue, its trace norm equals twice that eigenvalue.
- Optimization structure: The characteristic equation is a degree-d polynomial whose largest real root must be minimized over the diagonal entries of δ.Only for d = 2 do the roots have a simple closed form.
- Higher-dimensional difficulty: For d > 2, there is generally no simple characterization of the largest root and therefore no simple explicit formula for Ctr.The authors note that even pure qutrits are almost intractable analytically.
- Equivalent formulation: The optimization can nevertheless be reformulated as an equivalent problem whose solutions map between the optimal diagonal state and auxiliary variables.The correspondence is established in both directions for the formulations in Eqs. (9) and (10).
C. Arbitrary states
For arbitrary states, the paper formulates trace distance coherence as a semidefinite program. Numerical tests on random states found no strong-monotonicity violations, motivating a conjecture for all states.
- Semidefinite program: Ctr(ρ) for any arbitrary state ρ can be calculated as the optimal value of a semidefinite program.The formulation uses decompositions of Hermitian matrices into positive semidefinite parts and minimizes their combined trace.
- Semidefinite program: The SDP constrains the auxiliary matrices and δ to be positive semidefinite, with δ diagonal.These constraints encode the incoherent-state optimization.
- Strong monotonicity: Random-state tests using the SDP found no examples violating strong monotonicity, leading the authors to conjecture that Ctr is strongly monotone for all states.The incoherent channels were not generated uniformly.
III. RELATION BETWEEN Cl1 AND Cr
The paper studies how the l1-norm coherence measure relates to relative entropy coherence. Relative entropy coherence has an analytic entropy expression, while the section develops inequalities connecting it to Cl1.
- Definitions and motivation: Relative entropy coherence is defined analogously to relative entropy of entanglement and can be evaluated as Cr(ρ) = S(ρdiag) − S(ρ).Here S(ρ) is the von Neumann entropy using logarithm base 2.
- Definitions and motivation: The section aims to derive interrelations between the l1-norm-based measure Cl1 and relative entropy coherence Cr.The comparison is motivated by established relations between analogous entanglement measures.
A. Pure states
For pure states, the paper derives bounds relating Cl1 and Cr, including Cl1 ≥ Cr in every dimension. The bound is tight for maximally coherent states and improves a logarithmic-negativity bound on distillable entanglement.
- Pure-state inequalities: For pure qubit states, Cl1 ≥ Cr, matching the known upper bound for binary entropy.The relation is presented as the qubit specialization of the pure-state comparison.
- Tightness and visualization: Equality in the relevant bound holds for maximally coherent states in any dimension.For qutrits, the inequalities can be visualized over the two-parameter probability representation of the state.
- Pure-state inequalities: For all pure states, Cl1 ≥ Cr provides an independent inequality relating l1-norm and relative entropy coherence.The paper notes that this bound is generally not sharp.
- Entanglement connection: For pure states, Eq. (16) improves the known bound on distillable entanglement expressed through logarithmic negativity.The comparison uses the correspondence between logarithmic negativity and relative entropy of entanglement for pure states.
B. Arbitrary states
The paper derives bounds connecting Cl1, Ctr, and Cr, obtaining a sharp qubit bound while conjecturing a stronger inequality for all states.
- Cl1 provides an upper bound on Ctr, yielding rough bounds for mixed-state relations.
- Fannes’s inequality does not produce a useful bound for the mixed-state analysis.
- The Fannes–Audenaert bound sharpens the Ctr–Cr relation but is not monotonic in Ctr, preventing its application to Cl1.
- The proof uses an inequality between Holevo information and trace norm, together with diagonal-unitary constructions.
- For qubits, the resulting inequality is sharp and coincides with the pure-state bound.
- Numerical evidence motivates the conjecture Cl1 ≥ Cr for all states, despite an unresolved multiplicative factor in the derived bound.
IV. ALL OTHER lp-NORM AND SCHATTEN-p-NORM
The paper shows that l_p- and Schatten-p-norm coherence measures fail monotonicity properties for p∈(1,∞), including explicit strong-monotonicity violations and higher-dimensional counterexamples.
- For every p∈(1,∞), both Clp and Cp have states violating strong monotonicity.
- The authors note that an earlier apparent l2 violation instead satisfies strong monotonicity for the associated channel and all relevant qutrit states.
- A constructed choice violates the l_p strong-monotonicity inequality for all p∈(2,∞).
- The same counterexample transfers from Clp to Cp for Hermitian 2×2 matrices, so Schatten-p strong monotonicity also fails for all p∈(1,∞).
- The failure is stronger than strong-monotonicity violation: for p∈(1,∞), neither Cp nor Clp is a monotone.
- The monotonicity counterexample requires dimension higher than two, because Cp is a monotone for all qubit states and p∈[1,∞).
- Tensoring a coherent state with the maximally mixed state decreases Cp and Clp, directly producing monotonicity violations.
V. DISCUSSION AND CONCLUSION
The discussion contrasts strong-monotonicity requirements for coherence and entanglement, and connects a conjectured Cl1–Cr inequality to a possible physical interpretation of Cl1.
- Strong monotonicity is mandatory for coherence measures but only an extra feature for entanglement measures.
- For entanglement, strong monotonicity is linked to local-unitary invariance and a flag condition; trace-norm factorization would give Ctr the flag condition if strong monotonicity holds.
- Trace-distance entanglement remains unresolved because the closest separable state generally cannot be determined explicitly.
- Cl1 has been connected to unambiguous state discrimination, but the conjecture Cl1(ρ) ≥ Cr(ρ) would provide a stronger proposed interpretation.
- If the conjecture holds universally, Cl1 would analogously upper-bound distillable coherence, which equals Cr(ρ) for all ρ.