---
title: Basis-Independent Coherence in Quantum Theory
url: https://www.emergentmind.com/topics/basis-independent-coherence
type: topic
---

# Basis-Independent Coherence in Quantum Theory

Searching arXiv for the cited paper and closely related work on basis-independent coherence.
Basis-independent coherence is a family of quantum-coherence concepts designed to remove, or sharply reduce, the arbitrariness associated with choosing a reference basis. In the standard resource-theoretic formulation, coherence is defined relative to a fixed basis, so the same density operator may appear coherent or incoherent depending on representation. Basis-independent approaches instead quantify intrinsic quantumness by comparing states to basis-invariant reference objects, most commonly the maximally mixed state, or by asking whether a family of states admits a common diagonalizing basis. In relativistic and curved-spacetime settings, where preferred bases are often observer-dependent, this shift is especially consequential. In de Sitter spacetime, a recent formulation decomposes basis-independent coherence of two comoving detectors into total, collective, localized, and global parts, and studies how these quantities vary with Gibbons–Hawking temperature and vacuum squeezing [2510.02581].

## 1. Conceptual foundations

The primary motivation for basis-independent coherence is that conventional coherence measures are basis-dependent, unlike entanglement or discord, and this basis dependence can be problematic when no canonical measurement basis exists. This issue is particularly acute in relativistic and curved-spacetime scenarios, where different observers may disagree about the physically preferred basis. Basis-independent measures are therefore intended to provide a more intrinsic characterization of quantumness [2510.02581].

One influential route defines basis-independent coherence by selecting the maximally mixed state as the unique basis-independent incoherent state. In this approach, the reference state is
\[
\rho_{\mathrm{inc}}=\frac{I}{d},
\]
and the associated relative-entropy form is
\[
C(\rho)=S\!\left(\rho\bigg\|\frac{I}{d}\right)=\log_2 d-S(\rho).
\]
This formulation appears as the intrinsic basis-independent quantum coherence measure and also as a central definition in later applications and reviews [1701.05110; 2412.07449].

A distinct but related perspective uses the quantum Jensen–Shannon divergence (QJSD), whose square root is treated as a metric. In that framework, basis-independent coherence is the distance between a state and the maximally mixed state,
\[
C(\rho)=\sqrt{ S\left( \frac{\rho + I/d}{2} \right) - \frac{ S(\rho) + \log_2 d }{2} }.
\]
This QJSD-based construction is explicitly unitary-invariant and supports geometric inequality relations and multipartite decompositions [1805.09263].

A further conceptual branch does not focus on a single state but on families of states. In “set coherence,” a finite family is set incoherent exactly when all members are diagonal in one common basis, equivalently when they commute pairwise. This yields a genuinely relational, basis-independent notion of coherence [2010.10406; 2605.10003]. This suggests that “basis-independent coherence” is not a single notion but a cluster of closely related constructions whose common goal is to isolate observer-independent or representation-independent quantum structure.

## 2. Principal mathematical formulations

Several non-equivalent measures appear in the literature, and the distinction between them is substantive rather than terminological.

The relative-entropy construction gives
\[
C(\rho)=\log_2 d-S(\rho),
\]
with zero attained only by the maximally mixed state. In that sense, all non-maximally mixed states possess basis-independent coherence [1701.05110; 2412.07449]. Closely related optimization results show that the maximal relative entropy of basis-dependent coherence over all bases is also
\[
C_r^{\max}(\rho)=\log d-S(\rho),
\]
with the optimum achieved in a basis mutually unbiased to the eigenbasis of \(\rho\), constructed through complex Hadamard matrices or the Fourier matrix [1611.01740; 1707.09617]. This establishes a precise bridge between basis optimization and intrinsic coherence.

The QJSD formulation instead defines coherence as a metric-like distance to \(I/d\),
\[
C(\rho)=\sqrt{ S\left( \frac{\rho + I/d}{2} \right) - \frac{ S(\rho) + \log_2 d }{2} }.
\]
Because the square root of the QJSD satisfies the triangle inequality, this choice supports decomposition relations in multipartite systems [1805.09263].

For multipartite states \(\eta\), the de Sitter analysis introduces three QJSD-based quantities:
\[
C_T(\eta) = \sqrt{ S\left( \frac{\eta + \eta_I}{2} \right) - \frac{1}{2}\left[ S(\eta) + \log_2 d \right] },
\]
\[
C_C(\eta) = \sqrt{ S\left( \frac{\eta + \pi_\eta}{2} \right) - \frac{1}{2}\left[ S(\eta) + S(\pi_\eta) \right] },
\]
\[
C_L(\eta) = \sqrt{ S\left( \frac{\pi_\eta + \eta_I}{2} \right) - \frac{1}{2}\left[ S(\pi_\eta) + \log_2 d \right] }.
\]
Here \(\eta_I=I/d\), and \(\pi_\eta\) is the tensor product of the marginals. The decomposition satisfies
\[
C_T^2=C_C^2+C_L^2.
\]
The same work also considers global coherence or quantum consonance,
\[
C_G(\eta) = \sum_{k_n, l_n} \left| \eta_{k_n, l_n}^c \prod_m (1-\delta_{k_m, l_m}) \right|,
\]
which is designed to isolate genuinely nonlocal coherence after all local coherence has been removed by local unitaries [2510.02581].

Set-based formulations use yet another mathematical language. A family \(\vec\rho=(\rho_1,\ldots,\rho_n)\) is set incoherent iff \([\rho_i,\rho_j]=0\) for all \(i,j\). A universal pairwise criterion is
\[
(\rho,\sigma):=\operatorname{Tr}(\rho^2\sigma^2)-\operatorname{Tr}(\rho\sigma\rho\sigma)=\frac{1}{2}\|[\rho,\sigma]\|_2^2,
\]
so vanishing of this fourth-order Bargmann gap for every pair is equivalent to the existence of a common incoherent basis [2605.10003].

## 3. Distribution in multipartite systems

A central development in basis-independent coherence theory is the decomposition of total coherence into contributions associated with subsystems and correlations. In the QJSD framework, total coherence is defined relative to the maximally mixed state, while collective coherence compares the full state to the tensor product of its marginals, and localized coherence compares that product state to the maximally mixed state [1805.09263].

These quantities satisfy triangle-type relations. In the general multipartite QJSD treatment,
\[
C(\rho)\leq C_c(\rho)+C_l(\rho),
\]
and also
\[
C(\rho)\leq C_I(\rho)+C_L(\rho),
\]
where intrinsic coherence \(C_I\) is defined relative to the closest separable state, and \(C_c\) uses the product of marginals [1805.09263]. In relativistic motion, the relation is written as
\[
C_C(\rho)+C_L(\rho)\geq C_T(\rho),
\]
and is shown to persist under Unruh-type evolution [2408.12370].

The de Sitter study sharpens this decomposition by combining \(C_T\), \(C_C\), and \(C_L\) with the global quantity \(C_G\), thereby separating total coherence into localized and collective contributions while also tracking a genuinely nonlocal sector [2510.02581]. According to that analysis, total coherence is mostly due to localized contributions, but collective and global coherence can become dominant in symmetric, closely coupled setups. This suggests that distributional analysis is not merely bookkeeping; it identifies which portion of coherence is operationally attributable to subsystem structure and which portion is tied to inter-detector quantumness.

Beyond state-based decompositions, set coherence introduces a different notion of distribution. There the relevant question is whether multiple states or measurements can be made jointly incoherent by a single basis choice. Robustness-based measures \(R\) and \(R_1\) quantify the irreducible coherence of a whole ensemble rather than of one density operator [2010.10406]. A plausible implication is that basis-independent coherence has both intra-state and inter-state forms, with the former tied to mixedness or metric distance and the latter tied to incompatibility of simultaneous diagonalization.

## 4. Basis-independent coherence in de Sitter spacetime

In de Sitter spacetime, the problem is studied using two Unruh–DeWitt detectors interacting with a massless scalar field in the Bunch–Davies vacuum and in squeezed \(\alpha\)-vacua. The thermal sector is parameterized by the inverse Gibbons–Hawking temperature \(\beta=1/T_H\), while the vacuum family is labeled by the real squeezing parameter \(\alpha\). The equilibrium detector state is obtained from an open-system master equation, and the basis-independent coherence measures are evaluated from the resulting late-time density matrix [2510.02581].

The de Sitter analysis reports several distinct behaviors as functions of \(\beta\), \(|\alpha|\), detector gap \(\omega\), and initial-state purity parameter \(\tau\). For squeezed vacua with small \(|\alpha|\), total coherence \(C_T\) and localized coherence \(C_L\) display a U-shaped dependence on \(\beta\): they are high at low \(\beta\), attain a minimum at intermediate \(\beta\), and then rise to a plateau at high \(\beta\). By contrast, collective coherence \(C_C\) and global coherence \(C_G\) display a nonmonotonic profile with a dip at low \(\beta\), a peak at intermediate \(\beta\), and asymptotic saturation. For large \(|\alpha|\), approaching the Bunch–Davies vacuum, the curves become monotonic: thermality suppresses coherence before stabilization [2510.02581].

The same work identifies vacuum squeezing as a major control parameter. Squeezed \(\alpha\)-vacua with small \(|\alpha|\) produce substantial enhancement of extractable coherence, especially collective and global coherence, whereas in the Bunch–Davies limit the system becomes thermal and coherence is initially suppressed by Gibbons–Hawking radiation but can recover at low temperature [2510.02581]. Larger detector gap \(\omega\) prolongs the coherence plateau, allowing coherence to persist over a broader temperature range, and coherence cannot be generated from a fully mixed initial state.

Within this setting, global coherence \(C_G\) is singled out as the most robust and operationally meaningful resource, quantifying genuinely nonlocal quantumness between the detectors [2510.02581]. This is one of the clearest cases in which basis-independent coherence is not only a state descriptor but also a resource diagnostic in relativistic quantum information.

## 5. Relativistic acceleration and noninertial frames

The behavior of basis-independent coherence under acceleration has been studied in both field-mode and detector models. For two modes of a free Dirac field observed by relatively accelerated observers, the coherence between modes \(A\) and \(B_I\) decreases with increasing acceleration but remains finite even in the infinite-acceleration limit; the coherence between \(A\) and \(B_{II}\) is nonzero already at zero acceleration; and the coherence between \(B_I\) and \(B_{II}\) remains constant, exhibiting a freezing phenomenon [2510.11329].

In that analysis, the measure is
\[
C(\rho)=\sqrt{ S\left( \frac{\rho + \rho_M}{2} \right) - \frac{S(\rho) + \log_2 d}{2} },
\]
with \(\rho_M=I/d\). The nonzero value for \(C(\rho_{AB_{II}})\) at zero acceleration contrasts with basis-dependent coherence, which typically vanishes there because the reduced state is diagonal in the computational basis [2510.11329]. This difference is often used to illustrate that basis-independent coherence can capture quantumness missed by basis-dependent off-diagonal criteria.

A detector-based relativistic-motion study examines total, collective, and localized coherence under acceleration and coupling strength. In that setting, both total and collective coherence decrease monotonically with increasing acceleration and coupling and vanish at finite acceleration, whereas localized coherence decreases with acceleration but reaches zero only in the infinite-acceleration limit; it also increases with coupling strength [2408.12370]. The persistence of localized coherence relative to collective coherence indicates a hierarchy of robustness under Unruh thermal noise.

Taken together, these results support a recurrent theme: basis-independent coherence is more stable than basis-dependent coherence under relativistic degradation, but its components need not behave uniformly. Nonlocal sectors may be fragile in some models and robust in others, depending on the partition, the vacuum structure, and the precise coherence notion used [2510.11329; 2408.12370].

## 6. Relations to other quantum resources and operational tasks

Several works embed basis-independent coherence within a broader hierarchy of quantum resources. Using relative entropy, one formulation gives
\[
C(\rho)\geq C^{\mathcal B}(\rho)\geq D(\rho)\geq E(\rho),
\]
where \(C(\rho)=\log_2 d-S(\rho)\) is basis-independent coherence, \(C^{\mathcal B}(\rho)\) is basis-dependent coherence in basis \(\mathcal B\), \(D(\rho)\) is quantum discord, and \(E(\rho)\) is entanglement [2412.07449]. The same work proves the decomposition
\[
C(\rho)=C^{\mathcal B}(\rho)+C[\Phi^{\mathcal B}(\rho)],
\]
with \(\Phi^{\mathcal B}\) the dephasing channel in basis \(\mathcal B\). This identifies basis-independent coherence as the sum of basis-dependent coherence and the basis-independent coherence that survives dephasing.

An earlier “basis-free coherence” notion, defined as the minimum of basis-dependent relative entropy of coherence over all local bases,
\[
\mathcal C^{\mathrm{free}}(\rho)=\min_{\vec U}\mathcal C(\vec U\rho \vec U^\dagger),
\]
was shown to be exactly equal to relative-entropy quantum discord [1506.01773]. This is conceptually distinct from the maximally-mixed-state reference approach, where all non-maximally mixed states are coherent. The literature therefore contains two nonequivalent “basis-free” programs: one identifies irreducible coherence under local basis optimization with discord, while another identifies intrinsic coherence with distance from maximal mixing. Confusing these constructions is a common misconception.

Operational interpretations are diverse. In set coherence, robustness measures have an explicit meaning in discrimination games and quantify the advantage a set of states or measurements offers over incoherent ones [2010.10406]. In graph-based approaches, overlaps \(r_{ij}=\operatorname{Tr}(\rho_i\rho_j)\) are constrained by classicality polytopes, and violation of facet inequalities witnesses basis-independent coherence of a set [2209.02670]. In avian-inspired magnetic sensing, basis-independent coherence of the initial system–environment state, understood as non-maximal mixedness, is necessary for directional sensing in the stated model [2011.15016]. In quantum thermodynamics, the maximum average work extractable from \(\rho\) at temperature \(T\) is
\[
W=k_B T(\log_2 d-S(\rho))=k_B T\,C(\rho),
\]
linking basis-independent coherence directly to extractable work [2412.07449].

## 7. Extensions, trade-offs, and open issues

Several recent directions extend basis-independent coherence beyond its original formulations. One line studies coherence–mixedness trade-offs. For relative-entropy coherence optimized over bases,
\[
C_r^{\max}(\rho)+S(\rho)=\ln d,
\]
while for \(l_2\)-norm and Wigner–Yanase skew-information variants there are analogous equality-type basis-independent trade-offs with linear-entropy or geometric mixedness measures [2405.14337]. These relations quantify how environmental noise constrains the attainable intrinsic coherence of a state.

Another line concerns critical phenomena and many-body systems. QJSD-based basis-independent coherence has been used to analyze trade-off relations, singular behavior, scaling behavior, and monogamy in Ising systems, where local and collective coherence diagnose phase transitions and critical exponents without basis ambiguities [2001.10714]. This suggests that basis-independent coherence is useful not only for foundational consistency but also for diagnosing collective behavior when the relevant local basis is not fixed across phases.

Set coherence has also been refined through low-order Bargmann invariants. For qubits, second-order data suffice to decide pairwise set coherence; for qutrits, complete third-order data are sufficient; for \(d\geq 4\), fourth-order ordering-sensitive invariants provide the first universal pairwise criterion [2605.10003]. This development places basis-independent coherence alongside invariant theory and noncommutativity diagnostics.

A persistent controversy concerns terminology. Some papers use “basis-independent coherence” to mean distance from the maximally mixed state [1701.05110; 2412.07449]; others use “basis-free coherence” for the minimum over local basis choices and obtain discord [1506.01773]; still others reserve basis-independence for set coherence of multiple states [2010.10406; 2605.10003]. These are not interchangeable definitions. A plausible implication is that future work will need sharper taxonomies separating purity-like intrinsic coherence, irreducible local-basis coherence, and relational coherence of ensembles.

In de Sitter spacetime, the recent decomposition into total, collective, localized, and global contributions strengthens the case that basis-independent coherence can function as a robust relativistic quantum resource, especially when non-thermal squeezing is present [2510.02581]. More broadly, the literature indicates that basis-independent coherence is best understood not as a single scalar invariant but as a structured family of observer-robust coherence notions tailored to different operational and geometric questions.

Source: https://www.emergentmind.com/topics/basis-independent-coherence