Average Steering Coherence in Quantum Systems

• Average Steering Coherence (ASC) is a coherence functional that quantifies the average coherence induced on Bob’s subsystem via optimal local measurements by Alice.
• ASC leverages various coherence measures (e.g., relative entropy, ℓ1-norm) and is bounded by classical correlations, linking measurement-induced effects to quantum correlations.
• Operational formulations of ASC enable reconstruction using Pauli correlators, making it a robust witness for steering and non-local advantages in bipartite quantum systems.

Searching arXiv for recent and foundational papers on Average Steering Coherence and closely related formulations. Average Steering Coherence (ASC) denotes a family of steering-based coherence functionals for bipartite quantum systems. In the most direct formulation, it is the average coherence of Bob’s conditional states after Alice performs local measurements and communicates the outcomes; in the maximized form studied as steering-induced coherence, it is the maximal average coherence that Alice can remotely create on Bob, evaluated in the eigenbasis of $\rho_B=\mathrm{Tr}_A(\rho_{AB})$ and, when needed, minimized over degenerate eigenbases (Hu et al., 2015). Closely related constructions appear as the averaged conditional-state coherence underlying non-local advantage of quantum coherence (NAQC) (Mondal et al., 2015), as measurement-induced average coherence and maximal extra average coherence (Zhang et al., 2017), and as an operational steering witness admitting closed forms for two-qubit $X$ states with maximally mixed marginals (Thiyagarajan et al., 10 Jun 2026). The same acronym is not standard in activation-steering work on LLMs, where papers instead use judge-based coherence scores and, at most, suggest ASC-like averages as auxiliary aggregations rather than defining ASC as a formal metric (Cao et al., 7 Jun 2026).

1. Terminological scope and canonical formulations

The term is not used identically across the literature. In "Extracting quantum coherence via steering" (Hu et al., 2015), the central object is the steering-induced coherence (SIC), denoted $\bar{\mathcal C}(\rho_{AB})$, which is the maximal average coherence induced on Bob by Alice’s local projective measurements. In "Non-Local Advantage of Quantum Coherence" (Mondal et al., 2015), the paper does not introduce the term “Average Steering Coherence” explicitly, but it defines NAQC via averaging the coherence of Bob’s conditional states over Alice’s measurement settings and outcomes, with a normalization factor $1/2$. In "The classical correlation limits the ability of the measurement-induced average coherence" (Zhang et al., 2017), ASC corresponds to the measurement-induced average coherence $$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$. In "Quantum Correlation Hierarchy and Teleportation in Dephased Hydrogen Hyperfine System" (Thiyagarajan et al., 10 Jun 2026), ASC is an operational steering witness based on optimized averages of $\ell_1$-coherence.

Formulation	Defining object	Characteristic expression
SIC	Maximal induced coherence on Bob	$\bar{\mathcal C}(\rho_{AB})$
NAQC functional	Averaged coherence over settings, outcomes, and complementary bases	$$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$
Measurement-induced average coherence	Average coherence of Bob’s steered ensemble	$\sum_k p_k C(\rho_B(k))$
Operational ASC witness	Optimized Pauli-triad average of conditional $\ell_1$-coherence	$X$0

These formulations are not identical, but they share a common operational core: local measurement on one subsystem generates a conditional ensemble on the other subsystem, and the coherence of that ensemble is averaged, and often optimized, to quantify steering-enabled coherence generation.

2. Bipartite steering framework and induced coherence

For a bipartite state $X$1, Alice performs a local measurement $X$2 on $X$3. The outcome $X$4 occurs with probability

$X$5

and Bob’s normalized conditional state is

$X$6

In the SIC formulation, the reference incoherent basis on Bob is chosen to be the eigenbasis $X$7 of $X$8. When $X$9 is degenerate, the definition takes an infimum over all eigenbases $\bar{\mathcal C}(\rho_{AB})$0 satisfying $\bar{\mathcal C}(\rho_{AB})$1, where

$\bar{\mathcal C}(\rho_{AB})$2

The resulting steering-induced coherence is

$\bar{\mathcal C}(\rho_{AB})$3

Because $\bar{\mathcal C}(\rho_{AB})$4 by construction, the “net induced coherence” variant coincides with this quantity (Hu et al., 2015).

The optimization can be restricted to rank-1 projective measurements on $\bar{\mathcal C}(\rho_{AB})$5. This point is structurally important: ASC in this sense is not merely a property of $\bar{\mathcal C}(\rho_{AB})$6, but of $\bar{\mathcal C}(\rho_{AB})$7 together with the admissible local measurement class and the reference basis on the steered subsystem.

A more general averaged formulation dispenses with the eigenbasis construction and defines

$\bar{\mathcal C}(\rho_{AB})$8

where $\bar{\mathcal C}(\rho_{AB})$9 can be basis-dependent, such as relative entropy of coherence, or basis-free, such as total coherence. In this framework, the extra ASC generated by a given measurement is

$1/2$0

and the maximal extra ASC is

$1/2$1

For any convex coherence measure $1/2$2, one has $1/2$3, so $1/2$4 (Zhang et al., 2017).

3. Coherence measures, structural properties, and upper bounds

The main coherence measures used in the ASC literature are the relative entropy of coherence,

$1/2$5

the $1/2$6-norm of coherence,

$1/2$7

and, in basis-free settings, the total coherence

$1/2$8

(Hu et al., 2015, Zhang et al., 2017).

Several structural properties are established for SIC. It is nonnegative, and $1/2$9 iff $$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$0 is $$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$1-side classical, i.e.

$$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$2

in some $$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$3. It is monotone under local CPTP maps on $$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$4, and its behavior under Bob’s incoherent selective operations and convex mixing follows from the corresponding axioms of the underlying coherence measure (Hu et al., 2015).

The principal resource-theoretic upper bound links SIC to one-sided measurement-induced disturbance (MID). For a distance $$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$5 satisfying the axioms used in the paper,

$$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$6

and the main theorem gives

$$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$7

Thus, the average coherence Alice can remotely create on Bob is bounded by the one-sided quantumness of correlations measured by MID (Hu et al., 2015).

The bound is tight in important cases. For maximally correlated states

$$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$8

the relative-entropy version satisfies

$$C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k C(\rho_B(k))$$9

and pure bipartite states are included as a special case. For any two-qubit state,

$\ell_1$0

where $\ell_1$1 uses trace distance (Hu et al., 2015).

A distinct but complementary result concerns what limits the extra average coherence. For both basis-dependent ASC with $\ell_1$2 and basis-free ASC with $\ell_1$3, the achievable extra ASC is upper bounded by the Henderson–Vedral classical correlation

$\ell_1$4

The paper’s central conclusion is that classical correlation $\ell_1$5, not quantum correlation, limits the extra ASC. Quantitatively,

$\ell_1$6

and likewise after maximizing over measurements. For pure states, the extra basis-free ASC equals the entanglement entropy $\ell_1$7 (Zhang et al., 2017).

4. Complementarity, NAQC, and steering inequalities

A second major line of work treats ASC as the averaged coherence of steered conditional states across mutually unbiased bases (MUBs), especially the Pauli bases for qubits. For a single-qubit state

$\ell_1$8

the coherence complementarity relations are

$\ell_1$9

$\bar{\mathcal C}(\rho_{AB})$0

$\bar{\mathcal C}(\rho_{AB})$1

These constants become the LHS bounds for averaged coherence functionals in steering scenarios (Mondal et al., 2015).

For a general two-qubit state $\bar{\mathcal C}(\rho_{AB})$2, Alice measures $\bar{\mathcal C}(\rho_{AB})$3 with outcomes $\bar{\mathcal C}(\rho_{AB})$4, generating Bob’s conditional states $\bar{\mathcal C}(\rho_{AB})$5 with probabilities $\bar{\mathcal C}(\rho_{AB})$6. The NAQC functional corresponding to what is naturally called ASC is

$\bar{\mathcal C}(\rho_{AB})$7

with $\bar{\mathcal C}(\rho_{AB})$8 (Mondal et al., 2015).

For all local-hidden-state models,

$\bar{\mathcal C}(\rho_{AB})$9

Violation certifies NAQC and therefore steering. The implication is one-way: NAQC implies steerability, but not all steerable states achieve NAQC (Mondal et al., 2015).

For Werner states,

$$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$0

the resulting ASC values are explicit: $$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$1

$$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$2

$$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$3

The NAQC thresholds are therefore $$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$4 for $$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$5, $$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$6 for relative entropy, and $$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$7 for skew information, whereas the same state is steerable already for $$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$8 (Mondal et al., 2015).

The experimental study "Experimental demonstration of complementarity relations between quantum steering criteria" (Yang et al., 2020) tested related aggregate ASC sums

$$\frac{1}{2}\sum_{k,a}\sum_{i\neq k} p(a|k)\,C^{(i)}_M(\rho_B^{a|k})$$9

together with

$\sum_k p_k C(\rho_B(k))$0

For unsteerable states,

$\sum_k p_k C(\rho_B(k))$1

with $\sum_k p_k C(\rho_B(k))$2, $\sum_k p_k C(\rho_B(k))$3, and $\sum_k p_k C(\rho_B(k))$4, while the three-setting complementarity relation

$\sum_k p_k C(\rho_B(k))$5

holds for all two-qubit states. The experiment verified that skew-information coherence gives the strongest steering detection among the three measures (Yang et al., 2020).

5. Closed forms, dynamical hierarchy, and correlator reconstruction

A recent operational treatment defines ASC for two-qubit systems as

$\sum_k p_k C(\rho_B(k))$6

where $\sum_k p_k C(\rho_B(k))$7 rotates Alice’s measurement triad to the optimizing Pauli axes. For the entire two-qubit $\sum_k p_k C(\rho_B(k))$8-state family with maximally mixed marginals, the paper proves the simplification

$\sum_k p_k C(\rho_B(k))$9

with

$\ell_1$0

The paper adopts the steering-witness threshold $\ell_1$1 (Thiyagarajan et al., 10 Jun 2026).

In the dephased hydrogen hyperfine model, the electron and proton spins evolve under local Markovian dephasing with

$\ell_1$2

so that

$\ell_1$3

This yields a strict hierarchy

$\ell_1$4

where $\ell_1$5 is concurrence and $\ell_1$6 is trace-distance measurement-induced nonlocality. Entanglement is the most fragile resource; trace MIN can exhibit dephasing-immune freezing when $\ell_1$7; ASC is the most robust quantity and persists longest in every scenario studied (Thiyagarajan et al., 10 Jun 2026).

The same paper identifies four distinct dynamical regimes for $\ell_1$8: steerable and entangled, frozen-MIN entangled, non-steerable entangled, and frozen-discord regime. Operationally, ASC is directly reconstructible from three Pauli correlators,

$\ell_1$9

so no full state tomography is required. This correlator representation also makes the hierarchy experimentally accessible in spin systems (Thiyagarajan et al., 10 Jun 2026).

6. Vanishing conditions, interpretive caveats, and cross-domain ambiguity

Vanishing ASC has different meanings in different formulations. For SIC, $X$00 iff the state is $X$01-side classical (Hu et al., 2015). For maximal extra basis-dependent ASC with relative-entropy coherence, $X$02 iff $X$03 is block-diagonal in Bob’s computational basis or is a product state; for basis-free ASC with total coherence, $X$04 iff $X$05 is a product state (Zhang et al., 2017). These distinctions matter because SIC, NAQC-style ASC, and measurement-induced average coherence coincide only in special settings.

A recurring misconception is to identify ASC with steerability itself. The NAQC analysis explicitly shows that not all steerable states can achieve such advantage: Werner states furnish a standard example, being steerable for $X$06 while violating NAQC bounds only at substantially larger $X$07 (Mondal et al., 2015). Another misconception is that quantum correlation alone controls average remotely induced coherence. The measurement-induced average coherence results show instead that the upper bound is classical correlation $X$08, and that quantum correlation is neither sufficient nor necessary for nonzero extra ASC within a given measurement (Zhang et al., 2017).

The phrase also creates a cross-disciplinary ambiguity. In activation-steering studies of LLMs, papers on emergent misalignment and open-ended generation do not define “Average Steering Coherence.” The closest metric in one case is a per-response coherence score $X$09 from an automatic judge, with the paper using the threshold $X$10 to gate a coherent harmful EM rate,

$X$11

If one needs ASC in that framework, the paper states that the natural quantity is the mean coherence

$X$12

but also states that the authors do not report ASC (Cao et al., 7 Jun 2026). A related activation-steering paper likewise states that it does not define a metric called ASC and instead reports judged coherence $X$13, cross-entropy under the aligned model, embedding similarity, and repetition metrics (Herbster et al., 9 Apr 2026). In consequence, ASC is a standard technical notion in quantum steering and coherence theory, but not a standardized term in activation steering for LLMs.