---
title: Average Steering Coherence in Quantum Systems
url: https://www.emergentmind.com/topics/average-steering-coherence-asc
type: topic
---

# Average Steering Coherence in 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 [1508.01978]. Closely related constructions appear as the averaged conditional-state coherence underlying non-local advantage of quantum coherence (NAQC) [1508.03770], as measurement-induced average coherence and maximal extra average coherence [1703.02737], and as an operational steering witness admitting closed forms for two-qubit \(X\) states with maximally mixed marginals [2606.11731]. The same acronym is not standard in activation-steering work on large language models, 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 [2606.08682].

## 1. Terminological scope and canonical formulations

The term is not used identically across the literature. In "Extracting quantum coherence via steering" [1508.01978], 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" [1508.03770], 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" [1703.02737], 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" [2606.11731], 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 | \(\max_{U_A}\frac{1}{3}\sum_{k=1}^3 C_{\ell_1}(\rho_{B|k})\) |

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 \(\rho_{AB}\), Alice performs a local measurement \(\{M_k\}\) on \(A\). The outcome \(k\) occurs with probability
\[
p_k=\mathrm{Tr}\big[(M_k\otimes I_B)\rho_{AB}\big],
\]
and Bob’s normalized conditional state is
\[
\rho_{B|k}=\frac{\mathrm{Tr}_A\big[(M_k\otimes I_B)\rho_{AB}\big]}{p_k}.
\]
In the SIC formulation, the reference incoherent basis on Bob is chosen to be the eigenbasis \(\mathbb E_B\) of \(\rho_B=\mathrm{Tr}_A(\rho_{AB})\). When \(\rho_B\) is degenerate, the definition takes an infimum over all eigenbases \(\mathbb E_B\) satisfying \(\Lambda_B^{\mathbb E_B}(\rho_B)=\rho_B\), where
\[
\Lambda_B^{\mathbb E_B}(\cdot):=\sum_j |e_j^B\rangle\langle e_j^B|(\cdot)|e_j^B\rangle\langle e_j^B|.
\]
The resulting steering-induced coherence is
\[
\bar{\mathcal C}(\rho_{AB})
\equiv
\inf_{\mathbb E_B}
\left[
\sup_{\{M_k\}\ \text{(projective on A)}}
\sum_k p_k\,C(\rho_{B|k},\mathbb E_B)
\right].
\]
Because \(C(\rho_B,\mathbb E_B)=0\) by construction, the “net induced coherence” variant coincides with this quantity [1508.01978].

The optimization can be restricted to rank-1 projective measurements on \(A\). This point is structurally important: ASC in this sense is not merely a property of \(\rho_{AB}\), but of \(\rho_{AB}\) 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
\[
C_{\mathrm{avg}}(M_A;\rho_{AB})=\sum_k p_k\,C(\rho_B(k)),
\]
where \(C\) 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
\[
\Delta C(M_A)=C_{\mathrm{avg}}(M_A;\rho_{AB})-C(\rho_B),
\]
and the maximal extra ASC is
\[
\Delta C_{\max}(\rho_{AB})=\max_{M_A}\Delta C(M_A).
\]
For any convex coherence measure \(C\), one has \(C_{\mathrm{avg}}(M_A;\rho_{AB})\ge C(\rho_B)\), so \(\Delta C(M_A)\ge 0\) [1703.02737].

## 3. Coherence measures, structural properties, and upper bounds

The main coherence measures used in the ASC literature are the relative entropy of coherence,
\[
C_r(\rho,\Xi)=S(\Delta(\rho))-S(\rho),
\]
the \(\ell_1\)-norm of coherence,
\[
C_{l_1}(\rho,\Xi)=\sum_{i\neq j}|\rho_{ij}|,
\]
and, in basis-free settings, the total coherence
\[
C_T(\rho)=\log d-S(\rho)
\]
[1508.01978, 1703.02737].

Several structural properties are established for SIC. It is nonnegative, and \(\bar{\mathcal C}(\rho_{AB})=0\) iff \(\rho_{AB}\) is \(B\)-side classical, i.e.
\[
\rho_{AB}=\sum_j \rho_j^A\otimes |e_j^B\rangle\langle e_j^B|
\]
in some \(\mathbb E_B\). It is monotone under local CPTP maps on \(A\), and its behavior under Bob’s incoherent selective operations and convex mixing follows from the corresponding axioms of the underlying coherence measure [1508.01978].

The principal resource-theoretic upper bound links SIC to one-sided measurement-induced disturbance (MID). For a distance \(D\) satisfying the axioms used in the paper,
\[
\mathcal Q_B^D(\rho_{AB})
:=
\inf_{\mathbb E_B:\ \Lambda_B^{\mathbb E_B}(\rho_B)=\rho_B}
D\!\left(\rho_{AB},(I_A\otimes \Lambda_B^{\mathbb E_B})(\rho_{AB})\right),
\]
and the main theorem gives
\[
\bar{\mathcal C}(\rho_{AB})\le \mathcal Q_B^D(\rho_{AB}).
\]
Thus, the average coherence Alice can remotely create on Bob is bounded by the one-sided quantumness of correlations measured by MID [1508.01978].

The bound is tight in important cases. For maximally correlated states
\[
\rho^{\mathrm{mc}}_{AB}=\sum_{i,j}\rho_{ij}|ii\rangle\langle jj|,
\]
the relative-entropy version satisfies
\[
\bar{\mathcal C}^{r}(\rho_{AB})=\mathcal Q_B^{r}(\rho_{AB})=S(\rho_B)-S(\rho_{AB}),
\]
and pure bipartite states are included as a special case. For any two-qubit state,
\[
\bar{\mathcal C}^{l_1}(\rho_{AB})=\mathcal Q_B^{t}(\rho_{AB}),
\]
where \(\mathcal Q_B^t\) uses trace distance [1508.01978].

A distinct but complementary result concerns what limits the extra average coherence. For both basis-dependent ASC with \(C_r\) and basis-free ASC with \(C_T\), the achievable extra ASC is upper bounded by the Henderson–Vedral classical correlation
\[
J_A(\rho_{AB})=S(\rho_B)-\min_{\{\Omega_A\}}\sum_i q_i S(\varrho_B(i)).
\]
The paper’s central conclusion is that classical correlation \(J_A\), not quantum correlation, limits the extra ASC. Quantitatively,
\[
\Delta C_P(M_A)\le \Delta C_T(M_A)\le J_A(\rho_{AB}),
\]
and likewise after maximizing over measurements. For pure states, the extra basis-free ASC equals the entanglement entropy \(S(\rho_B)\) [1703.02737].

## 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
\[
\rho=\frac{1}{2}(I+\vec r\cdot \vec \sigma),
\]
the coherence complementarity relations are
\[
\sum_{i=x,y,z} C_i^{l_1}(\rho)\le \sqrt{6},
\]
\[
\sum_{i=x,y,z} C_i^{E}(\rho)\le C_2^m\approx 2.23,
\]
\[
\sum_{i=x,y,z} C_i^{S}(\rho)\le 2.
\]
These constants become the LHS bounds for averaged coherence functionals in steering scenarios [1508.03770].

For a general two-qubit state \(\rho_{AB}\), Alice measures \(\sigma_k\) with outcomes \(a\in\{0,1\}\), generating Bob’s conditional states \(\rho_B^{a|k}\) with probabilities \(p(a|k)\). The NAQC functional corresponding to what is naturally called ASC is
\[
\mathrm{ASC}_{M}(\rho_{AB})
=
\frac{1}{2}
\sum_{k\in\{x,y,z\}}
\sum_{a\in\{0,1\}}
\sum_{i\in\{x,y,z\}\setminus\{k\}}
p(a|k)\,C^{(i)}_{M}(\rho_B^{a|k}),
\]
with \(M\in\{l_1,E,S\}\) [1508.03770].

For all local-hidden-state models,
\[
\mathrm{ASC}_{l_1}(\rho_{AB})\le \sqrt{6},\qquad
\mathrm{ASC}_{E}(\rho_{AB})\le C_2^m\approx 2.23,\qquad
\mathrm{ASC}_{S}(\rho_{AB})\le 2.
\]
Violation certifies NAQC and therefore steering. The implication is one-way: NAQC implies steerability, but not all steerable states achieve NAQC [1508.03770].

For Werner states,
\[
\rho_w=p|\psi^-_{AB}\rangle\langle\psi^-_{AB}|+\frac{1-p}{4}I^A\otimes I^B,
\]
the resulting ASC values are explicit:
\[
\mathrm{ASC}_{l_1}(\rho_w)=3p,
\]
\[
\mathrm{ASC}_{E}(\rho_w)=3\left[1-\mathcal H\!\left(\frac{1+p}{2}\right)\right],
\]
\[
\mathrm{ASC}_{S}(\rho_w)=3\left(1-\sqrt{1-p^2}\right).
\]
The NAQC thresholds are therefore \(p>\sqrt{2/3}\approx 0.8165\) for \(\ell_1\), \(p\gtrsim 0.914\) for relative entropy, and \(p>2\sqrt{2}/3\approx 0.9428\) for skew information, whereas the same state is steerable already for \(p>1/2\) [1508.03770].

The experimental study "Experimental demonstration of complementarity relations between quantum steering criteria" [2007.11320] tested related aggregate ASC sums
\[
S_\ell^B(\rho_{AB})
=
\sum_{i\in\{x,y,z\}}\sum_a p(\rho_{B|M_a^i})\,C_{i+\ell}^q(\rho_{B|M_a^i}),
\qquad \ell\in\{0,1,2\},
\]
together with
\[
S_{12}^B=S_1^B+S_2^B,\qquad S_{012}^B=S_0^B+S_1^B+S_2^B.
\]
For unsteerable states,
\[
S_0^B\le \epsilon^q,\qquad
\frac{1}{2}S_{12}^B\le \epsilon^q,
\]
with \(\epsilon^{\ell_1C}=\sqrt6\), \(\epsilon^{REC}=2.23\), and \(\epsilon^{SIC}=2\), while the three-setting complementarity relation
\[
\frac{1}{3}S_{012}^B\le \epsilon^q
\]
holds for all two-qubit states. The experiment verified that skew-information coherence gives the strongest steering detection among the three measures [2007.11320].

## 5. Closed forms, dynamical hierarchy, and correlator reconstruction

A recent operational treatment defines ASC for two-qubit systems as
\[
\mathrm{ASC}(\rho_{AB})
=
\max_{U_A}\frac{1}{3}\sum_{k=1}^3 C_{\ell_1}(\rho_{B|k}),
\]
where \(U_A\) rotates Alice’s measurement triad to the optimizing Pauli axes. For the entire two-qubit \(X\)-state family with maximally mixed marginals, the paper proves the simplification
\[
\mathrm{ASC}(\rho)=|c_1|+|c_2|+|c_3|,
\]
with
\[
c_i=\mathrm{Tr}[\rho\,\sigma_i\otimes \sigma_i],\qquad i\in\{x,y,z\}.
\]
The paper adopts the steering-witness threshold \(\mathrm{ASC}>1\) [2606.11731].

In the dephased hydrogen hyperfine model, the electron and proton spins evolve under local Markovian dephasing with
\[
c_1(t)=b_1 e^{-2\kappa t},\qquad
c_2(t)=b_2 e^{-2\kappa t},\qquad
c_3(t)=b_3,
\]
so that
\[
\mathrm{ASC}(t)=\bigl(|b_1|+|b_2|\bigr)e^{-2\kappa t}+|b_3|.
\]
This yields a strict hierarchy
\[
C(t)\le \mathcal N_1(t)\le \mathrm{ASC}(t)
\quad \text{for all } t,
\]
where \(C(t)\) is concurrence and \(\mathcal N_1(t)\) is trace-distance measurement-induced nonlocality. Entanglement is the most fragile resource; trace MIN can exhibit dephasing-immune freezing when \(b_3\neq 0\); ASC is the most robust quantity and persists longest in every scenario studied [2606.11731].

The same paper identifies four distinct dynamical regimes for \(b_3\neq 0\): steerable and entangled, frozen-MIN entangled, non-steerable entangled, and frozen-discord regime. Operationally, ASC is directly reconstructible from three Pauli correlators,
\[
\mathrm{ASC}(t)=|\langle XX\rangle_t|+|\langle YY\rangle_t|+|\langle ZZ\rangle|,
\]
so no full state tomography is required. This correlator representation also makes the hierarchy experimentally accessible in spin systems [2606.11731].

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

Vanishing ASC has different meanings in different formulations. For SIC, \(\bar{\mathcal C}(\rho_{AB})=0\) iff the state is \(B\)-side classical [1508.01978]. For maximal extra basis-dependent ASC with relative-entropy coherence, \(\Delta C_P^{\max}=0\) iff \(\rho_{AB}\) is block-diagonal in Bob’s computational basis or is a product state; for basis-free ASC with total coherence, \(\Delta C_T^{\max}=0\) iff \(\rho_{AB}\) is a product state [1703.02737]. 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 \(p>1/2\) while violating NAQC bounds only at substantially larger \(p\) [1508.03770]. 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 \(J_A\), and that quantum correlation is neither sufficient nor necessary for nonzero extra ASC within a given measurement [1703.02737].

The phrase also creates a cross-disciplinary ambiguity. In activation-steering studies of large language models, 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 \(Q_i\in[0,100]\) from an automatic judge, with the paper using the threshold \(Q_i>50\) to gate a coherent harmful EM rate,
\[
Z_i^B=\mathbf 1\{Q_i>50\}H_i^B,\qquad
\mathrm{EMRate}^B=\frac{1}{N_B}\sum_{i=1}^{N_B} Z_i^B.
\]
If one needs ASC in that framework, the paper states that the natural quantity is the mean coherence
\[
\mathrm{ASC}_{\text{steer}}^B=\frac{1}{N_B}\sum_{i=1}^{N_B} Q_i^{(\text{steer})},
\]
but also states that the authors do not report ASC [2606.08682]. A related activation-steering paper likewise states that it does not define a metric called ASC and instead reports judged coherence \(\kappa\in[0,100]\), cross-entropy under the aligned model, embedding similarity, and repetition metrics [2604.08169]. In consequence, ASC is a standard technical notion in quantum steering and coherence theory, but not a standardized term in activation steering for LLMs.

Source: https://www.emergentmind.com/topics/average-steering-coherence-asc