Choi-Matrix CP-Divisibility Witness

• Choi-Matrix CP-Divisibility Witness is a diagnostic tool that verifies non-Markovian quantum dynamics by testing the positivity of the intermediate propagator’s Choi operator.
• It employs various techniques—including spectral analysis, uncertainty relations, entropy measures, and moment inequalities—to detect negative eigenvalues indicative of CP-indivisibility.
• The witness offers a unifying framework for evaluating non-Markovian behavior and complements conventional information-backflow diagnostics in open quantum systems.

Searching arXiv for recent and foundational papers on Choi-matrix CP-divisibility witnesses and related non-Markovianity diagnostics. A Choi-Matrix CP-Divisibility Witness is a criterion for detecting the breakdown of complete-positive divisibility in a quantum dynamical map by examining the Choi operator of an intermediate propagator. In the standard channel-state correspondence, a linear map is completely positive if and only if its Choi matrix is positive semidefinite. Applied to open-system dynamics, this converts CP-divisibility into a matrix-positivity condition on each intermediate map, so that any negative eigenvalue, violated moment inequality, negative entropy-based functional, or equivalent Gram-matrix non-positivity serves as a witness of CP-indivisibility and hence non-Markovianity in the Rivas–Huelga–Plenio sense. The general complete-positivity basis of this approach is the Choi representation of quantum channels (McCracken, 2013), while later work develops explicit witnesses based on uncertainty relations (Maity et al., 2019), linear entropy (Zheng et al., 2019), low-order moments (Mallick et al., 2023), characteristic-function Gram matrices (Nakagawa, 19 Apr 2026), and collision-model diagnostics (Bhoi et al., 12 Jun 2026).

1. Formal definition and conceptual setting

For a family of dynamical maps $\{\Lambda(t,0)\}_{t\ge 0}$, CP-divisibility means that for every $t\ge s$ there exists an intermediate map $V_{t,s}$ such that

$\Lambda(t,0)=V_{t,s}\Lambda(s,0),$

with $V_{t,s}$ completely positive. In invertible settings this is written as

$V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$

This is the operational criterion used across the non-Markovianity literature summarized in the cited works (Shrikant et al., 2018, Maity et al., 2019, Mallick et al., 2023).

The Choi-matrix formulation begins with the Choi–Jamiołkowski isomorphism. For a linear map $\Phi$, the Choi operator is constructed as

$$J(\Phi)=(\Phi\otimes I)(|\Omega\rangle\langle\Omega|),$$

with $|\Omega\rangle$ a maximally entangled state. Choi’s theorem states that

$$\Phi \text{ is completely positive} \iff J(\Phi)\ge 0.$$

This criterion is foundational for all Choi-based witnesses (McCracken, 2013, Nakagawa, 19 Apr 2026).

A Choi-matrix CP-divisibility witness is therefore any diagnostic that tests positivity of $t\ge s$0, or of an equivalent representation derived from it. If the intermediate Choi operator fails to be positive semidefinite, then $t\ge s$1 is not CP, CP-divisibility fails, and the evolution is non-Markovian in the CP-divisibility sense (Maity et al., 2019, Mallick et al., 2023).

A common misconception is that complete positivity of the full map $t\ge s$2 is sufficient for Markovianity. The cited works explicitly distinguish these notions: the full map may remain CPTP at every time while some intermediate map becomes not completely positive, which is precisely the signature detected by a Choi witness (Shrikant et al., 2018, Filippov et al., 2017, Bhoi et al., 12 Jun 2026).

2. Choi positivity as the witness mechanism

The witness mechanism is direct. For an intermediate map $t\ge s$3, one forms

$t\ge s$4

Then

$t\ge s$5

is equivalent to complete positivity of the intermediate propagator. Any failure of this condition witnesses CP-indivisibility (Maity et al., 2019, Nakagawa, 19 Apr 2026).

This witness can be phrased spectrally. If $t\ge s$6 has eigenvalues $t\ge s$7, then a negative eigenvalue certifies that $t\ge s$8 is not CP. In the uncertainty-relation formulation, the non-positive Choi operator is written as

$t\ge s$9

with $V_{t,s}$0; the existence of such negative spectral weight is the underlying witness content (Maity et al., 2019).

The same logic appears in explicit channel studies. In the non-Markovian dephasing construction of Chakraborty and collaborators, the Choi matrix of the intermediate map has two nonzero eigenvalues,

$V_{t,s}$1

$V_{t,s}$2

and the sign change of one eigenvalue marks the loss of complete positivity of the intermediate map (Shrikant et al., 2018).

This spectral picture also clarifies why Choi witnesses are stronger than some state-based information-backflow diagnostics. Information backflow often accompanies CP-indivisibility, but the two are not equivalent. A negative Choi eigenvalue detects non-CP intermediate dynamics even in parameter regimes where distinguishability-based backflow measures remain zero. This separation is explicit in recent collision-model numerics (Bhoi et al., 12 Jun 2026).

3. Analytic witness constructions derived from the Choi matrix

Several papers replace full spectral diagonalization of the intermediate Choi matrix by analytically simpler functionals whose negativity or inequality violation implies non-positivity.

The earliest group in the supplied material uses uncertainty relations. For Hermitian observables $V_{t,s}$3 and $V_{t,s}$4, the Robertson–Schrödinger relation

$V_{t,s}$5

holds for positive semidefinite operators. When the intermediate Choi state is not positive, this relation can be violated, and the violation becomes a sufficient witness of non-Markovianity (Maity et al., 2019). The same work also proves that the variance of a projector onto a negative-eigenvalue eigenvector,

$V_{t,s}$6

acts as a nonlinear witness (Maity et al., 2019).

A second construction uses entropy functionals. The linear entropy

$V_{t,s}$7

is nonnegative for a valid density operator. The paper "Detecting Non-Markovianity via Linear Entropy of Choi State" states that if

$V_{t,s}$8

then the Choi state is not positive semidefinite and the evolution is non-Markovian (Zheng et al., 2019). The same logic is extended there to Rényi entropy,

$V_{t,s}$9

with negative values taken as a witness of non-CP-divisible dynamics (Zheng et al., 2019).

A third approach uses low-order moments of the Choi matrix. For the Choi state $\Lambda(t,0)=V_{t,s}\Lambda(s,0),$0, define

$\Lambda(t,0)=V_{t,s}\Lambda(s,0),$1

For CP-divisible dynamics, the paper "Assessing non-Markovian dynamics through moments of the Choi state" derives the necessary inequality

$\Lambda(t,0)=V_{t,s}\Lambda(s,0),$2

Thus,

$\Lambda(t,0)=V_{t,s}\Lambda(s,0),$3

is a sufficient witness of CP-indivisibility (Mallick et al., 2023). This construction is technically significant because it replaces spectral analysis by experimentally more accessible low-order moments.

A plausible implication is that these different witnesses should be viewed not as competing definitions but as distinct projections of the same underlying object: positivity failure of the intermediate Choi operator. The supplied literature consistently supports that interpretation.

4. Gram-matrix and characteristic-function reformulations

A more recent reformulation embeds the Choi witness in a Bochner-type positive-definiteness framework. The paper "Map-Dependent Quantum Characteristic Functions and CP-Divisibility in Non-Markovian Quantum Dynamics" defines the normalized Choi operator

$\Lambda(t,0)=V_{t,s}\Lambda(s,0),$4

and, for a unitary operator basis $\Lambda(t,0)=V_{t,s}\Lambda(s,0),$5, the map-dependent characteristic function

$\Lambda(t,0)=V_{t,s}\Lambda(s,0),$6

From this, it constructs the Gram matrix

$\Lambda(t,0)=V_{t,s}\Lambda(s,0),$7

Its central Bochner–Choi positivity theorem is

$\Lambda(t,0)=V_{t,s}\Lambda(s,0),$8

(Nakagawa, 19 Apr 2026).

Applied to intermediate maps, this yields

$\Lambda(t,0)=V_{t,s}\Lambda(s,0),$9

where

$V_{t,s}$0

A negative eigenvalue of $V_{t,s}$1 is therefore an alternative witness of CP-divisibility breakdown (Nakagawa, 19 Apr 2026).

This reformulation is notable because it makes the witness resemble classical and quantum characteristic-function positivity criteria. The paper reports numerical examples for amplitude damping and pure dephasing in which negativity of the Gram matrix coincides with breakdown of CP-divisibility and with information backflow (Nakagawa, 19 Apr 2026).

At a more structural level, another line of work connects Choi positivity with Fourier positivity on inverse semigroups. For the inverse semigroup of matrix units, the Fourier transform at the identity representation becomes the Choi matrix,

$V_{t,s}$2

and positivity of this transform is equivalent to complete positivity (Sohail et al., 2 Sep 2025). That work does not address CP-divisibility as a time-dependent property, but it strengthens the conceptual status of Choi positivity as a special case of a broader positive-definiteness theorem (Sohail et al., 2 Sep 2025).

5. Model systems and explicit witness behavior

Concrete open-system models show how Choi-matrix witnesses behave in practice. In pure dephasing models with time-dependent rate

$V_{t,s}$3

negative $V_{t,s}$4 signals non-Markovianity in the CP-divisibility sense. The uncertainty-relation witness, the linear-entropy witness, and the moment witness are all reported to detect precisely those intervals in their respective formulations (Maity et al., 2019, Zheng et al., 2019, Mallick et al., 2023).

For amplitude damping,

$V_{t,s}$5

with

$V_{t,s}$6

the characteristic-function approach uses the intermediate parameter

$V_{t,s}$7

and notes that values exceeding unity indicate non-CP-divisible behavior. In the reported numerics, negative intervals of $V_{t,s}$8 coincide with negativity of the smallest Gram-matrix eigenvalue and with non-positivity of the intermediate Choi operator (Nakagawa, 19 Apr 2026).

In the explicitly constructed non-Markovian dephasing channel of Hall-type analysis, the Choi eigenvalue crossover occurs at $V_{t,s}$9, which corresponds simultaneously to maximal dephasing, singularity of the canonical decoherence rate, and momentary non-invertibility of the map (Shrikant et al., 2018). This is an instructive case because the witness does more than detect negativity: it localizes a structural transition where the intermediate-map reconstruction itself becomes singular.

For Pauli dynamical maps, geometric conditions on the parameter vector

$V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$0

encode infinitesimal CP-divisibility through inequalities

$V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$1

Although derived geometrically rather than through explicit Choi matrices, these are equivalent for Pauli maps to positivity of the Choi operator of the infinitesimal propagator (Filippov et al., 2017). This suggests that parameter-space divisibility tests can often be understood as compressed Choi witnesses.

6. Collision models, operational diagnostics, and partial reconstruction

Collision models provide an operationally transparent setting in which the Choi witness separates several dynamical regimes. In the finite-environment repeated-interaction model of a system qubit with the same environmental qubit, the intermediate map is

$V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$2

and the corresponding Choi matrix is

$V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$3

The paper defines the stepwise divisibility witness

$V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$4

with $V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$5 for CP intermediate dynamics and $V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$6 when at least one Choi eigenvalue is negative (Bhoi et al., 12 Jun 2026).

That work uses the Choi witness jointly with the BLP trace-distance measure to distinguish three regimes:

$V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$7	$V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$8	Classification
$V_{t,s}=\Lambda(t,0)\Lambda(s,0)^{-1}.$9	$\Phi$0	CP-divisible (Markovian)
$\Phi$1	$\Phi$2	CP-indivisible, P-divisible
$\Phi$3	$\Phi$4	CP-indivisible, non-P-divisible

This classification is technically important because it makes explicit that CP-indivisibility and information backflow are related but distinct. In both generalized depolarizing and generalized amplitude-damping reservoir maps, the Choi witness is reported to detect a broad CP-indivisible but P-divisible region that is invisible to BLP (Bhoi et al., 12 Jun 2026).

On the reconstruction side, partial standard quantum process tomography provides a route to experimentally probing selected Choi-matrix elements without full channel tomography. In the Choi operator basis $\Phi$5, the relation

$\Phi$6

gives direct access to process-matrix elements, and an arbitrary off-diagonal element can be obtained with sixteen measurements, while a diagonal element requires one measurement (Wu et al., 2011). That work does not itself provide a divisibility theorem, but it supports a plausible witness strategy based on reconstructing diagnostically relevant principal submatrices and checking their positivity constraints (Wu et al., 2011).

7. Relations, limitations, and scope

The Choi-matrix CP-divisibility witness is best understood as a family of criteria centered on one invariant fact: complete positivity of an intermediate map is equivalent to positivity of its Choi operator. Different witnesses merely expose this condition through different observables or functionals.

Several distinctions are essential. First, a witness is often only sufficient, not necessary. Violation of an uncertainty relation, negativity of a moment inequality, or negativity of a derived entropy certifies non-Markovianity, but absence of violation need not prove CP-divisibility (Maity et al., 2019, Mallick et al., 2023). By contrast, direct positivity of the full Choi matrix is necessary and sufficient, though potentially computationally or experimentally more demanding (McCracken, 2013).

Second, invertibility matters. The definition $\Phi$7 assumes the inverse exists. Explicit dephasing examples show that singular points can coincide with maximal dephasing and temporary non-invertibility, requiring care in interpreting intermediate-map witnesses and in normalizing some CP-divisibility measures (Shrikant et al., 2018).

Third, CP-divisibility is not identical to all notions of Markovianity. The supplied literature consistently uses the RHP/Hall framework, in which non-Markovianity is identified with failure of CP-divisibility (Shrikant et al., 2018, Mallick et al., 2023, Bhoi et al., 12 Jun 2026). This should not be conflated with weaker notions based only on positivity divisibility or with phenomenological memory effects.

Finally, not every Choi-based positivity theorem is automatically a CP-divisibility theorem. Structural works on complete positivity of a single map, including generalized Bochner–Choi connections and CP tests for static channels, strengthen the mathematical basis of Choi witnesses but do not by themselves address time-ordered divisibility (McCracken, 2013, Sohail et al., 2 Sep 2025). The divisibility witness emerges only when the construction is applied to intermediate propagators.

In this sense, the Choi-Matrix CP-Divisibility Witness is both a specific operational test and a unifying principle. Whether implemented through direct eigenvalue analysis, uncertainty-relation violation, entropy negativity, moment inequalities, Gram-matrix positivity, or collision-model diagnostics, its content remains the same: non-positivity of the Choi operator of an intermediate map is a certificate of CP-divisibility breaking.