1. Introduction
A number of works provide the theoretical and methodological foundations for the study of virtual quantum Markov chains (VQMCs), their recoverability properties, and their applications in quantum information theory. The notion of VQMCs was introduced in [1], laying the groundwork for subsequent investigations. Approximate quantum Markov chains were analyzed in [2], while the robustness of Markovianity under perturbations was studied in [3]. The operational meaning of quantum conditional mutual information was further explored in [4], and general conditions for recoverability in quantum information theory were developed in [5]. Necessary criteria for approximate recoverability were identified in [6], and a universal recovery map with broad applicability was constructed in [7]. In parallel, the framework of virtual quantum broadcasting was proposed in [8], while a celebrated inequality linking conditional mutual information to approximate Markov chains was provided in [9]. More recently, optimal tomography protocols for quantum Markov chains based on the continuity of Petz recovery were introduced in [10], directly inspiring our semidefinite programming (SDP)-based analysis. The relation between fidelity of recovery and entanglement measures was studied in [11], and efficient prediction of quantum properties from few measurements was developed in [12]. Practical perspectives were further advanced in [13], and quasiprobability sampling overhead in simulating non-local channels was investigated in [14], which closely relates to our definition of sampling overhead. In the broader context of virtual quantum resources, virtual quantum resource distillation was proposed in [15], and optimal unilocal broadcasting schemes were developed in [16]. Applications to multipartite systems appear in the dynamic quantum secret sharing protocol in [17], while experimental demonstrations of three-qubit entangled state control were reported in [18]. Finally, algorithmic tools for entropy estimation were developed in [19, 20], offering computational techniques to quantify correlations and recoverability costs. Together, these works establish the theoretical, experimental, and algorithmic foundation upon which our present study of four-qubit VQMCs is built.
Building upon the earlier study on VQMCs in tripartite systems, the present work extends the conceptual framework and computational methodology to the more intricate four-qubit setting. The first study established the foundational recovery-map formulation, the SDP characterization, and the kernel inclusion criterion as a necessary condition for the VQMC property. These elements provide the essential mathematical and algorithmic tools that the current work generalizes to accommodate the richer entanglement structures and higher-dimensional marginal dependencies inherent in four-body states. In particular, the prior framework guides the adaptation of constructive definitions and SDP tests for verifying VQMC conditions in the four-qubit case, as well as the formulation of virtual complexity measures through quasiprobability sampling overhead. This inheritance of both theoretical principles and computational techniques enables us to rigorously analyze representative examples—including GHZ, W, and mixed states—while addressing new challenges such as the limitations of reconstructing global states from low-order marginals.
2. Preliminaries
This section provides a brief overview of the main tools and results underlying our approach. Building on the framework of [1], where the concept of VQMCs was introduced, we recall the key definitions, structural characterizations, and computational formulations. These preliminaries will form the groundwork for the four-qubit extension developed in the subsequent sections.
2.1. Virtual quantum Markov chains
We begin with the formal definition.
(Virtual Quantum Markov Chain for three-qubit state)
A tripartite state ρABC is called a VQMC in order B if there exists a Hermitian-preserving, trace-preserving linear map ${ \mathcal R }:B\to BC$ such that
$\begin{eqnarray}\begin{array}{r}({I}_{A}\displaystyle \otimes { \mathcal R })({\rho }_{AB})={\rho }_{ABC}.\end{array}\end{eqnarray}$
Equivalently, the map ${ \mathcal R }$ can be expressed as a quasiprobability combination of two completely positive trace-preserving (CPTP) maps, $\begin{eqnarray}\begin{array}{l}{ \mathcal R }={c}_{1}{{ \mathcal N }}_{1}-{c}_{2}{{ \mathcal N }}_{2},\qquad {c}_{1},{c}_{2}\geqslant 0,\\ {{ \mathcal N }}_{1},{{ \mathcal N }}_{2}\in {\rm{CPTP}}(B,BC),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{r}{c}_{1}={c}_{2}+1.\end{array}\end{eqnarray}$
This formulation extends the conventional notion of quantum Markov chains, where a tripartite state ρABC is a quantum Markov chain in order B if and only if the conditional mutual information I(A: C∣B)ρ vanishes, equivalent to the existence of a CPTP recovery map ${ \mathcal R }:B\to BC$ such that $({I}_{A}\otimes { \mathcal R })({\rho }_{AB})={\rho }_{ABC}$ by allowing ${ \mathcal R }$ to be a quasiprobability mixture of CPTP maps. Thereby, it captures a richer set of correlations beyond exact conditional independence.2.2. Algebraic characterization
The fundamental structural property of VQMCs can be stated in terms of kernel inclusion. Expanding ρABC in a fixed orthonormal basis {∣j⟩} of subsystem C, we obtain operator blocks
$\begin{eqnarray}\begin{array}{rcl}{\hat{\rho }}_{AC| j} & := & {\mathrm{Tr}}_{B}\left[({I}_{A}\displaystyle \otimes \langle j{| }_{C})\,{\rho }_{ABC}\,({I}_{A}\displaystyle \otimes | j{\rangle }_{C})\right],\\ {\hat{\rho }}_{BC| j} & := & {\mathrm{Tr}}_{A}\left[({I}_{B}\displaystyle \otimes \langle j{| }_{C})\,{\rho }_{ABC}\,({I}_{B}\displaystyle \otimes | j{\rangle }_{C})\right].\end{array}\end{eqnarray}$
(Kernel-inclusion criterion [1])
The state ρABC is a VQMC in order B if and only if
$\begin{eqnarray}\begin{array}{r}\mathrm{Ker}({\hat{\rho }}_{AC| j})\subseteq \mathrm{Ker}({\hat{\rho }}_{BC| j})\quad \,\rm{for all}\,\,j.\end{array}\end{eqnarray}$
This theorem provides a simple operator-algebraic test for verifying whether a given tripartite state admits a virtual recovery map. It also underpins many of the examples considered in [1], such as the fact that the three-qubit W state is a VQMC, while the GHZ state is not.
(Support–kernel duality and rank monotonicity)
Let X, Y ≽ 0 act on the same finite-dimensional Hilbert space. Then
$\begin{eqnarray}\begin{array}{r}{\rm{Ker}}(X)\subseteq {\rm{Ker}}(Y)\quad \iff \quad {\rm{Supp}}(Y)\subseteq {\rm{Supp}}(X),\end{array}\end{eqnarray}$
where ${\rm{Supp}}(Z)={\rm{Ker}}{(Z)}^{\perp }$ denotes the support of Z. Consequently, $\begin{eqnarray}\begin{array}{r}{\rm{Ker}}(X)\subseteq {\rm{Ker}}(Y)\quad \Rightarrow \quad {\rm{rank}}(X)\geqslant {\rm{rank}}(Y).\end{array}\end{eqnarray}$
The equivalence follows from the fact that for any subspace S, ${\rm{Ker}}(Z)=S$ if and only if Supp(Z) = S⊥. Since ${\rm{Dim}}{\rm{Supp}}(Z)={\rm{rank}}(Z)$ for Z ≽ 0, the inclusion of supports directly implies the rank inequality (
2.3. Sampling overhead via semidefinite programming
A key computational tool introduced in [1] is the quantification of the sampling overhead, i.e., the quasiprobability cost required to realize a virtual recovery map. For a four-qubit state ρABCD with marginal ρABC, we define
$\begin{eqnarray}\begin{array}{rcl}\nu ({\rho }_{ABCD}) & := & \mathrm{log}\,{\rm{\min }}\left\{{c}_{1}+{c}_{2}\ | \ ({I}_{AB}\displaystyle \otimes { \mathcal R })({\rho }_{ABC})\right.\\ & = & \left.{\rho }_{ABCD}\right\},\end{array}\end{eqnarray}$
where ${ \mathcal R }={c}_{1}{{ \mathcal N }}_{1}-{c}_{2}{{ \mathcal N }}_{2}$ with ${{ \mathcal N }}_{1},{{ \mathcal N }}_{2}\in {\rm{CPTP}}(C,C\otimes D)$. This optimization problem can be reformulated as SDP. Let J1, J2 respectively denote the Choi matrices of ${{ \mathcal N }}_{1},{{ \mathcal N }}_{2}$, and define the effective Hermitian-preserving Choi operator ${J}_{CC^{\prime} D}:= {J}_{1}-{J}_{2}$, where $C^{\prime} $ is an isomorphic copy of C. The SDP is then given by $\begin{eqnarray}\begin{array}{r}\begin{array}{cl}\,\rm{minimize}\, & {c}_{1}+{c}_{2}\\ \,\rm{subject to}\, & {J}_{1}\succcurlyeq 0,\,{J}_{2}\succcurlyeq 0,\\ & {J}_{CC^{\prime} D}={J}_{1}-{J}_{2},\\ & {{\rm{Tr}}}_{C^{\prime} D}({J}_{1})={c}_{1}{I}_{C},\,{{\rm{Tr}}}_{C^{\prime} D}({J}_{2})={c}_{2}{I}_{C},\\ & {{\rm{Tr}}}_{C^{\prime} }\left[\left({\rho }_{ABC}^{{{\rm{T}}}_{C}}\displaystyle \otimes {I}_{C^{\prime} D}\right)\left({I}_{AB}\displaystyle \otimes {J}_{CC^{\prime} D}\right)\right]={\rho }_{ABCD}.\end{array}\end{array}\end{eqnarray}$
Here TC denotes the partial transpose on C in the computational basis associated with the Choi isomorphism, and ≽0 indicates positive semidefiniteness. The objective c1 + c2 captures the quasiprobability cost of simulating the virtual recovery, thus providing a quantitative measure of complexity for extending ρABC to ρABCD.The combination of definition 1 , lemma 2 , and the SDP formulation (9 ) constitutes the essential technical foundation for our analysis. These tools allow us to formalize what it means for a state to be a four-body VQMC, to provide an algebraic test for verification, and to quantify the associated sampling cost via convex optimization. In the remainder of the paper, we build upon this framework to explore structural properties, explicit examples, and computational aspects of four-qubit virtual quantum Markov chains.
3. Analytical tools for virtual quantum Markov chains
The identification of VQMC structures in multipartite states can be approached through different structural methods. We now present the main tools that will be central to our analysis.
3.1. Recovery-map condition: constructive definition
The most fundamental criterion for a four-body state ρABCD to be a VQMC is the existence of a CPTP map ${ \mathcal R }:C\to CD$ such that
$\begin{eqnarray}\begin{array}{r}{\rho }_{ABCD}=({I}_{AB}\displaystyle \otimes { \mathcal R })({\rho }_{ABC}),\end{array}\end{eqnarray}$
where ${\rho }_{ABC}={\mathrm{Tr}}_{D}{\rho }_{ABCD}$. This condition is both necessary and sufficient. In rare cases, such as symmetric or pure states (e.g., the four-qubit W state), it is possible to construct ${ \mathcal R }$ explicitly for directly certifying the VQMC property. However, in general, the map may not be accessible in a closed form.3.2. SDP formulation: computational test
The existence of such a CPTP map can be verified by casting the problem into SDP over the Choi matrix ${J}_{CC^{\prime} D}$ of ${ \mathcal R }$. Specifically, one solves for ${J}_{CC^{\prime} D}\succcurlyeq 0$ such that
$\begin{eqnarray}\begin{array}{rcl}{\mathrm{Tr}}_{C^{\prime} D}({J}_{CC^{\prime} D}) & = & {I}_{C}\quad \,\rm{and}\,\\ {\rho }_{ABCD} & = & {\mathrm{Tr}}_{C^{\prime} }\left[({\rho }_{ABC}^{{\rm{T}}}\displaystyle \otimes {I}_{C^{\prime} D})({I}_{AB}\displaystyle \otimes {J}_{CC^{\prime} D})\right].\end{array}\end{eqnarray}$
If the SDP is feasible, the state is a VQMC. This approach is fully general but requires numerical implementation and does not always yield analytic insight.3.3. Inclusion criterion: necessary but not sufficient
Recall from section 2 that, for each measurement outcome j on subsystem C, we defined the conditional states ${\hat{\rho }}_{AC| j},{\hat{\rho }}_{BC| j}$ and established the following necessary condition:
(Kernel-inclusion condition)
For each outcome j of a projective measurement on subsystem C, the conditional states must satisfy
$\begin{eqnarray}\begin{array}{r}\mathrm{Ker}({\hat{\rho }}_{AC| j})\subseteq \mathrm{Ker}({\hat{\rho }}_{BC| j}).\end{array}\end{eqnarray}$
This inclusion condition can be computed and efficiently eliminate candidate states that cannot support a virtual recovery channel. However, it is not a sufficient condition, and must be combined with other tools such as the recovery-map construction (10 ) or the SDP formulation (11 ) to confirm the VQMC structure.
3.4. Discussion and role in applications
The purpose of the above tools is to establish a systematic framework for analyzing VQMC states. In particular, the recovery-map condition in equation (10 ) provides the constructive definition of VQMCs. The SDP criterion in equation (11 ) offers a computational method for verifying the property, especially in cases where analytic recovery maps are not available. Finally, the kernel-inclusion condition in equation (12 ), which follows from Theorem 2 in section 2 , serves as an efficiently checkable necessary criterion.
With these preparations in place, section 4 builds directly on the analytical tools developed above by applying them to the setting of four-qubit quantum states. The recovery-map and SDP approaches will be employed to confirm explicit constructions, while the kernel-inclusion criterion will be used to rule out non-VQMC candidates. This transition from sections 3 to 4 thus reflects the shift from general structural principles to concrete applications.
4. Extension to four-qubit state
In this section we apply the analytical tools from section 3 to the study of four-qubit states. The recovery-map condition in equation (10 ) provides the constructive definition of a four-qubit VQMC, the SDP criterion in equation (11 ) serves as a computational test for the existence of a valid recovery channel, and the kernel-inclusion condition of Proposition 4 in equation (12 ) offers a tractable necessary test that helps to eliminate non-VQMC candidates. Using these tools, we analyze explicit examples such as the four-qubit W and GHZ states, mixed states interpolating between them, and more general convex combinations, thereby illustrating how the abstract principles of section 3 translate into concrete structural insights and constructive recovery maps in the four-qubit setting.
We consider a four-qubit pure state of system A, B, C, and D. Let ρABCD be a quantum state defined on the Hilbert space ${{ \mathcal H }}_{A}\otimes {{ \mathcal H }}_{B}\otimes {{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{D}$. We say that ρABCD is a VQMC state if there exists a CPTP (i.e., recovery) map
$\begin{eqnarray}\begin{array}{r}{ \mathcal R }:{ \mathcal L }({{ \mathcal H }}_{C})\to { \mathcal L }({{ \mathcal H }}_{C}\displaystyle \otimes {{ \mathcal H }}_{D}),\end{array}\end{eqnarray}$
such that $\begin{eqnarray}\begin{array}{r}{\rho }_{ABCD}=({I}_{AB}\displaystyle \otimes { \mathcal R })({\rho }_{ABC}).\end{array}\end{eqnarray}$
Here, the operator ${I}_{AB}\otimes { \mathcal R }$ denotes a composite quantum operation that applies the identity channel to systems A and B and applies the recovery map ${ \mathcal R }$ solely to system C. This models a situation where system D is generated virtually from system C via ${ \mathcal R }$. To facilitate practical verification of the VQMC condition, one can introduce the Choi matrix ${J}_{CC^{\prime} D}$ corresponding to the channel ${ \mathcal R }$ and reformulate the condition as a SDP. Specifically, we require
$\begin{eqnarray}\begin{array}{r}\begin{array}{l}{\rho }_{ABCD}={\mathrm{Tr}}_{C^{\prime} }\left[({\rho }_{ABC}^{{\rm{T}}}\displaystyle \otimes {I}_{C^{\prime} D})({I}_{AB}\displaystyle \otimes {J}_{CC^{\prime} D})\right],\\ \,\rm{with}\,\,{\mathrm{Tr}}_{C^{\prime} D}({J}_{CC^{\prime} D})={I}_{C}.\end{array}\end{array}\end{eqnarray}$
Let us fix an orthonormal basis {∣j⟩} for system C, and consider the projective measurement on C with outcomes labeled by j. For each outcome j, we define the unnormalized conditional states
$\begin{eqnarray}\begin{array}{r}{\hat{\rho }}_{AC| j}:= {\mathrm{Tr}}_{B}\left[({I}_{A}\displaystyle \otimes \langle j{| }_{C})\,{\rho }_{ABC}\,({I}_{A}\displaystyle \otimes | j{\rangle }_{C})\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{\hat{\rho }}_{BC| j}:= {\mathrm{Tr}}_{A}\left[({I}_{B}\displaystyle \otimes \langle j{| }_{C})\,{\rho }_{ABC}\,({I}_{B}\displaystyle \otimes | j{\rangle }_{C})\right].\end{array}\end{eqnarray}$
If a four-qubit quantum state ρABCD satisfies the VQMC condition in definition
$\begin{eqnarray}\begin{array}{r}\mathrm{Ker}({\hat{\rho }}_{AC| j})\subseteq \mathrm{Ker}({\hat{\rho }}_{BC| j}).\end{array}\end{eqnarray}$
Using (
Mathematically, this implies that
$\begin{eqnarray}\begin{array}{rcl}{\hat{\rho }}_{AC| j} & := & {\mathrm{Tr}}_{BD}\left[\left({I}_{A}\displaystyle \otimes \langle j{| }_{C}\displaystyle \otimes {I}_{D}\right)\,{\rho }_{ABCD}\,\right.\\ & & \left.\times \left({I}_{A}\displaystyle \otimes | j{\rangle }_{C}\displaystyle \otimes {I}_{D}\right)\right].\end{array}\end{eqnarray}$
Equivalently, by tracing out subsystem D, one obtains ( $\begin{eqnarray}\begin{array}{r}\mathrm{Ker}({\hat{\rho }}_{AC| j})\subseteq \mathrm{Ker}({\hat{\rho }}_{BC| j}).\end{array}\end{eqnarray}$
Equivalently, the support of ${\hat{\rho }}_{AC| j}$ is contained in the support of ${\hat{\rho }}_{BC| j}$, since the support is the orthogonal complement of the kernel. The portion of ${\hat{\rho }}_{AC| j}$ lying outside the support of ${\hat{\rho }}_{BC| j}$ would therefore be absent in ${\hat{\rho }}_{BC| j}$, and hence such a component could not be generated from ${\hat{\rho }}_{BC| j}$ by any local channel. This violates the basic data-processing inequality, and contradicts the assumption that ${ \mathcal R }$ exists. Thus, the kernel inclusion must hold for all j. The above proof shows that the inclusion $\mathrm{Ker}({\hat{\rho }}_{AC| j})\subseteq \mathrm{Ker}({\hat{\rho }}_{BC| j})$ is a necessary condition for VQMC. However, the inclusion is not sufficient. We deliver a counterexample in the following lemma.
There exists a non-VQMC four-qubit state that satisfies the kernel inclusion in (
We consider a mixed four-qubit state defined as a convex combination of the W and GHZ states
$\begin{eqnarray}\begin{array}{r}{\rho }_{p}:= p\,{\rho }_{{{\rm{W}}}_{4}}+(1-p)\,{\rho }_{{{\rm{GHZ}}}_{4}},\quad \,\rm{with}\,\,0\lt p\lt 1.\end{array}\end{eqnarray}$
Here, the two component states are given by $\begin{eqnarray}\begin{array}{rcl}{\rho }_{{{\rm{W}}}_{4}} & = & | {{\rm{W}}}_{4}\rangle \langle {{\rm{W}}}_{4}| ,\\ | {{\rm{W}}}_{4}\rangle & = & \frac{1}{2}(| 0001\rangle +| 0010\rangle +| 0100\rangle +| 1000\rangle ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{{\rm{GHZ}}}_{4}} & = & | {{\rm{GHZ}}}_{4}\rangle \langle {{\rm{GHZ}}}_{4}| ,\\ | {{\rm{GHZ}}}_{4}\rangle & = & \frac{1}{\sqrt{2}}(| 0000\rangle +| 1111\rangle ).\end{array}\end{eqnarray}$
We define the marginal three-party state by tracing out system D, $\begin{eqnarray}\begin{array}{r}{\rho }_{ABC}^{(p)}:= {\mathrm{Tr}}_{D}({\rho }_{p})=p\,{\rho }_{ABC}^{({\rm{W}})}+(1-p)\,{\rho }_{ABC}^{({\rm{GHZ}})},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{r}{\rho }_{ABC}^{({\rm{GHZ}})}=\frac{1}{2}(| 000\rangle \langle 000| +| 111\rangle \langle 111| ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{ABC}^{({\rm{W}})} & = & \frac{1}{4}\left(| 000\rangle \langle 000| +| \phi \rangle \langle \phi | \right),\\ | \phi \rangle & = & | 001\rangle +| 010\rangle +| 100\rangle .\end{array}\end{eqnarray}$
Using ( $\begin{eqnarray}\begin{array}{r}\mathrm{Ker}({\hat{\rho }}_{AC| j})\subseteq \mathrm{Ker}({\hat{\rho }}_{BC| j}),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{\hat{\rho }}_{AC| j} & := & {\mathrm{Tr}}_{B}\left[{\langle j{| }_{C}\,{\rho }_{ABC}^{(p)}\,| j\rangle }_{C}\right],\\ {\hat{\rho }}_{BC| j} & := & {\mathrm{Tr}}_{A}\left[{\langle j{| }_{C}\,{\rho }_{ABC}^{(p)}\,| j\rangle }_{C}\right].\end{array}\end{eqnarray}$
For a sufficiently large value of p, such as $p\geqslant \frac{1}{4}$, this inclusion condition holds. However, despite satisfying the inclusion condition, the state ρp in ( $\begin{eqnarray}\begin{array}{r}{\rho }_{p}\ne ({I}_{AB}\displaystyle \otimes { \mathcal R })({\rho }_{ABC}^{(p)}).\end{array}\end{eqnarray}$
We emphasize that in this construction, ρABCD = ρp as defined in (4.1. Sampling overhead and SDP formulation for four-qubit quantum states
For a four-qubit quantum state ρABCD, we define
$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}\nu ({\rho }_{ABCD}) & := & \mathrm{log}{\rm{\min }}\{{c}_{1}+{c}_{2}| ({c}_{1}{{ \mathcal N }}_{1}-{c}_{2}{{ \mathcal N }}_{2})\\ & & \times ({\rho }_{ABC})={\rho }_{ABCD},\\ & & {c}_{1},{c}_{2}\geqslant 0,\\ & & {{ \mathcal N }}_{1},{{ \mathcal N }}_{2}\in \,\rm{CPTP}\,(C,C\displaystyle \otimes D)\}.\end{array}\end{array}\end{eqnarray}$
We aim to recover the global state ρABCD from the marginal ρABC using a Hermitian-preserving transformation. The generalized sampling overhead is given in (30 ), where ${{ \mathcal N }}_{1},{{ \mathcal N }}_{2}$ are CPTP maps from subsystem C to C ⨂ D, and c1, c2 are sampling weights. The problem above can be equivalently reformulated as SDP with the following variables and constraints.
$\begin{eqnarray}\begin{array}{r}\begin{array}{cl}\,\rm{minimize}\, & {c}_{1}+{c}_{2}\\ \,\rm{subject to}\, & {J}_{1}\geqslant 0,\quad {J}_{2}\geqslant 0,\\ & {J}_{CC^{\prime} D}={J}_{1}-{J}_{2},\\ & {{\rm{Tr}}}_{C^{\prime} D}({J}_{1})={c}_{1}{I}_{C},\quad {{\rm{Tr}}}_{C^{\prime} D}({J}_{2})={c}_{2}{I}_{C},\\ & {{\rm{Tr}}}_{C^{\prime} }\left[({\rho }_{ABC}^{{\rm{T}}}\displaystyle \otimes {I}_{C^{\prime} D})({I}_{AB}\displaystyle \otimes {J}_{CC^{\prime} D})\right]={\rho }_{ABCD}.\end{array}\end{array}\end{eqnarray}$
Here J1, J2 are respectively the Choi matrices of the quantum channels ${{ \mathcal N }}_{1},{{ \mathcal N }}_{2}$. The operator ${J}_{CC^{\prime} D}$ represents the Hermitian-preserving effective map combining ${{ \mathcal N }}_{1}$ and ${{ \mathcal N }}_{2}$. The partial trace constraints ensure proper normalization, while the final constraint enforces that the reconstructed state exactly matches ρABCD. This SDP formulation allows us to quantify the minimum quasiprobability sampling cost required to virtually recover a four-body quantum state. Its structure mirrors the three-body case but incorporates the additional complexity and dimensionality of four-qubit systems.We consider a CPTP map from C → C ⨂ D in the four-qubit setting. Let qubits X = A, B, C, D with ${{ \mathcal H }}_{X}={{\mathbb{C}}}^{2}$. Define the CPTP map ${ \mathcal N }$ as
$\begin{eqnarray}\begin{array}{r}{ \mathcal N }({\rho }_{C})={\rho }_{C}\displaystyle \otimes | 0\rangle \langle 0{| }_{D},\end{array}\end{eqnarray}$
which appends a fixed pure state ∣0⟩ to qubit C, thereby embedding the system into ${{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{D}$. It is immediate that ${ \mathcal N }$ is CPTP. For all ρC, the output ρC ⨂ ∣0⟩⟨0∣is positive semidefinite, and ${\rm{Tr}}[{ \mathcal N }({\rho }_{C})]={\rm{Tr}}[{\rho }_{C}]=1$. For instance, let the initial three-qubit state be the GHZ state $\begin{eqnarray}\begin{array}{r}| \psi {\rangle }_{ABC}=\frac{1}{\sqrt{2}}(| 000\rangle +| 111\rangle ),\quad {\rho }_{ABC}=| \psi \rangle \langle \psi | .\end{array}\end{eqnarray}$
Applying ${ \mathcal N }$ to qubit C yields the four-qubit state $\begin{eqnarray}\begin{array}{r}{\rho }_{ABCD}=({I}_{AB}\displaystyle \otimes { \mathcal N })({\rho }_{ABC})={\rho }_{ABC}\displaystyle \otimes | 0\rangle \langle 0{| }_{D}.\end{array}\end{eqnarray}$
The Choi matrix ${J}_{CC^{\prime} D}\in { \mathcal B }({{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{C^{\prime} }\otimes {{ \mathcal H }}_{D})$ for ${ \mathcal N }$ is $\begin{eqnarray}\begin{array}{rcl}{J}_{CC^{\prime} D} & = & \displaystyle \sum _{i,j=0}^{1}| i\rangle \langle j{| }_{C}\displaystyle \otimes { \mathcal N }(| i\rangle \langle j| )\\ & = & \displaystyle \sum _{i,j=0}^{1}| i\rangle \langle j{| }_{C}\displaystyle \otimes | i\rangle \langle j{| }_{C^{\prime} }\displaystyle \otimes | 0\rangle \langle 0{| }_{D}.\end{array}\end{eqnarray}$
This operator is positive semidefinite and satisfies the trace-preserving condition $\begin{eqnarray}\begin{array}{r}{{\rm{Tr}}}_{C^{\prime} D}({J}_{CC^{\prime} D})={I}_{C}.\end{array}\end{eqnarray}$
4.2. GHZ state for VQMC
Let ρABCD = ∣GHZ4⟩⟨GHZ4∣be the four-qubit GHZ state in (
$\begin{eqnarray}\begin{array}{r}{\rho }_{ABCD}=({I}_{AB}\displaystyle \otimes { \mathcal R })({\rho }_{ABC}).\end{array}\end{eqnarray}$
We consider the genuine four-qubit entangled state in (
$\begin{eqnarray}\begin{array}{r}V| 0\rangle =| 00\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}V| 1\rangle =| 11\rangle .\end{array}\end{eqnarray}$
Then we define $\begin{eqnarray}\begin{array}{r}N({\rho }_{C})=V{\rho }_{C}{V}^{\dagger }.\end{array}\end{eqnarray}$
This is a valid CPTP map since V†V = I2 and the output lies in ${ \mathcal B }({{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{D})$. The violation ${\rm{Ker}}({\hat{\rho }}_{AC}) \nsubseteq {\rm{Ker}}({\hat{\rho }}_{BC})$ arises from the structural asymmetry in the conditional marginals of the GHZ4 state. Specifically, each post-measurement state ${\hat{\rho }}_{AC| j}$ has support only on subsystem A, as the C system has been projected out. In contrast, the corresponding marginals ${\hat{\rho }}_{BC| j}$ retain entangled support across both B and C. As a result, the overall support of ${\hat{\rho }}_{BC}$ spans a larger subspace than that of ${\hat{\rho }}_{AC}$, making the kernel inclusion condition fail. Therefore, the GHZ4 state does not satisfy the VQMC algebraic criterion. 4.3. W state for VQMC
Let ρABCD = ∣W4⟩⟨W4∣be the four-qubit W state in (
$\begin{eqnarray}\begin{array}{r}{\rho }_{ABCD}=({I}_{AB}\displaystyle \otimes { \mathcal R })({\rho }_{ABC}).\end{array}\end{eqnarray}$
Tracing out qubit D from the four-qubit W state yields a mixed three-qubit state ρABC in (
$\begin{eqnarray}\begin{array}{r}\begin{array}{l}| 1000\rangle ,\,| 0100\rangle ,\,| 0001\rangle \Rightarrow \,\,\rm{after tracing out}\,\,D:| 100\rangle ,\,| 010\rangle ,\,| 000\rangle .\end{array}\end{array}\end{eqnarray}$
Projecting onto C in state ∣0⟩ and tracing out B yields ${\hat{\rho }}_{AC| j}={\mathrm{Tr}}_{B}\left[({I}_{A}\otimes \langle j{| }_{C})\,{\rho }_{ABC}\,({I}_{A}\otimes | j{\rangle }_{C})\right]$ defined in (Thus, ${\hat{\rho }}_{AC| 1}$ = $\frac{1}{4}\,| 01\rangle \langle 01| $ = $\frac{1}{4}| {\psi }_{1}\rangle \langle {\psi }_{1}| $, where ∣ψ1⟩ = ∣01⟩. By direct computation, the conditional operators on subsystem BC are
$\begin{eqnarray}\begin{array}{rcl}{\hat{\rho }}_{BC| 0} & = & \frac{1}{4}\,(| 00\rangle +| 10\rangle )(\langle 00| +\langle 10| ),\\ {\hat{\rho }}_{BC| 1} & = & \frac{1}{4}\,| 01\rangle \langle 01| .\end{array}\end{eqnarray}$
Hence, we have $\begin{eqnarray}\begin{array}{rcl}\mathrm{Ker}({\hat{\rho }}_{AC| 0}) & = & \mathrm{Ker}({\hat{\rho }}_{BC| 0})\\ & = & {\rm{span}}\{| 01\rangle ,| 11\rangle ,| 00\rangle -| 10\rangle \},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rc}\mathrm{Ker}({\hat{\rho }}_{AC| 1})=\,\mathrm{Ker}({\hat{\rho }}_{BC| 1}) & ={\rm{span}}\{| 00\rangle ,| 10\rangle ,| 11\rangle \}.\end{array}\end{eqnarray}$
For $X={\hat{\rho }}_{AC| j}$ and $Y={\hat{\rho }}_{BC| j}$, the kernel-inclusion condition ${\rm{Ker}}({\hat{\rho }}_{AC| j})\subseteq {\rm{Ker}}({\hat{\rho }}_{BC| j})$ implies, by Lemma
To further confirm that the four-qubit W state satisfies the VQMC condition, we explicitly construct a CPTP map ${ \mathcal R }:C\to C\otimes D$ with Choi matrix
$\begin{eqnarray}\begin{array}{rcl}{J}_{CC^{\prime} D} & = & | 0\rangle \langle 0{| }_{C}\displaystyle \otimes | {\psi }_{0}\rangle \langle {\psi }_{0}{| }_{C^{\prime} D}\\ & & +| 1\rangle \langle 1{| }_{C}\displaystyle \otimes | {\psi }_{1}\rangle \langle {\psi }_{1}{| }_{C^{\prime} D},\end{array}\end{eqnarray}$
where $| {\psi }_{0}\rangle =\frac{1}{\sqrt{2}}(| 00\rangle +| 01\rangle )$ and ∣ψ1⟩ = ∣10⟩. This operator is positive semidefinite and satisfies ${\mathrm{Tr}}_{C^{\prime} D}({J}_{CC^{\prime} D})\,={I}_{C}$, thus defining a valid CPTP map ${ \mathcal R }$ via the Choi-Jamiolkowski isomorphism: ${ \mathcal R }(\rho )={\mathrm{Tr}}_{C^{\prime} }[({\rho }^{{\rm{T}}}\otimes {I}_{C^{\prime} D}){J}_{CC^{\prime} D}]$. Applying ${I}_{AB}\otimes { \mathcal R }$ to the marginal state ρABC in (4.4. Recover four-qubit states from two-qubit states
We now investigate whether a four-qubit quantum state ρABCD can be recovered directly from its two-qubit state ρAB via a CPTP channel ${ \mathcal R }:B\to BCD$. In other words, we ask whether
$\begin{eqnarray}\begin{array}{r}{\rho }_{ABCD}=({I}_{A}\displaystyle \otimes { \mathcal R })({\rho }_{AB})\end{array}\end{eqnarray}$
can hold for some fixed ${ \mathcal R }$, independently of the particular structure of ρABCD.We expand the global state in an operator basis on qubit A: ${\rho }_{ABCD}={\sum }_{i,j\in \{0,1\}}| i\rangle \langle j{| }_{A}\otimes {M}_{ij}^{BCD}$, where ${M}_{ij}^{BCD}:= {\,}_{A}{\langle i| {\rho }_{ABCD}| j\rangle }_{A}$ is an operator acting on BCD. Tracing out CD gives $\begin{eqnarray}\begin{array}{r}{\rho }_{AB}={\mathrm{Tr}}_{CD}{\rho }_{ABCD}=\displaystyle \sum _{i,j}| i\rangle \langle j{| }_{A}\displaystyle \otimes {\mathrm{Tr}}_{CD}\left(\right.{M}_{ij}^{BCD}\left)\right..\end{array}\end{eqnarray}$
If such a channel ${ \mathcal R }$ exists, linearity of quantum operations implies that for all i, j, 22 ) and let ρABCD = ∣W4⟩⟨W4∣. We group the terms by the value of qubit A:
$\begin{eqnarray}\begin{array}{r}{M}_{ij}^{BCD}={ \mathcal R }\left({\mathrm{Tr}}_{CD}{M}_{ij}^{BCD}\right).\end{array}\end{eqnarray}$
Thus, the action of ${ \mathcal R }$ on the B-operators ${\mathrm{Tr}}_{CD}{M}_{ij}^{BCD}$ must reproduce the full BCD-operators ${M}_{ij}^{BCD}$. We consider the W state defined as ( $\begin{eqnarray}\begin{array}{rcl}| {{\rm{W}}}_{4}\rangle & = & | 0{\rangle }_{A}\displaystyle \otimes \frac{1}{2}\left(| 001{\rangle }_{BCD}+| 010{\rangle }_{BCD}+| 100{\rangle }_{BCD}\right)\\ & & +| 1{\rangle }_{A}\displaystyle \otimes \frac{1}{2}| 000{\rangle }_{BCD}.\end{array}\end{eqnarray}$
From this decomposition we can write down the block operators ${M}_{ij}^{BCD}$ explicitly:
$\begin{eqnarray}\begin{array}{rcl}{M}_{00}^{BCD} & = & \frac{1}{4}\left(| 001\rangle +| 010\rangle +| 100\rangle \right)\\ & & \times \left(\langle 001| +\langle 010| +\langle 100| \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{M}_{01}^{BCD}=\frac{1}{4}\left(| 001\rangle +| 010\rangle +| 100\rangle \right)\langle 000| ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{M}_{10}^{BCD}=\frac{1}{4}| 000\rangle \left(\langle 001| +\langle 010| +\langle 100| \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{M}_{11}^{BCD}=\frac{1}{4}| 000\rangle \langle 000| .\end{array}\end{eqnarray}$
We now compute their partial traces over CD:
$\begin{eqnarray}\begin{array}{rcl}{\mathrm{Tr}}_{CD}{M}_{00}^{BCD} & = & \frac{1}{4}\left(| 0\rangle \langle 0{| }_{B}+| 1\rangle \langle 1{| }_{B}+| 0\rangle \langle 0{| }_{B}\right)\\ & = & \frac{1}{2}| 0\rangle \langle 0{| }_{B}+\frac{1}{4}| 1\rangle \langle 1{| }_{B},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{\mathrm{Tr}}_{CD}{M}_{01}^{BCD}=\frac{1}{4}| 0\rangle \langle 0{| }_{B},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{\mathrm{Tr}}_{CD}{M}_{10}^{BCD}=\frac{1}{4}| 0\rangle \langle 0{| }_{B},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{\mathrm{Tr}}_{CD}{M}_{11}^{BCD}=\frac{1}{4}| 0\rangle \langle 0{| }_{B}.\end{array}\end{eqnarray}$
Suppose a channel ${ \mathcal R }$ satisfying (49 ) exists. Then, in particular, we must have 59 ) and (60 ) are identical because the inputs are the same operator on B, yet the right-hand sides are different operators on BCD (one contains off-diagonal terms, the other is purely diagonal). This is impossible for a linear map ${ \mathcal R }$. Therefore, no CPTP channel ${ \mathcal R }:B\to BCD$ can satisfy (49 ) for all blocks ${M}_{ij}^{BCD}$ in the case of the W4 state. This shows that, in general, it is not possible to recover ρABCD from ρAB alone by acting only on subsystem B. Larger marginals, such as ρABC, are required for recovery in the context of virtual quantum Markov chains.
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal R }\left(\frac{1}{4}| 0\rangle \langle 0{| }_{B}\right) & = & {M}_{01}^{BCD}=\frac{1}{4}\left(| 001\rangle \right.\\ & & +\left.| 010\rangle +| 100\rangle \right)\langle 000| ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{ \mathcal R }\left(\frac{1}{4}| 0\rangle \langle 0{| }_{B}\right)={M}_{11}^{BCD}=\frac{1}{4}| 000\rangle \langle 000| .\end{array}\end{eqnarray}$
The left-hand sides of (4.5. Mixture of the four-qubit GHZ and W states for VQMC
We consider the convex mixture of the four-qubit GHZ and W states (21 ) where GHZ and W states are defined as (22 ) and (23 ), such that ${\rho }_{{{\rm{GHZ}}}_{4}}:= | {{\rm{GHZ}}}_{4}\rangle \langle {{\rm{GHZ}}}_{4}| $, ${\rho }_{{{\rm{W}}}_{4}}:= | {{\rm{W}}}_{4}\rangle \langle {{\rm{W}}}_{4}| $. We examine whether the resulting state ρp satisfies the virtual quantum Markov chain (VQMC) condition (7 ). We now compute the A-side block matrices ΘB∣A and ΘBC∣A of the marginal states ${\rho }_{AB}^{(p)}={\mathrm{Tr}}_{C}{\rho }_{ABC}^{(p)}$ and ${\rho }_{ABC}^{(p)}$, respectively. We consider (26 ). For the GHZ component, tracing out subsystem C gives
$\begin{eqnarray}\begin{array}{r}{\rho }_{AB}^{({\rm{GHZ}})}={\mathrm{Tr}}_{C}({\rho }_{ABC}^{({\rm{GHZ}})})=\frac{1}{2}(| 00\rangle \langle 00| +| 11\rangle \langle 11| ).\end{array}\end{eqnarray}$
For the W component we obtain $\begin{eqnarray}\begin{array}{r}{\rho }_{AB}^{({\rm{W}})}={\mathrm{Tr}}_{C}({\rho }_{ABC}^{({\rm{W}})})=\frac{1}{4}\left(| 00\rangle \langle 00| +{\sigma }_{AB}\right),\end{array}\end{eqnarray}$
where σAB denotes the partial trace of ∣φ⟩⟨φ∣ over C. Hence the convex mixture satisfies $\begin{eqnarray}\begin{array}{r}{\rho }_{AB}^{(p)}=p\,{\rho }_{AB}^{({\rm{W}})}+(1-p)\,{\rho }_{AB}^{({\rm{GHZ}})}.\end{array}\end{eqnarray}$
From ${\rho }_{AB}^{(p)}$ we extract the A-side block matrix ${{\rm{\Theta }}}_{B| A}^{(p)}\,={[{\langle i{| }_{A}{\rho }_{AB}^{(p)}| j\rangle }_{A}]}_{i,j=0,1}$. Similarly we have $\begin{eqnarray}\begin{array}{r}{{\rm{\Theta }}}_{BC| A}^{(p)}=p\,{{\rm{\Theta }}}_{BC| A}^{({\rm{W}})}+(1-p)\,{{\rm{\Theta }}}_{BC| A}^{({\rm{GHZ}})},\end{array}\end{eqnarray}$
where each block is given by $\begin{eqnarray}\begin{array}{r}{[{{\rm{\Theta }}}_{BC| A}]}_{ij}:= {\langle i{| }_{A}{\rho }_{ABC}^{(p)}| j\rangle }_{A}.\end{array}\end{eqnarray}$
These blocks are 4 × 4 matrices on BC. For instance $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Theta }}}_{BC| A}^{({\rm{GHZ}})} & = & \left[\begin{array}{cc}\frac{1}{2}| 00\rangle \langle 00| & 0\\ 0 & \frac{1}{2}| 11\rangle \langle 11| \end{array}\right],\\ {{\rm{\Theta }}}_{BC| A}^{({\rm{W}})} & = & \left[\begin{array}{cc}\frac{1}{4}| 00\rangle \langle 00| +X & Y\\ {Y}^{\dagger } & Z\end{array}\right],\end{array}\end{eqnarray}$
with X, Y, Z derived from ∣φ⟩⟨φ∣. The VQMC criterion requires: $\begin{eqnarray}\begin{array}{r}{\rm{Ker}}({{\rm{\Theta }}}_{B| A}^{(p)})\subseteq {\rm{Ker}}({{\rm{\Theta }}}_{BC| A}^{(p)}).\end{array}\end{eqnarray}$
Due to the rank structure of the GHZ blocks, which are diagonal and rank-1 in each block, their kernels are large. The W blocks introduce off-diagonal components which shrink the kernels. As p → 1, ${{\rm{\Theta }}}_{BC| A}^{(p)}$ approaches the fully supported W-type block matrix and the inclusion holds. However, for p small (GHZ-dominated), the off-diagonal support is too small to satisfy the inclusion, and the condition fails. Therefore, there exists a threshold value pc such that ρp satisfies the VQMC condition if and only if p ≥ pc.4.5.1. Analytic estimate of the critical threshold
To estimate the minimum weight pc required for the mixed state ${\rho }_{p}:= p{\rho }_{{{\rm{W}}}_{4}}+(1-p){\rho }_{{{\rm{GHZ}}}_{4}}$ to satisfy the virtual quantum Markov chain condition, we consider the kernel inclusion criterion:
$\begin{eqnarray}\begin{array}{r}{\rm{Ker}}({{\rm{\Theta }}}_{B| A}^{(p)})\subseteq {\rm{Ker}}({{\rm{\Theta }}}_{BC| A}^{(p)}).\end{array}\end{eqnarray}$
This condition fails when vectors in the kernel of ${{\rm{\Theta }}}_{B| A}^{(p)}$ are mapped non-trivially by ${{\rm{\Theta }}}_{BC| A}^{(p)}$. Using representative vectors and tracking their images under the W and GHZ components, we find that for small p, the GHZ support dominates and the inclusion fails. As p increases, the off-diagonal components contributed by ∣φ⟩ in ${\rho }_{ABC}^{(W)}$ begin to shrink the kernel of ${{\rm{\Theta }}}_{BC| A}^{(p)}$.There exists a sharp threshold pc such that ρp is a VQMC if and only if p ≥ pc. The lower bound pc satisfies
$\begin{eqnarray}\begin{array}{r}{p}_{c}\geqslant \frac{1}{4}.\end{array}\end{eqnarray}$
The exact value of pc can be further determined numerically by SDP feasibility checks. A natural question is whether the set of four-qubit states satisfying the VQMC condition is convex. That is, if two states ${\rho }_{1},{\rho }_{2}\in {{ \mathcal V }}_{4}$, does it follow that any convex combination ρλ ≔ λρ1 + (1 − λ)ρ2 also belongs to ${{ \mathcal V }}_{4}$ for all λ ∈ [0, 1]? This question arises naturally because many important sets in quantum information theory—such as separable states or PPT states—are convex. However, we will show that the VQMC condition is not preserved under convex mixing, thereby disproving this convexity conjecture. To that end, we construct a concrete counterexample. Let ρ1 be the pure four-qubit W state (22 ) and let ρ2 be the classical-conditional VQMC state
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{2} & := & \frac{1}{2}\left(| 00{\rangle }_{AB}{\langle 00| \displaystyle \otimes | 0\rangle }_{C}{\langle 0| \displaystyle \otimes | 0\rangle }_{D}\langle 0| \right.\\ & & \left.+| 11{\rangle }_{AB}{\langle 11| \displaystyle \otimes | 1\rangle }_{C}{\langle 1| \displaystyle \otimes | 1\rangle }_{D}\langle 1| \right).\end{array}\end{eqnarray}$
Both ρ1 and ρ2 satisfy the VQMC condition. For ρ1, this has been verified via direct evaluation of the inclusion criterion. For ρ2, the state is conditionally separable given system C, and can be trivially reconstructed via a recovery map that measures C and appends ${\rho }_{D}^{(j)}$ accordingly. Now consider the convex combination ρλ = λρ1 + (1 − λ)ρ2 with λ ∈ (0, 1), and examine the conditional post-measurement states given outcome C = 1. From ρ1, we have $\begin{eqnarray}\begin{array}{r}{\hat{\rho }}_{AC| 1}^{(1)}=| 01\rangle \langle 01| ,\quad {\hat{\rho }}_{BC| 1}^{(1)}=| 00\rangle \langle 00| .\end{array}\end{eqnarray}$
From ρ2, we have $\begin{eqnarray}\begin{array}{r}{\hat{\rho }}_{AC| 1}^{(2)}=| 11\rangle \langle 11| ,\quad {\hat{\rho }}_{BC| 1}^{(2)}=| 11\rangle \langle 11| .\end{array}\end{eqnarray}$
The convex mixture gives $\begin{eqnarray}\begin{array}{rcl}{\hat{\rho }}_{AC| 1} & = & \lambda | 01\rangle \langle 01| +(1-\lambda )| 11\rangle \langle 11| ,\\ {\hat{\rho }}_{BC| 1} & = & \lambda | 00\rangle \langle 00| +(1-\lambda )| 11\rangle \langle 11| .\end{array}\end{eqnarray}$
These conditional states have supports on orthogonal pairs, leading to the kernel spaces $\begin{eqnarray}\begin{array}{rcl}\mathrm{Ker}({\hat{\rho }}_{AC| 1}) & = & {\rm{span}}\{| 00\rangle ,| 10\rangle \},\\ \mathrm{Ker}({\hat{\rho }}_{BC| 1}) & = & {\rm{span}}\{| 01\rangle ,| 10\rangle \}.\end{array}\end{eqnarray}$
Since $| 00\rangle \in \mathrm{Ker}({\hat{\rho }}_{AC| 1})$ but $| 00\rangle \notin \mathrm{Ker}({\hat{\rho }}_{BC| 1})$, the inclusion condition fails: $\begin{eqnarray}\begin{array}{r}\mathrm{Ker}({\hat{\rho }}_{AC| 1}) \nsubseteq \mathrm{Ker}({\hat{\rho }}_{BC| 1}),\end{array}\end{eqnarray}$
hence ${\rho }_{\lambda }\notin {{ \mathcal V }}_{4}$ despite both endpoints belonging to ${{ \mathcal V }}_{4}$. This constitutes a strict counterexample to convexity of the VQMC state space. This result illustrates the geometric richness and structural complexity of ${{ \mathcal V }}_{4}$. The failure of the VQMC condition under convex mixing arises from a delicate mismatch between support spaces of conditional marginal states, which under interpolation can cause the kernel inclusion to break down. This suggests that ${{ \mathcal V }}_{4}$ is non-convex and may possess stratified or curved boundaries, unlike more familiar convex sets such as separable or PPT states.4.6. Choi matrix structure and interpretability
In cases where partial information about the recovery map is available (e.g., symmetry constraints or measurement structure), one may directly analyze the structure of the Choi matrix. This can reveal whether the state admits a classical conditioning structure or admits a measurement-based recovery mechanism. Such cases help bridge between operational interpretations and abstract recovery frameworks.
5. Conclusion
In this work, we have extended the framework of VQMCs from tripartite to four-qubit systems, establishing both structural characterizations and computational tools for their verification. We demonstrated that the kernel inclusion criterion provides a necessary condition, analyzed explicit examples such as the W and GHZ states, and developed an SDP-based approach for quantifying the sampling overhead required for virtual recovery. These results reveal the non-convexity of the VQMC set and highlight open directions, including extensions to higher-dimensional systems and more efficient SDP algorithms. Beyond their theoretical value, our findings have potential applications in quantum simulation, error mitigation, and quantum communication protocols.