Quantum Marginal Problems

The quantum marginal problem asks whether specified reduced states can come from a common global state satisfying stated conditions. Pure versus mixed global states, overlapping versus disjoint subsystems, symmetry, and particle statistics lead to different problems. Reviewed 11 October 2026, this page uses exact marginal results and semidefinite methods through the selected 2024 review as a baseline for bounded research projects.

Helpful background. Marginals and correlations and partial trace supply the stable definitions.

Compatibility is a global condition

Given candidate marginals $\rho_{AB}$ and $\rho_{BC}$, agreement on their shared system,

$$\operatorname{Tr}_A\rho_{AB} = \operatorname{Tr}_C\rho_{BC},$$

is necessary but not sufficient for a common tripartite state. For example, if $AB$ is in a pure entangled state, every compatible global extension factorizes as $\rho_{AB}\otimes\rho_C$. It cannot also make $BC$ a pure entangled pair.

For a mixed global state of fixed finite dimension, feasibility can be expressed as

$$\rho_{ABC}\ge0,\quad \operatorname{Tr}\rho_{ABC}=1,$$

with linear marginal constraints. Adding purity is nonconvex; adding fermionic structure changes the allowed Hilbert space and representability constraints.

Klyachko’s work resolves important spectral compatibility questions. It does not make arbitrary overlapping-marginal reconstruction or all N-representability instances easy.

Worked example: two noisy singlets sharing one qubit

Let both requested marginals be $\rho(p)=pP_s+(1-p)I_4/4$, for $0\le p\le1$, where $P_s$ projects onto the two-qubit singlet. They agree on $B$, whose marginal is $I_2/2$, at every $p$. Nevertheless, a common mixed three-qubit extension exists exactly for $p\le2/3$.

To prove necessity, consider $K=P_s^{AB}+P_s^{BC}$, with identities on omitted systems. Writing $\mathbf S_i=\boldsymbol\sigma_i/2$ gives

$$K=\tfrac12 I-\mathbf S_B\cdot(\mathbf S_A+\mathbf S_C).$$

Couple $A,C$ first. Their spin is zero or one. Coupling spin one to $B$ gives total spin $1/2$ or $3/2$, with $\mathbf S_B\cdot(\mathbf S_A+\mathbf S_C)=-1$ or $1/2$; the $AC$ singlet gives zero. Hence $\lambda_{\max}(K)=3/2$. Any extension must satisfy

$$\operatorname{Tr}(K\rho_{ABC}) =2\operatorname{Tr}[P_s\rho(p)] =\frac{1+3p}{2}\le\frac32.$$

For sufficiency, let $\rho_*$ be the normalized projector onto the two-dimensional subspace with $AC$ spin one and total spin $1/2$. Rotational invariance and $A\leftrightarrow C$ symmetry make its $AB$ and $BC$ marginals equal Werner states. Their singlet weights sum to $3/2$, so each has weight $3/4$, or $p=2/3$. Mixing $\rho_*$ with $I_8/8$ realizes every smaller nonnegative $p$. This is a hand-checkable instance of the extension geometry studied more generally by Johnson and Viola, not a new threshold conjecture.

The calculation supplies both a primal extension and a dual obstruction. It also distinguishes two questions: the pair states are entangled already for $p>1/3$, but compatible with a common mixed extension up to $2/3$. “Entangled” does not mean “impossible to share any correlations.”

Established algorithms and the unresolved target

For fixed finite dimensions, arbitrary mixed-state compatibility is one SDP in the global density matrix, whose size grows exponentially in particle number. Yu and colleagues supply a complete hierarchy for the additional pure-state requirement. Completeness of an infinite hierarchy is different from an efficient finite-level decision rule with a prescribed noise tolerance. The 2024 review by Tavakoli and colleagues gives the broader relaxation and certification framework.

Active targets therefore need a resource or inference restriction: a smaller certificate for a specified overlap graph, robustness under a specified confidence region, or a provably optimal bound on one property without reconstructing the full state. Do not propose “solve the pure-state marginal problem with an SDP hierarchy” as an untouched question. Nor does a hierarchy’s low-level feasibility prove that a pure completion exists.

Research opportunities

Robust data. Experimental estimates are uncertain and may be mutually inconsistent because of finite samples. A useful method distinguishes statistical incompatibility from a genuine model violation.

Physical restrictions. Locality, symmetry, particle number, and low-energy constraints can make a broad hard problem tractable for a useful subclass.

Certificates. A dual witness explaining incompatibility is more informative than a solver’s failure flag. It can often be translated into a measurable linear combination of local observables.

Reconstruction ambiguity. Compatible marginals need not determine a unique global state. A protocol should bound the target property across all compatible states rather than selecting one convenient completion.

An entry project: incompatible marginals with finite error bars

Baseline. Reproduce the pure-entangled-pair obstruction for three qubits analytically and with an SDP. Then add depolarizing noise to the two candidate pair states.

Resources. A laptop, partial traces, convex optimization, and elementary statistical confidence regions.

Deliverable. A compatibility threshold with primal extensions and dual witnesses. Replace exact marginals by uncertainty sets and report the range of a global observable over all consistent states.

Start by reproducing $p=2/3$ above with $8\times8$ matrices and the analytic witness $3I/2-K$. If each fitted pair density matrix is within trace distance $\varepsilon$ of its true value, each singlet expectation changes by at most $\varepsilon$. A fitted sum above $3/2+2\varepsilon$ therefore certifies incompatibility throughout those two error balls. This elementary margin is a useful solver and statistics control before optimizing a sharper witness. Use simultaneous confidence regions when the two estimates come from finite data; separate confidence levels do not automatically give the desired joint coverage.

Failure criterion. An optimizer returning one global state does not establish uniqueness. Incompatibility claimed at a noise level smaller than tomography or solver error is not certified.

Research extension. Add a physically motivated symmetry or local-energy constraint and derive an analytic threshold or a scalable witness. Another direction compares which additional local measurement most reduces the uncertainty of a desired global property.

For an independent ambiguity check, the states $(|000\rangle\pm|111\rangle)/\sqrt2$ have identical two-body marginals but opposite $\langle X_AX_BX_C\rangle$. Optimizing this observable over their shared marginal constraints must return the full interval $[-1,1]$. Stop and repair the formulation if the program reports a unique completion or a narrower interval without an additional assumption. A useful extension chooses extra experimentally accessible data that provably shrink such an interval; recovering one preferred state numerically is not enough.

Choosing a useful target

Marginal methods are valuable when full tomography is impossible but selected correlations are accessible. Choose the global question first: entanglement, energy, entropy, or compatibility with a model. Reconstructing an entire state when only one property matters can waste resources and obscure uncertainty.

Progress is a sharper certificate, a class with provable efficient structure, or a robust inference procedure validated on known instances. Avoid claiming that pairwise consistency proves global consistency or that spectral constraints characterize all matrix-valued marginal data.

References

• P. D. Johnson and L. Viola, “Compatible quantum correlations: Extension problems for Werner and isotropic states,” Physical Review A 88, 032323 (2013). Exact extension geometry.
• A. A. Klyachko, “Quantum marginal problem and N-representability,” Journal of Physics: Conference Series 36, 72–86 (2006). arXiv:quant-ph/0511102.
• J. Preskill, “Quantum Shannon Theory,” lecture notes, arXiv:1604.07450 (2016). Background on entropy and correlations.
• A. Tavakoli, A. Pozas-Kerstjens, P. Brown, and M. Araújo, “Semidefinite programming relaxations for quantum correlations,” Reviews of Modern Physics 96, 045006 (2024). Methods and limitations.
• X.-D. Yu, T. Simnacher, N. Wyderka, H. C. Nguyen, and O. Gühne, “A complete hierarchy for the pure state marginal problem in quantum mechanics,” Nature Communications 12, 1012 (2021). Pure-state theorem.

Entry points

Prove the pure-marginal factorization statement before running a hierarchy. Read a marginal theorem’s input carefully: eigenvalues, full matrices, and particle-reduced density matrices are different data.