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Compatible-Marginal Spin Alignment for Three Qubits

Haoran Wang

Published Oct 9, 2026• 0 citations
Publisher

Abstract

The strong spin-alignment conjecture fails for three qubits when the subset states are chosen independently. We prove that it holds when they are marginals of one global state: for every three-qubit state, every positive semidefinite qubit operator Q, and every nonnegative set of subset weights, the aligned output majorizes the corresponding marginal mixture. The proof classifies the extreme rays of cones on which a fixed set of aligned eigenvalues is largest. This reduces arbitrary weights to two-term sums and the complete one- and two-body layers. For arbitrary numbers of subsystems, we also characterize when one unital quantum channel simultaneously realizes all marginal replacements. This criterion proves alignment for fully PPT states. A separate argument proves alignment for locally rotated stabilizer states in every prime local dimension, at maximally mixed background. These results determine the minimum output Rényi entropies of the associated three-qubit replacement channels. Under either of the two input restrictions above, they also determine the many-use classical capacity and the exact strong-converse exponent for correlated replacement channels of arbitrary size.

DOI
10.5281/zenodo.23253914

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The strong spin-alignment conjecture fails for three qubits when the subset states are chosen independently. We prove that it holds when they are marginals of one global state: for every three-qubit state, every positive semidefinite qubit operator Q, and every nonnegative set of subset weights, the aligned output majorizes the corresponding marginal mixture. The proof classifies the extreme rays of cones on which a fixed set of aligned eigenvalues is largest. This reduces arbitrary weights to two-term sums and the complete one- and two-body layers. For arbitrary numbers of subsystems, we also characterize when one unital quantum channel simultaneously realizes all marginal replacements. This criterion proves alignment for fully PPT states. A separate argument proves alignment for locally rotated stabilizer states in every prime local dimension, at maximally mixed background. These results determine the minimum output Rényi entropies of the associated three-qubit replacement channels. Under either of the two input restrictions above, they also determine the many-use classical capacity and the exact strong-converse exponent for correlated replacement channels of arbitrary size.
We prove a spectral majorization law for compatible three-qubit marginals: for every global state, every common single-qubit replacement state, and arbitrary nonnegative weights on all eight supports, the weighted marginal sum is majorized by its perfectly aligned counterpart. For normalized weights, this simultaneously minimizes every Schur-concave output entropy. We characterize equality, establish stability bounds and an incompatibility witness, and determine the single-block Holevo information for correlated unbiased replacement. We also develop input-certification protocols with explicit measurement, noise, calibration, and sample-complexity assumptions, together with rigorously delimited four-qubit results and a channel-compatibility example. The unrestricted noncommuting four-qubit middle-rank problem remains open. Complete proofs, exact certificates, and reproducible source are included.
The spin alignment conjecture was originally formulated in connection with the additivity of coherent information for a class of quantum channels known as platypus channels. Recently, a stronger majorization-based version was proposed by M. A. Alhejji and E. Knill [Commun. Math. Phys. 405, 119, 2024], asserting that the spectrum of the alignment operator is always majorized by that of the perfectly aligned configuration. In this letter, we show that this strong spin alignment conjecture is false in general by constructing an explicit counterexample in the smallest unresolved case, namely three qubits. The example uses two-body states that are not jointly compatible with any single three-qubit global state, which naturally leads to a compatibility-constrained variant of the conjecture.
Determining whether a mixed quantum state is uniquely determined among all states by its k-body marginals (k-UDA) is a fundamental problem in quantum system certification. We develop a range-based approach to this problem by analyzing the structure of the range of the global state. For three-qubit states, we show that states with GHZ-SLOCC-free ranges are 2-UDA at ranks one, three, and four. We derive a necessary and sufficient range criterion for rank-two 2-UDA states and reduce it to a finite quadratic-form test. To cover the remaining range configurations, we formulate an exact range-restricted semidefinite programming criterion and extend it to arbitrary finite-dimensional tripartite states. We also show that every three-qubit state of rank at least five is not 2-UDA, and further extend high-rank obstructions to multipartite systems. For a channel-based multipartite family, we characterize exactly when a state is (n−1)(n-1)(n−1)-UDA and show that lower-order marginals never suffice. Finally, we apply these results to the certification of genuine multipartite entanglement.
Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In 2⊗22\otimes22⊗2 and 2⊗32\otimes32⊗3 systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition
The von Neumann entropy plays a vital role in quantum information theory. The von Neumann entropy determines, e.g., the capacities of quantum channels. Also, entropies of composite quantum systems are important for future quantum networks, and their characterization is related to the quantum marginal problem. Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropies of multipartite quantum systems is studied. The problem of its characterization is not new -- however, progress has been sparse, indicating that the problem might be hard and that new methods might be needed. Here, a variety of different and complementary approaches are taken. First, I look at global properties. It is known that the von Neumann entropy region -- just like its classical counterpart -- forms a convex cone. I describe the symmetries of this cone and highlight geometric similarities and differences to the classical entropy cone. In a different approach, I utilize the local geometric properties of extremal rays of a cone. I show that quantum states whose entropy lies on such an extremal ray of the quantum entropy cone have a very simple structure. As the set of all quantum states is very complicated, I look at a simple subset called stabilizer states. I improve on previously known results by showing that under a technical condition on the local dimension, entropies of stabilizer states respect an additional class of information inequalities that is valid for random variables from linear codes. In a last approach I find a representation-theoretic formulation of the classical marginal problem simplifying the comparison with its quantum mechanical counterpart. This novel correspondence yields a simplified formulation of the group characterization of classical entropies (IEEE Trans. Inf. Theory, 48(7):1992-1995, 2002) in purely combinatorial terms.
We prove an exponential strong converse for the unassisted private capacity of every finite-dimensional degradable quantum channel. We use a single error criterion: the infidelity between the actual message-estimate-environment state and an ideal state consisting of uniformly distributed, perfectly correlated messages that are independent of the environment. At every rate strictly above the channel's maximum coherent information, the optimized fidelity to this ideal state decays exponentially in the number of channel uses. The same bound holds when the ideal environment state is fixed to the actual environment marginal. The proof applies to arbitrary mixed-state encoders and arbitrary collective decoders. Its main operator ingredient is an overlap bound for privacy tests: two receiver placements with a common subsystem of dimension dCd_CdC​ have overlap at most dC/Md_C/MdC​/M, where MMM is the key size, independently of both shield dimensions. We combine this bound with the signed averaging construction presented in arXiv:2608.01308 and a single-filter reduction to symmetric states. A Gaussian filter replaces the common receiver subsystem by a system of controlled dimension from which the original correlations can be approximately recovered using quantum side information. The filter preserves exact exchange symmetry, and a uniform smooth-entropy estimate bounds its output dimension for arbitrary correlated channel inputs. We also obtain an exponential strong converse at zero for antidegradable channels under the same joint criterion and, consequently, for the private capacity of the quantum erasure channel throughout its parameter range.