What is the conjecture
This is problem #41 in the Open Quantum Problems collection.
Problem (Open Quantum Problem #41: “All rank inequalities for reduced states of quadripartite quantum states”).
Given any tripartite density matrix ρ_{ABC}, let
r_{AB} = rank(Tr_C(ρ_{ABC}))
denote the rank of the corresponding marginal. Prove or find a counterexample to the following hypothesis:
r_{AB} ≤ r_{AC} r_{BC}.
A standard formalization is as follows:
- Fix finite-dimensional complex Hilbert spaces
H_A, H_B, H_C.
- A tripartite mixed state is a density operator
ρ_{ABC} on H_A ⊗ H_B ⊗ H_C (positive semidefinite, trace 1).
- For each subsystem
X ⊆ {A,B,C}, write ρ_X for the reduced state obtained by partial trace over the complementary parties, and define
r_X := rank(ρ_X).
- The singled-out conjectured inequality is
r_{AB} ≤ r_{AC} r_{BC}.
- Since the 0-Rényi entropy is
S_0(ρ) = log rank(ρ), the same conjecture can be written additively as
S_0(AB) ≤ S_0(AC) + S_0(BC).
The problem is often viewed through purifications to four-party pure states:
- every tripartite mixed state
ρ_{ABC} admits a purification |ψ⟩_{ABCD},
- for pure four-party states, complementary subsystem ranks agree (
r_X = r_{X^c}),
- and the conjecture is equivalently expressible (up to relabeling/complements) in the form highlighted by Cadney–Huber–Linden–Winter:
r_{BC} ≤ r_{AB} r_{AC} for every pure state |ψ⟩_{ABCD}.
This is why the OQP title is about reduced states of quadripartite quantum states, even though the explicit hypothesis is stated for a tripartite mixed state.
The broader classification problem is:
- Main question / rank-inequality classification problem: determine all universal multiplicative inequalities among the ranks of reduced states of four-party quantum states, equivalently all universal linear inequalities among the 0-Rényi entropies
S_0(X)=log r_X.
Cadney–Huber–Linden–Winter showed that several families of four-party linear S_0-inequalities are always valid, including
S_0(A) ≥ 0,S_0(A) + S_0(B) ≥ S_0(AB),S_0(AB) + S_0(AC) ≥ S_0(A),S_0(AB) + S_0(AC) + S_0(BC) ≥ 2 S_0(A),
together with all versions obtained by permuting parties (and, more generally, substituting pairwise disjoint subsets of parties). They further showed that completeness of the four-party linear rank-inequality picture would follow from the conjectured inequality above.
Where to find the details / references
Primary sources:
Important later update / likely resolution:
- Z. Song, L. Chen, Y. Sun, and M. Hu, “A complete picture of the four-party linear inequalities in terms of the 0-entropy”
(claims to prove the conjectured inequality and complete the classification):
Key related references (as listed on the OQP page):
- E. H. Lieb and M.-B. Ruskai, “Proof of the strong subadditivity of quantum-mechanical entropy” (J. Math. Phys. 14:1938–1941 (1973))
- N. Linden, M. Mosonyi, and A. Winter, “The structure of Renyi entropic inequalities” (Proc. R. Soc. A 469(2158):20120737 (2013); arXiv: 1212.0248)
- N. Pippenger, “The inequalities of quantum information theory” (IEEE Trans. Inf. Theory 49(4):773–789 (2003))
- R. W. Yeung and Z. Zhang, “On Characterization of Entropy Function via Information Inequalities” (IEEE Trans. Inf. Theory 44(4):1440–1452 (1998))
- N. Linden and A. Winter, “A New Inequality for the von Neumann Entropy” (Commun. Math. Phys. 259:129–138 (2005); arXiv: quant-ph/0406162)
Prerequisites needed
- Finite-dimensional quantum mechanics: density matrices, tensor products, partial trace, purification
- Reduced states and their ranks; Schmidt rank across bipartitions of pure states
- Rényi-0 entropy (
S_0 = log rank) and the entropy-vector viewpoint
- Linear algebra / matrix analysis: rank inequalities, tensor/Kronecker products, support subspaces
- (Optional) Convex-cone / extremal-ray language for entropy or log-rank vectors
- ams-81 (Quantum theory)
- ams-15 (Linear and multilinear algebra; matrix theory)
- ams-47 (Operator theory)
- ams-94 (Information and communication theory)
Choose either option
What is the conjecture
This is problem #41 in the Open Quantum Problems collection.
A standard formalization is as follows:
H_A,H_B,H_C.ρ_{ABC}onH_A ⊗ H_B ⊗ H_C(positive semidefinite, trace1).X ⊆ {A,B,C}, writeρ_Xfor the reduced state obtained by partial trace over the complementary parties, and definer_X := rank(ρ_X).r_{AB} ≤ r_{AC} r_{BC}.S_0(ρ) = log rank(ρ), the same conjecture can be written additively asS_0(AB) ≤ S_0(AC) + S_0(BC).The problem is often viewed through purifications to four-party pure states:
ρ_{ABC}admits a purification|ψ⟩_{ABCD},r_X = r_{X^c}),r_{BC} ≤ r_{AB} r_{AC}for every pure state|ψ⟩_{ABCD}.This is why the OQP title is about reduced states of quadripartite quantum states, even though the explicit hypothesis is stated for a tripartite mixed state.
The broader classification problem is:
S_0(X)=log r_X.Cadney–Huber–Linden–Winter showed that several families of four-party linear
S_0-inequalities are always valid, includingS_0(A) ≥ 0,S_0(A) + S_0(B) ≥ S_0(AB),S_0(AB) + S_0(AC) ≥ S_0(A),S_0(AB) + S_0(AC) + S_0(BC) ≥ 2 S_0(A),together with all versions obtained by permuting parties (and, more generally, substituting pairwise disjoint subsets of parties). They further showed that completeness of the four-party linear rank-inequality picture would follow from the conjectured inequality above.
Where to find the details / references
Primary sources:
(introduces the conjectured missing four-party rank inequality / 0-entropy inequality):
Important later update / likely resolution:
(claims to prove the conjectured inequality and complete the classification):
Key related references (as listed on the OQP page):
Prerequisites needed
S_0 = log rank) and the entropy-vector viewpointAMS categories
Choose either option