Information shared by three systems
Entropy measures uncertainty in a specified state. For a quantum state, von Neumann entropy is calculated from the state's eigenvalues, rather than by assigning a definite hidden classical value to every measurement. Lieb and Ruskai proved a fundamental constraint for three parts A, B and C: the entropy of ABC plus that of B is no greater than the entropy of AB plus that of BC. This is strong subadditivity.
A classical example you can count
Use two independent fair bits A and C. Define B to be zero when they agree and one when they differ. There are four equally likely triples. In this invented example, ABC has two bits of entropy, B has one, and each pair AB and BC has two. The inequality becomes 2 + 1 less than or equal to 2 + 2. All four entropies refer to the same joint distribution.
Why conditioning can reveal a connection
Before B is known, the bits A and C are independent. If B is zero, learning A tells you C; if B is one, learning A tells you the opposite of C. The difference between the two sides of the inequality is one bit here. This example helps visualize conditional correlation. Counting four classical outcomes does not prove the quantum theorem, whose states and operators require a different argument.
What the original proof supplies
The original uses operator trace inequalities and links the entropy result with a convexity statement. It also addresses infinite-dimensional extensions through approximation, including an appendix contributed by Barry Simon. Simon's appendix contribution remains distinct from the two-author main paper. Expressions involving infinite entropies require care; an undefined subtraction of infinities is not an application of the finite-dimensional formula.
A bounded way to continue
Recalculate the classroom example from its four rows, then replace B with a fixed bit and compare the entropies. Keep the distribution and units visible. A source explanation can distinguish strong subadditivity from later equality and recovery results, entropy-power inequalities and other entropy conjectures. Similar terminology does not make these the same mathematical contribution.
Scientific scope
The displayed entropy combination is recorded where its terms are well-defined; it is not interpreted by subtracting divergent infinities in arbitrary infinite-dimensional states. This theorem is not a Shannon/Wehrl minimum-entropy conjecture, entropy-power inequality, or an equality/recovery characterization from later work.
Read the original work
Sources and reading locations for this explanation
- Proof of the Strong Subadditivity of Quantum-Mechanical Entropy
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People and systems
Elliott H. Lieb; Mary Beth Ruskai
Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Barry Simon
Author of the appendix on entropy convergence used for infinite-dimensional extensions
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