There are states about which the whole is completely known and the parts are not known at all.
Paper 88 wrote that under the single word entropy there are two different quantities, the coarse-grained one and the horizon-area one. This paper adds a third address: the von Neumann entropy. NO NEW PHYSICAL LAW IS CLAIMED. The von Neumann entropy, subadditivity, the Araki-Lieb triangle inequality and the negativity of the conditional entropy are all standard. Nothing in Paper 88 is retracted: the separation of the two entropies stands, and a third is simply added.
First, it is not Shannon’s entropy quantised. Shannon’s entropy requires a probability distribution, and a quantum state does not come with one, since a different distribution appears for each basis one measures in. Collecting the Shannon quantity over twenty thousand measurement bases for random single-qubit mixed states, the value measured in the eigenbasis equalled the von Neumann entropy to machine precision, with a difference of zero in all four cases, while random bases only approached it from above. The von Neumann entropy is therefore the smallest Shannon quantity obtainable by measurement: in any other basis, more disorder is reported than the state actually holds. The two start from different data – Shannon is handed a distribution and returns a number, while von Neumann is handed a state and chooses the distribution himself.
Second, unitary evolution does not change it at all. Over four thousand random four-dimensional density matrices and random unitaries, the largest change was 2.33 times ten to the minus fifteen. It is exactly conserved for a closed system. So the von Neumann entropy cannot by itself be the arrow of time: the arrow requires something non-unitary, a partial trace, a measurement or a coarse-graining (Papers 77, 78 and 88).
Third, and this is the core. The whole can carry less than a part. For the singlet state the entropy of the pair is zero, being a pure state, while either half alone has entropy 0.6931471806, that is the natural logarithm of two. Classically this cannot happen: drawing two hundred thousand random two-by-two joint distributions and measuring the entropy of the whole minus the larger of the two parts, the smallest value found was 0.007226 and it was never once negative. Classically, knowing the whole includes knowing the parts; quantum mechanically it does not, and there are states about which the pair is completely known and either half alone is not known at all. This is not a statement that quantum systems are more disordered: an entropy of zero for the pair says there is no disorder whatever. It says only that the information has been placed in the correlation rather than in the parts.
Fourth, the conditional entropy therefore turns negative. For the singlet it is minus 0.6931471806, that is minus the logarithm of two, while over the same two hundred thousand classical draws the smallest value was 0.009555. The classical conditional entropy is bounded below by zero and the quantum one reaches minus the logarithm of two – the two do not even share a sign convention, being written with the same symbol and called by the same name while having different ranges. Negative entropy is not a figure of speech but has an operational meaning, readable as a resource for future use in quantum state merging; this paper does not enter that interpretation, since what can be written is that the value is negative, and its address.
Fifth, what stands in its place. Over twenty thousand random two-qubit states of random rank, subadditivity and the Araki-Lieb triangle inequality were violated zero times, and the singlet lies exactly on the boundary of the triangle inequality, the difference between the parts being zero and the whole being zero as well. The classical inequality is not weakened but replaced.
The census. The coarse-grained entropy lives in the observer’s resolution and is moved by changing the coarse-graining (Papers 77, 78, 88). The horizon-area entropy lives in the boundary area and dimension and is moved by changing the area (Papers 73, 88). The von Neumann entropy lives in the spectrum of the density matrix and is moved only by a non-unitary operation. Three quantities under one word: the first lives in the observer, the second in the boundary, and the third in the state itself. Of the three, only the third is unchanged by the passage of time.
Entanglement entropy is known to obey an area law in field theory, so there is a rhyme between the third and the second. This paper does not write that they share a root: one is fixed by the choice of subsystem, the other by geometry. The line Paper 88 drew, that the only relation is a bound, is not crossed here either.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new physical law (there is none) / any retraction of Paper 88 (there is none) / an interpretation of negative entropy (not entered). What is added: measuring that the eigenbasis Shannon quantity equals the von Neumann entropy, checking unitary invariance numerically, confirming on two hundred thousand samples that the classical inequality never fails and setting it against the quantum minus logarithm of two, and writing for each of the three entropies what must be moved for it to move. What remains: there are states about which the whole is completely known and the parts are not known at all. In classical language that sentence contradicts itself. It does not contradict itself because the information has been placed not in the parts but between them.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Whole Can Be Better Known Than Its Parts [Paper 162]. Zenodo.
DOI: 10.5281/zenodo.22092568
ORCID: 0009-0006-8587-3450