dorsal/arxiv
View SchemaInfluence-free states on compound quantum systems
| Authors | Howard Barnum, Christopher A. Fuchs, Joseph M. Renes, Alexander Wilce |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0507108 |
| URL | https://arxiv.org/abs/quant-ph/0507108 |
Abstract
Let Alice and Bob be able to make local quantum measurements and communicate classically. The set of mathematically consistent joint probability assignments (``states'') for such measurements is properly larger than the set of quantum-mechanical mixed states for the Alice-Bob system. It is canonically isomorphic to the set of positive (not necessarily completely positive) linear maps Phi from the bounded linear operators on Alice's space to those on Bob's, for which Tr Phi(I)=1. We review the fact that allowing classical communication is equivalent to enforcing ``no-instantaneous-signalling'' (``no--influence'') in the direction opposite the communication. We establish that in the subclass of ``decomposable'' states, i.e. convex combinations of positive states with "PTP" ones whose partial transpose is positive, the extremal states are just the extremal positive and extremal PTP states. We show that two such states, shared by the same pair of parties, cannot necessarily be combined as independent states (their tensor product) if the full set of quantum operations is allowed locally to each party. We use a framework of ``test spaces'' and states on these, suited for exhibiting the analogies and deviations of empirical probabilistic theories from classical probability theory. This leads to a deeper understanding of analogies between quantum mechanics and Bayesian probability theory. The existence of a ``most Bayesian'' quantum rule for updating states after measurement, and its association with the situation when information on one system is gained by measuring another, is a case of a general proposition holding for test spaces combined subject to the no-signalling requirement.
{
"annotation_id": "6bbe87c6-c68a-48af-a569-f31cb61c73dc",
"date_created": "2026-03-02T18:02:17.051000Z",
"date_modified": "2026-03-02T18:02:17.051000Z",
"file_hash": "0dd661b93216d77df0dfeb5bbcea44a7cd8dfbf0ffadb94b21d40d70cc15ddde",
"private": false,
"record": {
"abstract": "Let Alice and Bob be able to make local quantum measurements and communicate\nclassically. The set of mathematically consistent joint probability assignments\n(``states\u0027\u0027) for such measurements is properly larger than the set of\nquantum-mechanical mixed states for the Alice-Bob system. It is canonically\nisomorphic to the set of positive (not necessarily completely positive) linear\nmaps Phi from the bounded linear operators on Alice\u0027s space to those on Bob\u0027s,\nfor which Tr Phi(I)=1. We review the fact that allowing classical communication\nis equivalent to enforcing ``no-instantaneous-signalling\u0027\u0027 (``no--influence\u0027\u0027)\nin the direction opposite the communication. We establish that in the subclass\nof ``decomposable\u0027\u0027 states, i.e. convex combinations of positive states with\n\"PTP\" ones whose partial transpose is positive, the extremal states are just\nthe extremal positive and extremal PTP states. We show that two such states,\nshared by the same pair of parties, cannot necessarily be combined as\nindependent states (their tensor product) if the full set of quantum operations\nis allowed locally to each party. We use a framework of ``test spaces\u0027\u0027 and\nstates on these, suited for exhibiting the analogies and deviations of\nempirical probabilistic theories from classical probability theory. This leads\nto a deeper understanding of analogies between quantum mechanics and Bayesian\nprobability theory. The existence of a ``most Bayesian\u0027\u0027 quantum rule for\nupdating states after measurement, and its association with the situation when\ninformation on one system is gained by measuring another, is a case of a\ngeneral proposition holding for test spaces combined subject to the\nno-signalling requirement.",
"arxiv_id": "quant-ph/0507108",
"authors": [
"Howard Barnum",
"Christopher A. Fuchs",
"Joseph M. Renes",
"Alexander Wilce"
],
"categories": [
"quant-ph"
],
"title": "Influence-free states on compound quantum systems",
"url": "https://arxiv.org/abs/quant-ph/0507108"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "955e34d0-e464-4c16-8f94-003e5652e48f",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}