Characterization of linear maps on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>M</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>whose multiplicity maps have maximal norm, with an application in quantum information
Bibliographic record
Abstract
Given a linear map<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Φ</mml:mi><mml:mo>:</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math>, its multiplicity maps are defined as the family of linear maps<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Φ</mml:mi><mml:mo>⊗</mml:mo><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>id</mml:mtext></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math>, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>id</mml:mtext></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math>denotes the identity on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>M</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math>. Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mo>⋅</mml:mo><mml:msub><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mn>1</mml:mn></mml:msub></mml:math>denote the trace-norm on matrices, as well as the induced trace-norm on linear maps of matrices, i.e.<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mi mathvariant="normal">Φ</mml:mi><mml:msub><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mo movablelimits="true" form="prefix">max</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mi mathvariant="normal">Φ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mn>1</mml:mn></mml:msub><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mi>X</mml:mi><mml:msub><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo fence="false" stretchy="false">}</mml:mo></mml:math>. A fact of fundamental importance in both operator algebras and quantum information is that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mi mathvariant="normal">Φ</mml:mi><mml:mo>⊗</mml:mo><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>id</mml:mtext></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mn>1</mml:mn></mml:msub></mml:math>can grow with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>. In general, the rate of growth is bounded by<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mi mathvariant="normal">Φ</mml:mi><mml:mo>⊗</mml:mo><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>id</mml:mtext></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mn>1</mml:mn></mml:msub><mml:mo>≤</mml:mo><mml:mi>k</mml:mi><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mi mathvariant="normal">Φ</mml:mi><mml:msub><mml:mo fence="false" stretchy="false">‖</mml:mo><mml:mn>1</mml:mn></mml:msub></mml:math>, and matrix transposition is the canonical example of a map achieving this bound. We prove that, up to an equivalence, the transpose is the unique map achieving this bound. The equivalence is given in terms of complete trace-norm isometries, and the proof relies on a particular characterization of complete trace-norm isometries regarding preservation of certain multiplication relations.We use this result to characterize the set of single-shot quantum channel discrimination games satisfying a norm relation that, operationally, implies that the game can be won with certainty using entanglement, but is hard to win without entanglement. Specifically, we show that the well-known example of such a game, involving the Werner-Holevo channels, is essentially the unique game satisfying this norm relation. This constitutes a step towards a characterization of single-shot quantum channel discrimination games with maximal gap between optimal performance of entangled and unentangled strategies.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; both teacher heads agree on what is shown here.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".