Analysis of SM quantum information

Caleb J O'Loan

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Analysis of SM quantum information

Caleb J O'Loan

Quantum Physics · Submitted 1 May 2008

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that the SM quantum information is not a well-defined metric on the space of density matrices: different choices of phase of the eigenvectors lead to different metrics.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.87 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.91 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Morozova and Chentsov (Morozova and Chentsov 90) studied Riemannian metrics on the set of probability measures. They showed that, up to a constant factor, the Fisher information is the only Riemannian metric which is monotone under stochastic transformation. Sarovar and Milburn (Sarovar and Milburn 06) computed an upper bound on the Fisher information for one-parameter channels. In (O'Loan 07) we extended their bound to an upper bound on the Fisher information of multi-parameter families of states; we call this the SM quantum information. Petz and Sud\'ar (Petz 95) characterized fully the set of monotone metrics on the space of all density matrices. We analyse the SM quantum information in light of their work. We show that the SM quantum information is not a well-defined metric on the space of density matrices: different choices of phase of the eigenvectors lead to different metrics. We define a new metric C_L as a lower bound among the SM quantum informations. We look at properties of C_L and show that it is invariant but not monotone.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Morozova and Chentsov (Morozova and Chentsov 90) studied Riemannian metrics on the set of probability measures.	background 1.00	0.02	0.01	asserted 0.98
s1	They showed that, up to a constant factor, the Fisher information is the only Riemannian metric which is monotone under stochastic transformation.	background 0.98	0.03	0.04	asserted 1.00
s2	Sarovar and Milburn (Sarovar and Milburn 06) computed an upper bound on the Fisher information for one-parameter channels.	background 1.00	0.03	0.02	asserted 1.00
s3	In (O'Loan 07) we extended their bound to an upper bound on the Fisher information of multi-parameter families of states; we call this the SM quantum information.	contribution 0.80	0.90	0.06	asserted 1.00
s4	Petz and Sud\'ar (Petz 95) characterized fully the set of monotone metrics on the space of all density matrices.	background 1.00	0.02	0.02	asserted 1.00
s5	We analyse the SM quantum information in light of their work.	contribution 0.79	0.55	0.04	asserted 0.60
s6	We show that the SM quantum information is not a well-defined metric on the space of density matrices: different choices of phase of the eigenvectors lead to different metrics.	result 0.99	0.95	0.66	asserted 1.00
s7	We define a new metric C_L as a lower bound among the SM quantum informations.	contribution 0.97	0.95	0.17	asserted 1.00
s8	We look at properties of C_L and show that it is invariant but not monotone.	result 0.96	0.95	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.91.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Morozova and Chentsov (Morozova and Chentsov 90) studied Riemannian metrics on the set of probability measures. They showed that, up to a constant factor, the Fisher information is the only Riemannian metric which is monotone under stochastic transformation. Sarovar and Milburn (Sarovar and Milburn 06) computed an upper bound on the Fisher information for one-parameter channels. In (O'Loan 07) we extended their bound to an upper bound on the Fisher information of multi-parameter families of states; we call this the SM quantum information. Petz and Sud\'ar (Petz 95) characterized fully the set of monotone metrics on the space of all density matrices. We analyse the SM quantum information in light of their work. We show that the SM quantum information is not a well-defined metric on the space of density matrices: different choices of phase of the eigenvectors lead to different metrics. We define a new metric C_L as a lower bound among the SM quantum informations. We look at properties of C_L and show that it is invariant but not monotone.

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