A QIG review of the paper “Entropy uncertainty and measurement reversibility”

I’d like to highlght a video we’ve recently uploaded of a recent talk by Kais Abdelhkalek who presented a review and overview of the recent paper: “Entropic uncertainty and measurement reversibility” by Mario Berta, Stephanie Wehner, Mark M. Wilde, arXiv:1511.00267

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One Response to A QIG review of the paper “Entropy uncertainty and measurement reversibility”

Nice video Kais and thanks for presenting our work! To answer some of the questions that came up at the end, the connection to measurement uncertainty is not so clear to me at the moment. Usually in these kinds of uncertainty relations, there is some relation or trade-off between “noise” and “disturbance”. I think the measurement reversibility term is clearly related to disturbance, but then there is not clearly a term in there related to noise. However, one could argue that H(X|B) is connected to noise, because it captures how well one could guess the outcome of the measurement. But I think there is more to understand here, because the usual measurement uncertainty relations are about how well one can guess the outcome of an ideal measurement of X using an actual instrument compared to how much disturbance this instrument causes to a future measurement of Z (and various ways of handling this). It could be interesting to apply the relative entropy recovery theorem and the approach in http://arxiv.org/abs/1310.6603 and see what comes out. I haven’t tried this yet…

Regarding the other question, “Is it related to the map that Manny and I came up with?”, I think this sounded like Howard Barnum! The answer is “yes!”, it is very much so related to this map, but we call it the Petz recovery map because it was discovered about 15 years before the work of Barnum and Knill by Denes Petz (Barnum and Knill seem to be unaware of Petz’s work if you read their 2001 paper). It would be ideal to have the Petz recovery map in there as the recovery channel in the remainder term for data processing, but no one currently knows how to prove it (even though it is a running conjecture with lots of numerical evidence supporting it at this point). The best thing we can come up with to put in there is a “rotated, twirled” Petz recovery map, which comes about from an application of the Stein-Hirschman complex interpolation theorem. You can find details of all of this explained in Chapter 12 of http://arxiv.org/abs/1106.1445

Nice video Kais and thanks for presenting our work! To answer some of the questions that came up at the end, the connection to measurement uncertainty is not so clear to me at the moment. Usually in these kinds of uncertainty relations, there is some relation or trade-off between “noise” and “disturbance”. I think the measurement reversibility term is clearly related to disturbance, but then there is not clearly a term in there related to noise. However, one could argue that H(X|B) is connected to noise, because it captures how well one could guess the outcome of the measurement. But I think there is more to understand here, because the usual measurement uncertainty relations are about how well one can guess the outcome of an ideal measurement of X using an actual instrument compared to how much disturbance this instrument causes to a future measurement of Z (and various ways of handling this). It could be interesting to apply the relative entropy recovery theorem and the approach in http://arxiv.org/abs/1310.6603 and see what comes out. I haven’t tried this yet…

Regarding the other question, “Is it related to the map that Manny and I came up with?”, I think this sounded like Howard Barnum! The answer is “yes!”, it is very much so related to this map, but we call it the Petz recovery map because it was discovered about 15 years before the work of Barnum and Knill by Denes Petz (Barnum and Knill seem to be unaware of Petz’s work if you read their 2001 paper). It would be ideal to have the Petz recovery map in there as the recovery channel in the remainder term for data processing, but no one currently knows how to prove it (even though it is a running conjecture with lots of numerical evidence supporting it at this point). The best thing we can come up with to put in there is a “rotated, twirled” Petz recovery map, which comes about from an application of the Stein-Hirschman complex interpolation theorem. You can find details of all of this explained in Chapter 12 of http://arxiv.org/abs/1106.1445