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abstract: 'We define an infinite dimensional modification of lower-semicomputability of density operators by Gács with an attempt to fix some problem in the paper.  Our attempt is partly achieved by showing the existence of universal operator under some additional assumption.  It is left as a future task to eliminate this assumption.  We also see some properties and examples which stimulate further research.  In particular, we show that universal operator has certain nontrivial form if it exists.' 
author:
- Toru Takisaka
title: 'On Gács’ quantum algorithmic entropy [^1] '
---

[^1]: Most part of this research was carried out without knowing about Tadaki’s work \[15\].  Quite recently, in March 2014, Prof. Tadaki draw our attention to his work and we noticed that there are substantial overlaps between Tadaki’s work and ours.  As far as we know, his work seems to be the first one providing an extension of Gacs’ work to the infinite-dimensional setting.  In the present paper, we tried to reflect Tadaki’s results as possible as we can.  Interested readers should consult Tadaki’s article \[15\].  We are very grateful to Prof. Tadaki for his advices.