Quantum Inf Process (2010) 9:61–73
Partitioned trace distances
Alexey E. Rastegin
Received: 8 May 2009 / Accepted: 12 August 2009 / Published online: 29 August 2009
© Springer Science+Business Media, LLC 2009
Abstract New quantum distance is introduced as a half-sum of several singular
values of difference between two density operators. This is, up to factor, the met-
ric induced by so-called Ky Fan norm. The partitioned trace distances enjoy similar
properties to the standard trace distance, including the unitary invariance, the strong
convexity and the close relations to the classical distances. The partitioned distances
cannot increase under quantum operations of certain kind including bistochastic maps.
All the basic properties are re-formulated as majorization relations. Possible applica-
tions to quantum information processing are brieﬂy discussed.
Keywords Trace distance · Ky Fan’s norm · Metric properties · Strong convexity ·
Majorization · Monotonicity · Quantum operation
PACS 03.67.-a · 03.65.Ta · 02.10.Ud
A quantiﬁcation of closeness of quantum states is inevitable task for quantum informa-
tion processing . On the other hand, the spaces of quantum states are very interesting
mathematical subjects . If states are pure then comparison of them is not difﬁcult.
But all the real devices are exposed to noise. So the pure states used will eventually
evolve to mixed states. This is not a unique reason for consideration of mixed states.
As it is shown in , the cloning machine which can use any mixed states in symmetric
space is very important in quantum computation. There are many ways to compare
two mixed quantum states. It has been found that two measures, the trace distance
A. E. Rastegin (
Department of Theoretical Physics, Irkutsk State University, Gagarin Bv. 20, 664003 Irkutsk, Russia