The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation

Abstract

Stinespring's dilation theorem is the basic structure theorem for quantum channels: it states that any quantum channel arises from a unitary evolution on a larger system. Here we prove a continuity theorem for Stinespring's dilation: if two quantum channels are close in cb-norm, then it is always possible to find unitary implementations which are close in operator norm, with dimensionindependent bounds. This result generalizes Uhlmann's theorem from states to channels and allows to derive a formulation of the information-disturbance tradeoff in terms of quantum channels, as well as a continuity estimate for the no-broadcasting theorem. We briefly discuss further implications for quantum cryptography, thermalization processes, and the black hole information loss puzzle.

Bures Distance for Completely Positive Maps

“…We provide several examples with explicit computations of the Bures distance for CP-maps. In particular, we show that the infimum in the definition of Bures metric may not be attained in all common representation modules, answering a question raised in [12,13]. It turns out that the example is quite simple involving CP-maps on 2 × 2 matrix algebra.…”

“…This is true for states ( [4]). The question in the general case was asked by [12,13]. Here we resolve it in the negative through a simple counter example.…”

Weak Decoupling Duality and Quantum Identification

“…The difference between Eqs. (2) and (3) is precisely the difference between two norms on superoperators, the naïve one inherited from the trace norm, and the so-called completely bounded norm [10], [11], [7]. Not surprisingly, Eq.…”

“…Here we will use a clean approximate formulation due to Renes [26]. The second result is the continuity of the Stinespring dilation of a quantum channel, established by Kretschmann et al [7]. Here we only need a corollary, which can be interpreted as a bound on the information-disturbance trade-off.…”

“…But by Lemma 23 this means that the channel from the code qubit to the environment is arbitrarily close to a constant map in the diamond norm. At this point we can then invoke Theorem 3 on information-disturbance [7] to conclude that the channel from the code qubit to B n can be arbitrarily well error-corrected. (Note that this argument is following our proof of the weak decoupling duality; in particular, any 2-dimensional subspace of a quantum-ID code, and in fact any subspace of sufficiently small dimension, is a quantum errorcorrecting code!)…”