  Quantum measurement is a basic tool to manifest intrinsic quantum effects
from fundamental tests to quantum information applications. While a measurement
is typically performed to gain information on a quantum state, its role in
quantum technology is indeed manifold. For instance, quantum measurement is a
crucial process element in measurement-based quantum computation. It is also
used to detect and correct errors thereby protecting quantum information in
error-correcting frameworks. It is therefore important to fully characterize
the roles of quantum measurement encompassing information gain, state
disturbance and reversibility, together with their fundamental relations.
Numerous efforts have been made to obtain the trade-off between information
gain and state disturbance, which becomes a practical basis for secure
information processing. However, a complete information balance is necessary to
include the reversibility of quantum measurement, which constitutes an integral
part of practical quantum information processing. We here establish all pairs
of trade-off relations involving information gain, disturbance, and
reversibility, and crucially the one among all of them together. By doing so,
we show that the reversibility plays a vital role in completing the information
balance. Remarkably, our result can be interpreted as an
information-conservation law of quantum measurement in a nontrivial form. We
completely identify the conditions for optimal measurements that satisfy the
conservation for each tradeoff relation with their potential applications. Our
work can provide a useful guideline for designing a quantum measurement in
accordance with the aims of quantum information processors.
