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### Online identification of symmetric pure states

#### Gael Sentís, EMV, Ramon Muñoz-Tapia

##### arXiv:2107.02127

ArXiv**Abstract: **We consider online strategies for discriminating between
symmetric pure states with zero error when n copies of the states are
provided. Optimized online strategies involve local, possibly adaptive
measurements on each copy and are optimal at each step, which makes them
robust in front of particle losses or an abrupt termination of the
discrimination process. We first review previous results on binary
minimum and zero error discrimination with local measurements that achieve
the maximum success probability set by optimizing over global measurements,
highlighting their online features. We then extend these results to the
case of zero error identification of three symmetric states with constant
overlap. We provide optimal online schemes that attain global performance
for any n if the state overlaps are positive, and for odd n if overlaps
have a negative value. For arbitrary complex overlaps, we show compelling
evidence that online schemes fail to reach optimal global performance.
The online schemes that we describe only require to store the last
outcome obtained in a classical memory, and adaptiveness of the
measurements reduce to at most two changes, regardless of the value of n.

### Quantum Sequential Hypothesis Testing

#### EMV, Christoph Hirche, Gael Sentís, Michalis Skotiniotis, Marta Carrizo, Ramon Muñoz-Tapia, and John Calsamiglia

##### Phys. Rev. Lett. 126, 180502 (2021)

ArXiv PRL**Abstract: **We introduce sequential analysis in quantum information
processing, by focusing on the fundamental task of quantum hypothesis testing.
In particular, our goal is to discriminate between two arbitrary quantum
states with a prescribed error threshold ε when copies of the states can be
required on demand. We obtain ultimate lower bounds on the average number of
copies needed to accomplish the task. We give a block-sampling strategy that
allows us to achieve the lower bound for some classes of states. The bound is
optimal in both the symmetric as well as the asymmetric setting in the sense
that it requires the least mean number of copies out of all other procedures,
including the ones that fix the number of copies ahead of time. For qubit states
we derive explicit expressions for the minimum average number of copies and
show that a sequential strategy based on fixed local measurements outperforms
the best collective measurement on a predetermined number of copies. Whereas
for general states the number of copies increases as log(1/ε), for pure states
sequential strategies require a finite average number of samples even in the
case of perfect discrimination, i.e., ε=0.

### Quantum measurement optimization by decomposition of measurements into extremals

#### EMV, Carlos Pineda, Pablo Barberis-Blostein

##### Scientific Reports volume 10, Article number: 9375 (2020)

Scientific Reports**Abstract: **Using the convex structure of positive operator
value measurements and several quantities used in quantum
metrology, such as quantum Fisher information or the quantum
Van Trees information, we present an efficient numerical method
to find the best strategy allowed by quantum mechanics to
estimate a parameter. This method explores extremal measurements
thus providing a significant advantage over previously used methods.
We exemplify the method for different cost functions in a qubit and
in a harmonic oscillator and find a strong numerical advantage when
the desired target error is sufficiently small.

### Certified answers for ordered quantum discrimination problems

#### EMV and Ramon Muñoz-Tapia

##### Phys. Rev. A 100, 042331 (2019)

ArXiv PRA**Abstract: **We investigate the quantum state discrimination task for
sets of linear independent pure states with an intrinsic ordering. These
structured discrimination problems allow for a scheme that provides a
certified level of error; that is, answers that never deviate from the
true value by more than a specified distance and hence control the desired
quality of the results. We obtain an efficient semidefinite program and also
find a general lower bound valid for any error distance that only requires
the knowledge of an optimal minimum-error scheme. We apply our results to
the cases of quantum change point and quantum state anomaly detection.

### Online optimal exact identification of a quantum change point

#### Gael Sentís, EMV, Ramon Muñoz-Tapia

##### Phys. Rev. A 98, 052305 (2018)

ArXiv PRA**Abstract: **We consider online detection strategies for identifying
a change point in a stream of quantum particles allegedly prepared in
identical states. We show that the identification of the change point
can be done without error via sequential local measurements while
attaining the optimal performance bound set by quantum mechanics. In
this way, we establish the task of exactly identifying a quantum change
point as an instance where local protocols are as powerful as global ones.
The optimal online detection strategy requires only one bit of memory
between subsequent measurements, and it is amenable to experimental
realization with current technology.

### Quantum estimation of unknown parameters

#### EMV, Carlos Pineda, François Leyvraz, and Pablo Barberis-Blostein

##### Phys. Rev. A 95, 012136 (2017)

ArXiv PRA**Abstract: **We discuss the problem of finding the
best measurement strategy for estimating the value of
a quantum system parameter. In general the optimum
quantum measurement, in the sense that it maximizes
the quantum Fisher information and hence allows one
to minimize the estimation error, can only be
determined if the value of the parameter is already
known. A modification of the quantum Van Trees
inequality, which gives a lower bound on the error in
the estimation of a random parameter, is proposed. The
suggested inequality allows us to assert if a
particular quantum measurement, together with an
appropriate estimator, is optimal. An adaptive strategy
to estimate the value of a parameter, based on our
modified inequality, is proposed.