Uncertainty Relation Revisited from Quantum Estimation Theory Yu Watanabe Collaborators: Takahiro Sagawa, Masahito Ueda The University of Tokyo
arXiv:1010.3571 (2010).
2/17/2011
Nagoya Winter Workshop 2011 (NWW2011)
Outline
Two types of uncertainty relation Our setup of estimation process Our results Relation to information geometry
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Two types of uncertainty relation
Indeterminacy --- property of quantum states --Trade-off relation on state preparation Non-commutable observables cannot have definite value simultaneously.
Complementarity --- property of measurements (operations) --Trade-off relation between measurement errors We cannot perform measurement without back-action. We cannot perform precise measurement of non-commutable observables simultaneously.
We cannot know
and
precisely, if
.
There must exist trade-off relations obtained information about non-commutable observables. However, the optimal trade-off relation is not found so far. optimal: the equality is achievable for all states and observables. What is the optimal bound of complementarity? 4
Estimation Theory -- Essential Part of Complementarity
: estimator (estimated value) of
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Example
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Example
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Example
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Example
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Example
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Cramer-Rao inequality
For all consistent estimators such that
for all
and
: Fisher information matrix
: parameter of 11
Quantum Cramer-Rao inequality
For all POVM
for all : Quantum Fisher information matrix Ex.) SLD, RLD, etc…
Combining with classical Cramer-Rao inequality,
Projection measurement of
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Measurement Error
Measurement error of observable
in POVM
Projection measurement of
and
cannot vanish simultaneously, if
There must exist trade-off relations between and ..
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Theorem 1
For all • POVM measurements • observables • quantum states
Proof From quantum Cramer-Rao inequality, for all
: RLD Fisher information matrix
then the discriminant of quadratic polynomial is always negative. 14
Arthurs and Goodman’s inequality Unbiased measurement Target
Probe : projection measurement of for all : unbiasedness condition For all unbiased measurements,
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Estimation Scheme on Unbiased Measurement
One of the simplest (may not be optimal)way to estimate and is : eigenvalue of : number of times we obtain outcome In this case,
then
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Theorem 1
For all • POVM measurements • observables • quantum states
However, the bound is not optimal. Example. For for all
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Conjecture •
The optimal bound of the trade-off relation is
: “intrinsic” quantum fluctuation
: “intrinsic” quantum correlation function
• •
2-dimensional Hilbert space : proved Higher dimension : not proved but numerical evidences exist.
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What is intrinsic quantum fluctuation and correlation ? : simultaneous irreducible invariant subspace of
and
: projection on
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Example
and
cannot be simultaneous block diagonalizable
and
are commutable
In general
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Theorem 2 --- achievability of the optimal bound --For all observables and quantum states, there exist a measurement
such that
The measurement that attain the inequality is performing projection measurements of with probability
.
Condition for attaining the bound is the off-diagonal element of
equals to zero. 21
Not Optimal Scheme
In this case,
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Optimal Scheme
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Example
Simultaneous measurement of two spin observables.
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Theorem 3 For the case of
,
is satisfied.
Lemma
: SLD Fisher information matrix
From Gill and Massar bound,
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Proof of Theorem 3 If
,
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Numerical Evidence of the Conjecture POVMs are plotted.
Nagaoka’s inequality
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Relation to Information Geometry •
Cramer-Rao inequality • Fisher metric: monotone metric on probability distribution manifold • Quantum Fisher metric: monotone metric on quantum state space • Quantum Cramer-Rao inequality • Complementarity on quantum information geometry 28
Fisher metric and Cencov’s theorem
Monotonicity of statistical model Markov mapping
Monotone metric on statistical model is uniquely determined to Fisher metric .
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Quantum Fisher metric and Petz’s theorem
Monotonicity of quantum statistical manifold Quantum process
Monotone metric on quantum statistical manifold has minimum and maximum. •
SLD is the minimum and RLD is the maximum. 30
Quantum Cramer-Rao inequality
What is quantum measurement? Quantum measurement
Quantum Cramer-Rao inequality for all
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Complementarity from geometrical point of view
Equality of
cannot be achievable.
Equality of with real vector attained by the projection measurement of
can be .
Complementarity on quantum information geometry Trade-off relation between and Our conjecture
is important to understand the effect on quantum state manifold by the measurement
. 32
Summary and future studies
Summary • We conjecture that the optimal bound of measurement errors is
•
This is important for understanding quantum information geometry.
Future studies • Prove the conjecture. It may come soon!
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Thank you!
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Quantum Estimation Theory
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