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.

Kennard-Robertson’s inequality (1927, 1929) : quantum fluctuation

Schrodinger’s inequality (1930)

: symmetrized quantum corelation function 3

Two types of uncertainty relation

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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