Editing Uncertainty principle (section)

===Systematic and statistical errors===

The inequalities above focus on the ''statistical imprecision'' of observables as quantified by the standard deviation <math>\sigma</math>. Heisenberg's original version, however, was dealing with the ''systematic error'', a disturbance of the quantum system produced by the measuring apparatus, i.e., an observer effect.

If we let <math>\varepsilon_A</math> represent the error (i.e., [[accuracy|inaccuracy]]) of a measurement of an observable ''A'' and <math>\eta_B</math> the disturbance produced on a subsequent measurement of the conjugate variable ''B'' by the former measurement of ''A'', then the inequality proposed by Masanao Ozawa − encompassing both systematic and statistical errors - holds:<ref name="Ozawa2003"/>
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|equation = <math> \varepsilon_A\, \eta_B + \varepsilon_A \, \sigma_B + \sigma_A \, \eta_B \,\ge\, \frac{1}{2} \, \left| \Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right|</math>
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Heisenberg's uncertainty principle, as originally described in the 1927 formulation, mentions only the first term of Ozawa inequality, regarding the ''systematic error''. Using the notation above to describe the ''error/disturbance'' effect of ''sequential measurements'' (first ''A'', then ''B''), it could be written as
{{Equation box 1
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|equation = <math> \varepsilon_{A} \, \eta_{B} \, \ge \, \frac{1}{2} \, \left| \Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right|</math>
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The formal derivation of the Heisenberg relation is possible but far from intuitive. It was ''not'' proposed by Heisenberg, but formulated in a mathematically consistent way only in recent years.<ref>{{Cite journal | doi = 10.1103/PhysRevLett.111.160405| title = Proof of Heisenberg's Error-Disturbance Relation| journal = Physical Review Letters| volume = 111| issue = 16| year = 2013| last1 = Busch | first1 = P. | last2 = Lahti | first2 = P. | last3 = Werner | first3 = R. F. |arxiv = 1306.1565 |bibcode = 2013PhRvL.111p0405B | pmid=24182239 | page=160405| s2cid = 24507489}}</ref><ref>{{Cite journal | doi = 10.1103/PhysRevA.89.012129| title = Heisenberg uncertainty for qubit measurements| journal = Physical Review A| volume = 89| issue = 1| pages = 012129| year = 2014| last1 = Busch | first1 = P. | last2 = Lahti | first2 = P. | last3 = Werner | first3 = R. F. |arxiv = 1311.0837 |bibcode = 2014PhRvA..89a2129B | s2cid = 118383022}}</ref>
Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errors <math>\sigma_A</math> and <math>\sigma_B</math>. There is increasing experimental evidence<ref name="Rozema"/><ref>{{Cite journal| last1 = Erhart | first1 = J.| last2 = Sponar | first2 =S.| last3 = Sulyok | first3 = G. | last4 = Badurek | first4 = G. | last5 = Ozawa | first5 = M. | last6 = Hasegawa | first6 = Y.| title = Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements | journal = Nature Physics | volume=8 | pages=185–189 | year=2012 | doi=10.1038/nphys2194 | arxiv = 1201.1833 | bibcode = 2012NatPh...8..185E | issue=3 | s2cid = 117270618}}</ref><ref>{{Cite journal| last1 = Baek | first1 = S.-Y. | last2 = Kaneda | first2 = F. | last3 = Ozawa | first3 = M. | last4 = Edamatsu | first4 = K. | title = Experimental violation and reformulation of the Heisenberg's error-disturbance uncertainty relation |journal = Scientific Reports |volume= 3 |pages= 2221 |year= 2013 |doi= 10.1038/srep02221 |bibcode = 2013NatSR...3.2221B | pmid=23860715 | pmc=3713528}}</ref><ref>{{Cite journal| last1 = Ringbauer | first1 = M. | last2 = Biggerstaff | first2 = D.N. | last3 = Broome | first3 = M.A. | last4 = Fedrizzi | first4 = A. | last5 = Branciard | first5 = C. | last6 = White | first6 = A.G. | title = Experimental Joint Quantum Measurements with Minimum Uncertainty |journal = Physical Review Letters |volume= 112 | issue = 2 |pages= 020401 |year= 2014 |doi= 10.1103/PhysRevLett.112.020401 |arxiv = 1308.5688 |bibcode = 2014PhRvL.112b0401R | pmid=24483993| s2cid = 18730255 }}</ref> that the total quantum uncertainty cannot be described by the Heisenberg term alone, but requires the presence of all the three terms of the Ozawa inequality.

Using the same formalism,<ref name="Sen2014"/> it is also possible to introduce the other kind of physical situation, often confused with the previous one, namely the case of ''simultaneous measurements'' (''A'' and ''B'' at the same time):
{{Equation box 1
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|equation = <math> \varepsilon_A \, \varepsilon_B \, \ge \, \frac{1}{2} \, \left| \Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right|</math>
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The two simultaneous measurements on ''A'' and ''B'' are necessarily<ref>{{Cite journal | last1 = Björk | first1 = G. | last2 = Söderholm | first2 = J. | last3 = Trifonov | first3 = A. | last4 = Tsegaye | first4 = T. | last5 = Karlsson | first5 = A. |
title = Complementarity and the uncertainty relations | doi = 10.1103/PhysRevA.60.1874 | journal = Physical Review | volume = A60 | issue = 3 | year = 1999| page = 1878 |arxiv = quant-ph/9904069 |bibcode = 1999PhRvA..60.1874B | s2cid = 27371899 }}</ref> ''unsharp'' or [[weak measurement|''weak'']].

It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality. By adding Robertson<ref name="Sen2014"/>
{{Equation box 1
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|equation = <math> \sigma_{A} \, \sigma_{B} \, \ge \, \frac{1}{2} \, \left| \Bigl\langle \bigl[\hat{A},\hat{B} \bigr] \Bigr\rangle \right|</math>
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and Ozawa relations we obtain
<math display="block">\varepsilon_A \eta_B + \varepsilon_A \, \sigma_B + \sigma_A \, \eta_B + \sigma_A \sigma_B \geq \left|\Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right| .</math>
The four terms can be written as:
<math display="block">(\varepsilon_A + \sigma_A) \, (\eta_B + \sigma_B) \, \geq \, \left|\Bigl\langle\bigl[\hat{A},\hat{B} \bigr] \Bigr\rangle \right| .</math>
Defining:
<math display="block">\bar \varepsilon_A \, \equiv \, (\varepsilon_A + \sigma_A)</math>
as the ''inaccuracy'' in the measured values of the variable ''A'' and
<math display="block">\bar \eta_B \, \equiv \, (\eta_B + \sigma_B)</math>
as the ''resulting fluctuation'' in the conjugate variable ''B'', Kazuo Fujikawa<ref>{{Cite journal|last = Fujikawa|first = Kazuo|title = Universally valid Heisenberg uncertainty relation|journal = Physical Review A|volume=85|year=2012|doi=10.1103/PhysRevA.85.062117|arxiv = 1205.1360 |bibcode = 2012PhRvA..85f2117F|issue=6 |pages = 062117|s2cid = 119640759}}</ref> established an uncertainty relation similar to the Heisenberg original one, but valid both for ''systematic and statistical errors'':
{{Equation box 1
|indent =:
|equation = <math> \bar \varepsilon_A \, \bar \eta_B \, \ge \, \left| \Bigl\langle \bigl[\hat{A},\hat{B}\bigr] \Bigr\rangle \right|</math>
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