Skip to main content

Obstacles on the Way Toward an Entropic Uncertainty Relation

Abstract

Deutsch’s entropic uncertainty relation is examined by using two experiments in which the spin of a single spin-half particle is detected after passing through two Stern-Gerlach apparatuses in two successive times. Two experiments differ only in the angle settings of the Stern-Gerlach apparatuses. In the general case where the angles are arbitrarily arranged, Deutsch’s inequality is violated.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. Heisenberg, W.: Z. Phys. 43, 172–198 (1927). English translation in Wheeler and Zurek (1983) Quantum theory and measurement 62–84

    Article  MATH  ADS  Google Scholar 

  2. Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Z. Phys. 44, 326–352 (1927)

    MATH  ADS  Google Scholar 

  3. Condon, E.U.: Remarks on uncertainty principles. Science 69, 573–574 (1929)

    Article  ADS  Google Scholar 

  4. Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163–164 (1929)

    Article  ADS  Google Scholar 

  5. Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  6. Bialynicki-Birula, I.: Formulation of the uncertainty relations in terms of the Rényi entropies. Phys. Rev. A 74, 52101 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  7. Ghirardi, G., Marinatto, L., Romano, R.: An optimal entropic uncertainty relation in a two-dimensional Hilbert space. Phys. Lett. A 317, 32–36 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Shannon, C.E.: Bell Syst. Tech. J. 27, 379 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  9. Everett, E.: The many-world interpretation of quantum mechanics. PhD thesis (1957) (Princeton University Press, Princeton, 1973)

  10. Hirschman, I.: A note on entropy. Am. J. Math. 79, 152–156 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  12. Shafiee, A., Golshani, M.: Single-particle Bell-type inequality. Ann. Fond. Louis Broglie 28, 105–118 (2003)

    MathSciNet  Google Scholar 

  13. Taylor, S., Cheung, S., Brukner, Č., Vedral, V.: Entanglement in time and temporal communication complexity. AIP Conf. Proc. 734, 281–284 (2004)

    Article  ADS  Google Scholar 

  14. Brukner, C., Zukowski, M.: Bell’s inequalities: foundations and quantum communication (2009). arXiv:0909.2611 [quantph]

  15. Zukowski, M.: Temporal Leggett-Garg-Bell inequalities for sequential multi-time actions in quantum information processing, and a re-definition of macroscopic realism (2010). arXiv:1009.1749 [quantph]

  16. Maassen, H., Uffink, J.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  17. Wehner, S., Winter, A.: Entropic uncertainty relations—a survey. New J. Phys. 12(2), 025009 (2010)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Afshin Shafiee.

Appendices

Appendix A: Detailed Calculations of the First Examination

For the spin-state of the particle along the x or y directions, we have:

$$\begin{aligned} \vert x+ \rangle =&\frac{1}{\sqrt{2}} \bigl( \vert z+ \rangle +\vert z- \rangle \bigr) \end{aligned}$$
(A.1)
$$\begin{aligned} \vert x- \rangle =&\frac{1}{\sqrt{2}} \bigl( \vert z+ \rangle -\vert z- \rangle \bigr) \end{aligned}$$
(A.2)
$$\begin{aligned} \vert y+ \rangle =&\frac{1}{\sqrt{2}} \bigl( \vert z+ \rangle +i\vert z- \rangle \bigr) \end{aligned}$$
(A.3)
$$\begin{aligned} \vert y- \rangle =&\frac{1}{\sqrt{2}} \bigl( \vert z+ \rangle -i\vert z- \rangle \bigr) \end{aligned}$$
(A.4)

with single probabilities:

$$ P_{x}^{+}=\frac{1}{2}, \quad P_{x}^{-}=\frac{1}{2}, \quad P_{y}^{+}=\frac{1}{2}, \quad P_{y}^{-}=\frac{1}{2} $$
(A.5)

Now, where, e.g., \(P_{x}^{+}\) is the probability of finding the result \(\frac{\hbar}{2}\) for the spin component along the x-direction of particle, the other probabilities are defined similarly.

For left-hand side of inequality (7), one finds:

$$ -\frac{1}{2}\ln\frac{1}{2}-\frac{1}{2}\ln\frac{1}{2}- \frac{1}{2}\ln\frac{1}{2}-\frac{1}{2}\ln \frac{1}{2}=2\ln2=1.386 $$
(A.6)

For the right hand side of (7), one gets:

(A.7)

where, 1.386>0.317 as was expected.

Appendix B: Detailed Calculations of the Second Examination

Here we define the single probabilities at t 1 and t 2 as:

$$\begin{aligned} P_{+}^{t_{1}} =&\frac{1}{2} ( 1+\sin\theta_{a} ) \end{aligned}$$
(B.1)
$$\begin{aligned} P_{-}^{t_{1}} =&\frac{1}{2} ( 1-\sin\theta_{a} ) \end{aligned}$$
(B.2)
$$\begin{aligned} P_{+}^{t_{2}} =&\frac{1}{2} ( 1+\sin\theta_{a} \cos\theta_{ab} ) \end{aligned}$$
(B.3)
$$\begin{aligned} P_{-}^{t_{2}} =&\frac{1}{2} ( 1-\sin\theta_{a} \cos\theta_{ab} ) \end{aligned}$$
(B.4)

where, e.g., \(P_{+}^{t_{1}}(P_{+}^{t_{2}})\) is the probability of finding the result \(+\frac{\hbar}{2}\) for the spin component of the particle along the direction \(\hat{a}(\hat{b})\) at t 1(t 2). Other probabilities are defined similarly. For spin-states of the particle at t 1 and t 2, we have:

$$\begin{aligned} \bigl \vert \varphi_{+}^{t_{1}} \bigr\rangle =&\cos\frac{\theta_{a}}{2}\vert z+ \rangle +\sin\frac{\theta_{a}}{2}\vert z- \rangle \end{aligned}$$
(B.5)
$$\begin{aligned} \bigl \vert \varphi_{+}^{t_{2}} \bigr\rangle =&\cos\frac{\theta_{b}}{2}\vert z+ \rangle +\sin\frac{\theta_{b}}{2}\vert z- \rangle \end{aligned}$$
(B.6)
$$\begin{aligned} \bigl \vert \varphi_{-}^{t_{1}} \bigr\rangle =&-\sin\frac{\theta_{a}}{2}\vert z+ \rangle +\cos\frac{\theta_{a}}{2}\vert z- \rangle \end{aligned}$$
(B.7)
$$\begin{aligned} \bigl \vert \varphi_{-}^{t_{2}} \bigr\rangle =&-\sin\frac{\theta_{b}}{2}\vert z+ \rangle +\cos\frac{\theta_{b}}{2}\vert z- \rangle \end{aligned}$$
(B.8)

Now, the probability amplitudes for the same or different results at successive times can be obtained as:

$$\begin{aligned} \bigl\langle \varphi_{+}^{t_{1}}\bigm|\varphi_{+}^{t_{2}} \bigr\rangle =&\cos\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2}+\sin \frac{\theta_{a}}{2} \sin\frac{\theta_{b}}{2}=\cos\frac{\theta_{ab}}{2} \end{aligned}$$
(B.9)
$$\begin{aligned} \bigl\langle \varphi_{+}^{t_{1}}\bigm|\varphi_{-}^{t_{2}} \bigr\rangle =&\sin\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2}-\cos \frac{\theta_{a}}{2} \sin\frac{\theta_{b}}{2}=\sin\frac{\theta_{ab}}{2} \end{aligned}$$
(B.10)
$$\begin{aligned} \bigl\langle \varphi_{-}^{t_{1}}\bigm|\varphi_{+}^{t_{2}} \bigr\rangle =&- \biggl[ \sin\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2}-\cos \frac{\theta_{a}}{2} \sin\frac{\theta_{b}}{2} \biggr] =-\sin \frac{\theta_{ab}}{2} \end{aligned}$$
(B.11)
$$\begin{aligned} \bigl\langle \varphi_{-}^{t_{1}}\bigm|\varphi_{-}^{t_{2}} \bigr\rangle =&\cos\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2}+\sin \frac{\theta_{a}}{2}\sin\frac{\theta_{b}}{2}=\cos\frac{\theta_{ab}}{2} \end{aligned}$$
(B.12)

So, for the left-hand side of the inequality (18) we have,

(B.13)

Similarly, for the right hand side of (18), one obtains:

(B.14)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Khatam, I., Shafiee, A. Obstacles on the Way Toward an Entropic Uncertainty Relation. Int J Theor Phys 53, 3370–3379 (2014). https://doi.org/10.1007/s10773-013-1707-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-013-1707-z

Keywords

  • Entropic uncertainty relations
  • Quantum entanglement in time
  • Information