### Entanglement-Assisted Entropic Uncertainty Principle

[From left to right] Xiao-Ye Xu, Guang-Can Guo, Chuan-Feng Li and Jin-Shi Xu

Researchers at the University of Science and Technology of China (USTC) and National University of Singapore have verified the entanglement-assisted entropic uncertainty principle and demonstrated its practical usage to witness entanglement [1].

In quantum mechanics, the ability to predict the precise outcomes of two conjugate observables, such as the position and momentum, for a particle is restricted by the uncertainty principle. For example, the more precisely the location of the particle is determined, the less accurate the momentum determination will be. Originally given by Heisenberg, the uncertainty relation expressed in the form of standard deviation is further extended to the entropic form to precisely reflect its physical meanings.

The uncertainty principle is the essential characteristic of quantum mechanics. However, the possibility of violating the Heisenberg’s uncertainty relation has been considered early. In 1935, Einstein, Podolsky, and Rosen published the famous paper in which they considered using two particles entangled in position and momentum freedoms to violate the Heisenberg’s uncertainty relation and to challenge the correctness of quantum mechanics (EPR paradox) [2]. Popper also proposed a practical experiment using entangled pairs to demonstrate the violation of the Heisenberg’s uncertainty relation [3]. After a long debate and many experimental works, it is known that these violations do not contradict the quantum theory and they are now implemented as a signature of entanglement which is the fundamental feature of quantum mechanics and the important source of quantum information processing.

Recently, a stronger entropic uncertainty relation, which uses previously determined quantum information, was proved by Berta et al. [4], whose equivalent form was previously conjectured by Renes and Boileau [5]. By initially entangling the particle of interest to another particle that acts as a quantum memory, the uncertainty associated with the outcomes of two conjugate observables can be reduced to zero. The lower bound of the uncertainty is essentially dependent on the entanglement between the particle of interest and the quantum memory. This novel entropic uncertainty relation greatly extends the uncertainty principle.

Image: The practical setup to demonstrate the uncertainty game proposed by Berta et al. (also see the past 2Physics article dated August 29, 2010).

The experimental group leaded by Prof. Chuan-Feng Li at USTC prepared a special kind of entangled photon state, called as the Bell diagonal state, in an entirely optical setup. One of the photons is sent for measurement and the other one acts as an assisted particle carrying the quantum information of the one of interest. The assisted photon is stored in a spin-echo based quantum memory which consisted of two polarization maintaining fibers each of 120 m length and two half-wave plates. The storing time can reach as long as 1.2 us. The lower bound of the uncertainty related to the outcomes of two conjugate observables is measured, which can be reduced to arbitrarily small values when the two particles share quasi-maximal entanglement. As a result, the entropic form of Heisenberg’s uncertainty relation is violated and the novel one is confirmed. By measuring observables on both particles, the group further used the novel entropic uncertainty relation to witness entanglement and to compare with other entanglement measurements. The novel entanglement witness can be obtained by a few separate measurements on each of the entangled particles, which shows its ease of accessibility.

The verified entropic uncertainty principle implies that the uncertainty principle is not only observable-dependent, but is also observer-dependent, providing a particularly intriguing perspective [6]. The method used to estimate uncertainties by directly performing measurements on both photons has practical application in verifying the security of quantum key distribution. This novel uncertainty relation would also find practical use in the area of quantum engineering.

The experimental investigation of the novel entropic uncertainty principle has caused great interests. Another relevant experimental work was performed independently by Prevedel and colleagues [7] and both papers are published in the same issue of Nature Physics.

References:

[1] C.-F. Li, J.-S. Xu, X.-Y. Xu, K. Li and G.-C. Guo, "Experimental investigation of the entanglement-assisted entropic uncertainty principle". Nature Physics, 7, 752-756 (2011). Abstract.

[2] A. Einstein, B. Podolsky and N. Rosen, "Can quantum mechanical description of physical reality be considered complete?" Phys. Rev. 47, 777-780 (1935). Abstract.

[3] K. R. Popper, "Zur Kritik der Ungenauigkeitsrelationen", Naturwiss. 22, 807-808 (1934). Article.

[4] M. Berta, M. Christandl, R. Colbeck, J. M. Renes and R. Renner, "The uncertainty principle in the presence of quantum memory". Nature Physics, 6, 659-662 (2010). Abstract. 2Physics Article.

[5] J. M. Renes and J. C. Boileau, "Conjectured strong complementary information tradeoff". Physical Review Letters, 103, 020402 (2005). Abstract.

[6] A. Winter, "Coping with uncertainty", Nature Phys. 6, 640-641 (2010). Abstract.

[7] R. Prevedel, D. R. Hamel, R. Colbeck, K. Fisher and K. J. Resch, Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement, Nature Physics, 7, 757-761 (2011). Abstract.

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