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Sci Rep. 2017; 7: 44764.
Published online 2017 March 20. doi:  10.1038/srep44764
PMCID: PMC5357915

A Stronger Multi-observable Uncertainty Relation

Abstract

Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for N incompatible observables stronger than the simple generalization of an existing uncertainty relation for two observables. Further comparisons of our uncertainty relation with other related ones for spin-An external file that holds a picture, illustration, etc.
Object name is srep44764-m1.jpg and spin-1 particles indicate that the obtained uncertainty relation gives a better lower bound.

Uncertainty relation is one of the fundamental building blocks of quantum theory, and now plays an important role in quantum mechanics and quantum information1,2,3,4. It is introduced by Heisenberg5 in understanding how precisely the simultaneous values of conjugate observables could be in microspace, i.e., the position X and momentum P of an electron. Kennard6 and Weyl7 proved the uncertainty relation

An external file that holds a picture, illustration, etc.
Object name is srep44764-m2.jpg

where the standard deviation of an operator X is defined by An external file that holds a picture, illustration, etc.
Object name is srep44764-m3.jpg. Later, Robertson proposed the well-known formula of uncertainty relation8

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Object name is srep44764-m4.jpg

which is applicable to arbitrary incompatible observables, and the commutator is defined by [A, B] = AB  BA. The uncertainty relation was further strengthed by Schrödinger9 with the following form

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Object name is srep44764-m5.jpg

Here the commutator defined as {A, B} [equivalent] AB + BA.

It is realized that the traditional uncertainty relations may not fully capture the concept of incompatible observables as the lower bound could be trivially zero while the variances are not. An important question in uncertainty relation is how to improve the lower bound and immune from triviality problem10,11. Various attempts have been made to find stronger uncertainty relations. One typical kind of relation is that of Maccone and Pati, who derived two stronger uncertainty relations

An external file that holds a picture, illustration, etc.
Object name is srep44764-m6.jpg
An external file that holds a picture, illustration, etc.
Object name is srep44764-m7.jpg

where left angle bracketψ|ψ[perpendicular]right angle bracket = 0, An external file that holds a picture, illustration, etc.
Object name is srep44764-m8.jpg, and the sign on the right-hand side of the inequality takes + (−) while ileft angle bracket[A, B]right angle bracket is positive (negative). The basic idea behind these two relations is adding additional terms to improve the lower bound. Along this line, more terms12,13,14 and weighted form of different terms15,16 have been put into the uncertainty relations. It is worth mentioning that state-independent uncertainty relations can immune from triviality problem17,18,19,20. Recent experiments have also been performed to verify the various uncertainty relations21,22,23,24.

Besides the conjugate observables of position and momentum, multiple observables also exist, e.g., three component vectors of spin and angular momentum. Hence, it is important to find uncertainty relation for multiple incompatible observables. Recently, some three observables uncertainty relations were studied, such as Heisenberg uncertainty relation for three canonical observables25, uncertainty relations for three angular momentum components26, uncertainty relation for three arbitrary observables14. Furthermore, some multiple observables uncertainty relations were proposed, which include multi-observable uncertainty relation in product27,28 and sum29,30 form of variances. It is worth noting that Chen and Fei derived an variance-based uncertainty relation30

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Object name is srep44764-m9.jpg

for arbitrary N incompatible observables, which is stronger than the one such as derived from the uncertainty inequality for two observables10.

In this paper, we investigate variance-based uncertainty relation for multiple incompatible observables. We present a new variance-based sum uncertainty relation for multiple incompatible observables, which is stronger than an uncertainty relation from summing over all the inequalities for pairs of observables10. Furthermore, we compare the uncertainty relation with existing ones for a spin-An external file that holds a picture, illustration, etc.
Object name is srep44764-m10.jpg and spin-1 particle, which shows our uncertainty relation can give a tighter bound than other ones.

Results

Theorem 1. For arbitrary N observables A1, A2, …, AN, the following variance-based uncertainty relation holds

An external file that holds a picture, illustration, etc.
Object name is srep44764-m11.jpg

The bound becomes nontrivial as long as the state is not common eigenstate of all the N observables.

Proof: To derive (7), start from the equality

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Object name is srep44764-m12.jpg

then using the inequality

An external file that holds a picture, illustration, etc.
Object name is srep44764-m13.jpg

we obtain the uncertainty relation (7) QED.

When N = 2 we have the following corollary

Corollary 1.1. For two incompatible observables A and B, we have

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Object name is srep44764-m14.jpg

which is derived from Theorem 1 for N = 2, and stronger than uncertainty relation (5).

To show that our relation (7) has a stronger bound, we consider the result in ref. 10, the relation (5) is derived from the uncertainty equality

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Object name is srep44764-m15.jpg

Using the above uncertainty equality, one can obtain two inequalities for arbitrary N observables, namely

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Object name is srep44764-m16.jpg

and

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Object name is srep44764-m17.jpg

The bound in (6) is tighter than the one in (12)30. However, the lower bound in (6) is not always tighter than the one in (13) (see Fig. 1).

Figure 1
Example of comparison between our relation (7) and the ones (6), (12), (13).

Example 1. To give an overview that the relation (7) has a better lower bound than the relations (6), (12), (13), we consider a family of qubit pure states given by the Bloch vector An external file that holds a picture, illustration, etc.
Object name is srep44764-m18.jpg, and choose the Pauli matrices

An external file that holds a picture, illustration, etc.
Object name is srep44764-m19.jpg

Then we have (Δσx)2 + σy)2 + σz)2 = 2, An external file that holds a picture, illustration, etc.
Object name is srep44764-m20.jpg, and An external file that holds a picture, illustration, etc.
Object name is srep44764-m21.jpg. Similarly, [Δ(σx  σy)]2 = 2, and An external file that holds a picture, illustration, etc.
Object name is srep44764-m22.jpg. The comparison between the lower bounds (6), (12), (13) and (7) is given in Fig. 1. Apparently, our bound is tighter than (6), (12) and (13). We shall show with detailed proofs and examples that our uncertainty relation (7) has better lower bound than that of (6), (12), (13) in the following sections.

Comparison between the lower bound of our uncertainty relation (7) with that of inequality (12)

First, we compare our relation (7) with the one (12). Note that An external file that holds a picture, illustration, etc.
Object name is srep44764-m23.jpg, the relation (12) becomes

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Object name is srep44764-m24.jpg

Simplify the above inequality, we obtain

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Object name is srep44764-m25.jpg

which is equal to the relation (12).

Similarly, by using An external file that holds a picture, illustration, etc.
Object name is srep44764-m26.jpg, our relation (7) becomes

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Object name is srep44764-m27.jpg

Simplify the above inequality, we get

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Object name is srep44764-m28.jpg

which is equal to the relation (7). It is easy to see that the right-hand side of (17) is greater than the right-hand side of (15). Hence, the relation (7) is stronger than the relation (12).

Comparison between the lower bound of our uncertainty relation (7) with that of inequalities (6) and (13)

Here, we will show the uncertainty relation (7) is stronger than inequalities (13) and (6) for a spin-An external file that holds a picture, illustration, etc.
Object name is srep44764-m29.jpg particle and measurement of Pauli-spin operators σx, σy, σz. Then the uncertainty relation (7) has the form

An external file that holds a picture, illustration, etc.
Object name is srep44764-m30.jpg

the relation (13) is given by

An external file that holds a picture, illustration, etc.
Object name is srep44764-m31.jpg

and the relation (6) says that

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Object name is srep44764-m32.jpg

We consider a qubit state and its Bloch sphere representation

An external file that holds a picture, illustration, etc.
Object name is srep44764-m33.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep44764-m34.jpg are Pauli matrices and the Bloch vector An external file that holds a picture, illustration, etc.
Object name is srep44764-m35.jpg is real three-dimensional vector such that An external file that holds a picture, illustration, etc.
Object name is srep44764-m36.jpg. Then we have An external file that holds a picture, illustration, etc.
Object name is srep44764-m37.jpg, An external file that holds a picture, illustration, etc.
Object name is srep44764-m38.jpg. The relation (18) has the form

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Object name is srep44764-m39.jpg

where we define An external file that holds a picture, illustration, etc.
Object name is srep44764-m40.jpg. And the relation (19) becomes

An external file that holds a picture, illustration, etc.
Object name is srep44764-m41.jpg

Let us compare the lower bound of (22) with that of (23). The difference of these two bounds is

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Object name is srep44764-m42.jpg

for all An external file that holds a picture, illustration, etc.
Object name is srep44764-m43.jpg. When An external file that holds a picture, illustration, etc.
Object name is srep44764-m44.jpg, the above inequality becomes equality, then the Eq. (24) has the minimum value An external file that holds a picture, illustration, etc.
Object name is srep44764-m45.jpg. This illustrates that the uncertainty relation (7) is stronger that the one (13) for a spin-An external file that holds a picture, illustration, etc.
Object name is srep44764-m46.jpg particle and measurement of Pauli-spin operators σx, σy, σz.

Let us compare the uncertainty relation (18) with (20). The relation (20) has the form

An external file that holds a picture, illustration, etc.
Object name is srep44764-m47.jpg

where we define An external file that holds a picture, illustration, etc.
Object name is srep44764-m48.jpg. Then the difference of these two bounds of relation (22) and (25) becomes

An external file that holds a picture, illustration, etc.
Object name is srep44764-m49.jpg

where we have twice used Cauchy’s inequality. When An external file that holds a picture, illustration, etc.
Object name is srep44764-m50.jpg and An external file that holds a picture, illustration, etc.
Object name is srep44764-m51.jpg, the above inequality becomes equality, then the Eq. (26) has the minimum value An external file that holds a picture, illustration, etc.
Object name is srep44764-m52.jpg. This illustrates that the uncertainty relation (7) is stronger that the one (6) for a spin-An external file that holds a picture, illustration, etc.
Object name is srep44764-m53.jpg particle and measurement of Pauli-spin operators σx, σy, σz.

Example 2. For spin-1 systems, we consider the following quantum state characterized by θ and ϕ

An external file that holds a picture, illustration, etc.
Object name is srep44764-m54.jpg

with 0  θ  π, 0  ϕ  2π. By choosing the three angular momentum operators (An external file that holds a picture, illustration, etc.
Object name is srep44764-m55.jpg)

An external file that holds a picture, illustration, etc.
Object name is srep44764-m56.jpg

the comparison between the lower bounds (6), (12), (13) and (7) is shown by Fig. 2. The results suggest that the relation (7) can give tighter bounds than other ones (6), (12), (13) for a spin-1 particle and measurement of angular momentum operators An external file that holds a picture, illustration, etc.
Object name is srep44764-m57.jpg, An external file that holds a picture, illustration, etc.
Object name is srep44764-m58.jpg, An external file that holds a picture, illustration, etc.
Object name is srep44764-m59.jpg.

Figure 2
Example of comparison between our relation (7) and ones (6), (12), (13).

Conclusion

We have provided a variance-based sum uncertainty relation for N incompatible observables, which is stronger than the simple generalizations of the uncertainty relation for two observables derived by Maccone and Pati [Phys. Rev. Lett. 113, 260401 (2014)]. Furthermore, our uncertainty relation gives a tighter bound than the others by comparison for a spin-An external file that holds a picture, illustration, etc.
Object name is srep44764-m60.jpg particle with the measurements of spin observables σx, σy, σz. And also, in the case of spin-1 with measurement of angular momentum operators Lx, Ly, Lz, our uncertainty relation predicts a tighter bound than other ones.

Additional Information

How to cite this article: Song, Q.-C. et al. A Stronger Multi-observable Uncertainty Relation. Sci. Rep. 7, 44764; doi: 10.1038/srep44764 (2017).

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Acknowledgments

This work was supported in part by the Ministry of Science and Technology of the People’s Republic of China (2015CB856703); by the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB23030100; and by the National Natural Science Foundation of China(NSFC) under the grants 11375200 and 11635009.

Footnotes

The authors declare no competing financial interests.

Author Contributions Q.-C.S. and J.-L.L. and G.-X.P. and C.-F.Q. contribute equally to this work, and agree with the manuscript submitted.

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