A Stronger Multi-observable Uncertainty Relation
AA Stronger Multi-observable Uncertainty Relation
Qiu-Cheng Song , Jun-Li Li , Guang-Xiong Peng , and Cong-Feng Qiao Department of Physics, University of Chinese Academy of Sciences, YuQuan Road 19A, Beijing 100049, China Department of Physics & Astronomy, York University, Toronto, ON M3J 1P3, Canada Key Laboratory of Vacuum Physics, University of Chinese Academy of Sciences * To whom correspondence should be addressed; E-mail: [email protected]
ABSTRACT
Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables inthe preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here wepresent a variance-based sum uncertainty relation for N incompatible observables stronger than the simple generalization ofan existing uncertainty relation for two observables. Further comparisons of our uncertainty relation with other related ones forspin- and spin- particles indicate that the obtained uncertainty relation gives a better lower bound. Introduction
Uncertainty relation is one of the fundamental building blocks of quantum theory, and now plays an important role in quantummechanics and quantum information . It is introduced by Heisenberg in understanding how precisely the simultaneousvalues of conjugate observables could be in microspace, i.e., the position X and momentum P of an electron. Kennard andWeyl proved the uncertainty relation ∆ X ∆ P ≥ ¯ h , (1)where the standard deviation of an operator X is defined by ∆ X = (cid:112) (cid:104) ψ | X | ψ (cid:105) − (cid:104) ψ | X | ψ (cid:105) . Later, Robertson proposed thewell-known formula of uncertainty relation ( ∆ A ) ( ∆ B ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) ψ | [ A , B ] | ψ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) , (2)which is applicable to arbitrary incompatible observables, and the commutator is defined by [ A , B ] = AB − BA . The uncertaintyrelation was further strengthed by Schr¨odinger with the following form ( ∆ A ) ( ∆ B ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) [ A , B ] (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) (cid:104){ A , B }(cid:105) − (cid:104) A (cid:105)(cid:104) B (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) . (3)Here the commutator defined as { A , B } ≡ AB + BA .It is realized that the traditional uncertainty relations may not fully capture the concept of incompatible observables as thelower bound could be trivially zero while the variances are not. An important question in uncertainty relation is how to improvethe lower bound and immune from triviality problem . Various attempts have been made to find stronger uncertaintyrelations. One typical kind of relation is that of Maccone and Pati, who derived two stronger uncertainty relations ( ∆ A ) + ( ∆ B ) ≥ ± i (cid:104) ψ | [ A , B ] | ψ (cid:105) + |(cid:104) ψ | A ± iB | ψ ⊥ (cid:105)| , (4) ( ∆ A ) + ( ∆ B ) ≥ |(cid:104) ψ ⊥ A + B | A + B | ψ (cid:105)| = [ ∆ ( A + B )] , (5)where (cid:104) ψ | ψ ⊥ (cid:105) = | ψ ⊥ A + B (cid:105) ∝ ( A + B − (cid:104) A + B (cid:105) ) | ψ (cid:105) , and the sign on the right-hand side of the inequality takes +( − ) while i (cid:104) [ A , B ] (cid:105) is positive (negative). The basic idea behind these two relations is adding additional terms to improve the lower bound.Along this line, more terms and weighted form of different terms have been put into the uncertainty relations. It isworth mentioning that state-independent uncertainty relations can immune from triviality problem . Recent experimentshave also been performed to verify the various uncertainty relations .Besides the conjugate observables of position and momentum, multiple observables also exist, e.g., three component vectorsof spin and angular momentum. Hence, it is important to find uncertainty relation for multiple incompatible observables. a r X i v : . [ qu a n t - ph ] M a y ecently, some three observables uncertainty relations were studied, such as Heisenberg uncertainty relation for three canonicalobservables , uncertainty relations for three angular momentum components , uncertainty relation for three arbitraryobservables . Furthermore, some multiple observables uncertainty relations were proposed, which include multi-observableuncertainty relation in product and sum form of variances. It is worth noting that Chen and Fei derived an variance-based uncertainty relation N ∑ i = ( ∆ A i ) ≥ N − ∑ ≤ i < j ≤ N [ ∆ ( A i + A j )] − ( N − ) (cid:34) ∑ ≤ i < j ≤ N ∆ ( A i + A j ) (cid:35) , (6)for arbitrary N incompatible observables, which is stronger than the one such as derived from the uncertainty inequality for twoobservables .In this paper, we investigate variance-based uncertainty relation for multiple incompatible observables. We present a newvariance-based sum uncertainty relation for multiple incompatible observables, which is stronger than an uncertainty relationfrom summing over all the inequalities for pairs of observables . Furthermore, we compare the uncertainty relation withexisting ones for a spin- and spin-1 particle, which shows our uncertainty relation can give a tighter bound than other ones. Results
Theorem 1
For arbitrary N observables A , A , . . . , A N , the following variance-based uncertainty relation holds N ∑ i = ∆ ( A i ) ≥ N (cid:34) ∆ ( N ∑ i = A i ) (cid:35) + N ( N − ) (cid:34) ∑ ≤ i < j ≤ N ∆ ( A i − A j ) (cid:35) . (7) 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 ∑ ≤ i < j ≤ N [ ∆ ( A i − A j )] = N N ∑ i = ( ∆ A i ) − (cid:34) ∆ ( N ∑ i = A i ) (cid:35) , (8)then using the inequality N ( N − ) ∑ ≤ i < j ≤ N [ ∆ ( A i − A j )] ≥ (cid:34) ∑ ≤ i < j ≤ N ∆ ( A i − A j ) (cid:35) , (9)we obtain the uncertainty relation (7).QED.To show that our relation (7) has a stronger bound, we consider the result in Ref. 10, the relation (5) is derived from theuncertainty equality ∆ A + ∆ B = [ ∆ ( A + B )] + [ ∆ ( A − B )] . (10)Using the above uncertainty equality, one can obtain two inequalities for arbitrary N observables, namely N ∑ i = ( ∆ A i ) ≥ ( N − ) ∑ ≤ i < j ≤ N [ ∆ ( A i + A j )] (11)and N ∑ i = ( ∆ A i ) ≥ ( N − ) ∑ ≤ i < j ≤ N [ ∆ ( A i − A j )] . (12)The bound in (6) is tighter than the one in (11) . However, the lower bound in (6) is not always tighter than the one in (12) (seeFigure 1). igure 1. (color online). Example of comparison between our relation (7) and the ones (6),(11),(12). The upper line is thesum of the variances (SV) ( ∆ σ x ) + ( ∆ σ y ) + ( ∆ σ z ) . The black line is the lower bound (LB) given by our relation (7). Thesolid green line is the bound (6) (FB). The dashed green line is the bound (11) (PB1). The blue line is the bound (12) (PB2). Example 1
To give an overview that the relation (7) has a better lower bound than the relations (6),(11),(12), we consider afamily of qubit pure states given by the Bloch vector (cid:126) r = ( √ cos θ , √ cos θ , sin θ ) , and choose the Pauli matrices σ x = (cid:18) (cid:19) , σ y = (cid:18) − ii (cid:19) , σ z = (cid:18) − (cid:19) . Then we have ( ∆ σ x ) + ( ∆ σ y ) + ( ∆ σ z ) = [ ∆ ( σ x + σ y )] = ( sin θ ) , and [ ∆ ( σ y + σ z )] = [ ∆ ( σ x + σ z )] = + cos 2 θ − √ sin 2 θ . Similarly, [ ∆ ( σ x − σ y )] =
2, and [ ∆ ( σ y − σ z )] = [ ∆ ( σ x − σ z )] = + cos 2 θ + √ sin 2 θ . The comparison betweenthe lower bounds (6),(11),(12) and (7) is given in Figure 1. Apparently, our bound is tighter than (6) ,(11) and (12). We shallshow with detailed proofs and examples that our uncertainty relation (7) has better lower bound than that of (6),(11),(12) in thefollowing sections. Comparison between the lower bound of our uncertainty relation (7) with that of inequality (11)
First, we compare our relation (7) with the one (11). Note that ∆ ( A i + A j ) = ∆ A i + ∆ A j + (cid:104){ A i , A j }(cid:105) − (cid:104) A i (cid:105)(cid:104) A j (cid:105) , the relation(11) becomes N ∑ i = ( ∆ A i ) ≥ ( N − ) (cid:40) ( N − ) N ∑ i = ( ∆ A i ) + ∑ ≤ i < j ≤ N [ (cid:104){ A i , A j }(cid:105) − (cid:104) A i (cid:105)(cid:104) A j (cid:105) ] (cid:41) . (13)Simplify the above inequality, we obtain N ∑ i = ( ∆ A i ) ≥ ( N − ) ∑ ≤ i < j ≤ N [ (cid:104){ A i , A j }(cid:105) − (cid:104) A i (cid:105)(cid:104) A j (cid:105) ] , (14)which is equal to the relation (11).Similarly, by using ∆ ( A i − A j ) = ∆ A i + ∆ A j − (cid:104){ A i , A j }(cid:105) + (cid:104) A i (cid:105)(cid:104) A j (cid:105) , our relation (7) becomes N ∑ i = ( ∆ A i ) ≥ N (cid:34) N ∑ i = ( ∆ A i ) + ∑ ≤ i < j ≤ N [ (cid:104){ A i , A j }(cid:105) − (cid:104) A i (cid:105)(cid:104) A j (cid:105) ] (cid:35) + N ( N − ) ( N − ) N ∑ i = ( ∆ A i ) − ∑ ≤ i < j ≤ N [ (cid:104){ A i , A j }(cid:105) − (cid:104) A i (cid:105)(cid:104) A j (cid:105) ] + i (cid:54) = i (cid:48) or j (cid:54) = j (cid:48) ∑ ≤ i < j ≤ N ≤ i (cid:48) < j (cid:48) ≤ N ∆ ( A i − A j ) ∆ ( A i (cid:48) − A j (cid:48) ) . (15)Simplify the above inequality, we get N ∑ i = ( ∆ A i ) ≥ N − ∑ ≤ i < j ≤ N [ (cid:104){ A i , A j }(cid:105) − (cid:104) A i (cid:105)(cid:104) A j (cid:105) ] + ( N − )( N − ) i (cid:54) = i (cid:48) or j (cid:54) = j (cid:48) ∑ ≤ i < j ≤ N ≤ i (cid:48) < j (cid:48) ≤ N ∆ ( A i − A j ) ∆ ( A i (cid:48) − A j (cid:48) ) , (16) hich is equal to the relation (7). It is easy to see that the right-hand side of (16) is greater than the right-hand side of (14).Hence, the relation (7) is stronger than the relation (11). Comparison between the lower bound of our uncertainty relation (7) with that of inequalities (6) and (12)
Here, we will show the uncertainty relation (7) is stronger than inequalities (12) and (6) for a spin- particle and measurementof Pauli-spin operators σ x , σ y , σ z . Then the uncertainty relation (7) has the form ( ∆ σ x ) + ( ∆ σ y ) + ( ∆ σ z ) ≥ [ ∆ ( σ x + σ y + σ z )] + (cid:32) ∑ ≤ i < j ≤ ∆ ( σ i − σ j ) (cid:33) , (17)the relation (12) is given by ( ∆ σ x ) + ( ∆ σ y ) + ( ∆ σ z ) ≥ ∑ ≤ i < j ≤ [ ∆ ( σ i − σ j )] , (18)and the relation (6) says that ( ∆ σ x ) + ( ∆ σ y ) + ( ∆ σ z ) ≥ ∑ ≤ i < j ≤ [ ∆ ( σ i + σ j )] − (cid:32) ∑ ≤ i < j ≤ ∆ ( σ i + σ j ) (cid:33) . (19)We consider a qubit state and its Bloch sphere representation ρ = ( I + (cid:126) r · (cid:126) σ ) , (20)where (cid:126) σ = ( σ x , σ y , σ z ) are Pauli matrices and the Bloch vector (cid:126) r = ( x , y , z ) is real three-dimensional vector such that (cid:107) (cid:126) r (cid:107) ≤ ( ∆ σ x ) = Tr [ ρσ x σ x ] − Tr [ ρσ x ] = − x , ( ∆ σ x ) + ( ∆ σ y ) + ( ∆ σ z ) = − ( x + y + z ) . The relation (17) hasthe form ( ∆ σ x ) + ( ∆ σ y ) + ( ∆ ρ σ z ) ≥ α + (cid:0) − ( x + y + z ) (cid:1) , (21)where we define α = (cid:112) − ( x − y ) + (cid:112) − ( x − z ) + (cid:112) − ( y − z ) . And the relation (18) becomes ( ∆ σ x ) + ( ∆ σ y ) + ( ∆ σ z ) ≥ (cid:0) − ( x + y + z ) + xy + xz + yz (cid:1) . (22)Let us compare the lower bound of (21) with that of (22). The difference of these two bounds is19 α + (cid:0) x + y + z (cid:1) − ( xy + xz + zy ) −
12 (23) ≥ (cid:18)(cid:113) − ( x − y ) (cid:113) − ( x − z ) (cid:113) − ( y − z ) (cid:19) − (cid:0) x + y + z (cid:1) − ≥ (cid:18)(cid:113) − ( x − y ) (cid:113) − ( x − z ) (cid:113) − ( y − z ) (cid:19) − , for all x , y , z ∈ [ − , ] . When x = y = z = ± / √
3, the above inequality becomes equality, then the Eq.(23) has the minimumvalue 1 / >
0. This illustrates that the uncertainty relation (7) is stronger that the one (12) for a spin- particle and measurementof Pauli-spin operators σ x , σ y , σ z .Let us compare the uncertainty relation (17) with (19). The relation (19) has the form ( ∆ σ x ) + ( ∆ σ y ) + ( ∆ σ z ) ≥ − ( xy + xz + yz ) − ( x + y + z ) − β , (24)where we define β = (cid:112) − ( x + y ) + (cid:112) − ( x + z ) + (cid:112) − ( y + z ) . Then the difference of these two bounds of relation (21)and (24) becomes19 α + β + ( x + y + z ) + ( x + y + z ) − ≥ (cid:32) √ α + √ + √ β (cid:33) (cid:18) + √ (cid:19) + ( x − z ) − , here we have twice used Cauchy’s inequality. When α = √ , β = √ + √ , x + y + z = x = − y = ± / √ , z = or x = − z = ± / √ , y = or y = − z = ± / √ , x =
0, the above inequality becomes equality, then the Eq.(25) has the minimum value √ − >
0. This illustrates that the uncertainty relation (7) is stronger that the one (6) for a spin- particle and measurementof Pauli-spin operators σ x , σ y , σ z . Figure 2. (color online). Example of comparison between our relation (7) and ones (6),(11),(12). We choose A = L x , A = L y and A = L y three components of the angular momentum for spin-1 particle, and a family of states parametrized by θ and φ as | ψ (cid:105) = sin θ cos φ | (cid:105) + sin θ sin φ | (cid:105) + cos θ | − (cid:105) . [A] For φ = π /
4, the comparison between our relation (7) and ones(6),(11),(12). The upper line is the sum of the variances (SV) ( ∆ L x ) + ( ∆ L y ) + ( ∆ L z ) . The black line is the lower bound(LB) given by our relation (7). The solid green line is the bound (6) (FB). The dashed green line is the bound (11) (PB1). Theblue line is the bound (12) (PB2). [B] The comparison between our relation (7) and (6), which shows that our relation (7) (LB)has stronger bound than (6) (FB). [C] The comparison between our relation (7) and (12), which shows that our relation (7) (LB)has stronger bound than (12) (PB2). [D] The lower bound of the relation (6) minus the lower bound of the relation (12). Example 2
For spin-1 systems, we consider the following quantum state characterized by θ and φ | ψ (cid:105) = sin θ cos φ | (cid:105) + sin θ sin φ | (cid:105) + cos θ | − (cid:105) , (26)with 0 ≤ θ ≤ π , ≤ φ ≤ π . By choosing the three angular momentum operators (¯ h = L x = √ , L y = √ − i i − i i , L z = − , the comparison between the lower bounds (6),(11),(12) and (7) is shown by Figure 2. The results suggest that the relation (7)can give tighter bounds than other ones (6),(11),(12) for a spin-1 particle and measurement of angular momentum operators L x , L y , L z . Conclusion
We have provided a variance-based sum uncertainty relation for N incompatible observables, which is stronger than the simplegeneralizations of the uncertainty relation for two observables derived by Maccone and Pati [Phys. Rev. Lett. , 260401(2014)]. Furthermore, our uncertainty relation gives a tighter bound than the others by comparison for a spin- particle with the easurements of spin observables σ x , σ y , σ z . And also, in the case of spin-1 with measurement of angular momentum operators L x , L y , L z , our uncertainty relation predicts a tighter bound than other ones. References Busch, P., Heinonen, T. & Lahti, P. J. Heisenberg’s uncertainty principle.
Phys. Rep. , 155 (2007). Hofmann, H. F. & Takeuchi, S. Violation of local uncertainty relations as a signature of entanglement.
Phys. Rev. A ,032103 (2003). G¨uhne, O. Characterizing entanglement via uncertainty relations.
Phys. Rev. Lett. , 117903 (2004). Fuchs, C. A. & Peres, A. Quantum-state disturbance versus information gain: Uncertainty relations for quantum information.
Phys. Rev. A , 2038 (1996). Heisenberg, W. ¨Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik.
Z. Phys. , 172(1927). Kennard, E. H. Zur Quantenmechanik einfacher Bewegungstypen.
Z. Phys. , 326 (1927). Weyl, H. Gruppentheorie and Quantenmechanik (Hirzel, Leipzig). (1928). Robertson, H. P. The uncertainty principle.
Phys. Rev. , 163 (1929). Schr¨odinger, E. Situngsberichte der Preussischen Akademie der Wissenschaften.
Physikalisch-mathematische Klasse ,296 (1930). Maccone, L. & Pati, A. K. Stronger uncertainty relations for all incompatible observables.
Phys. Rev. Lett. , 260401(2014).
Coles, P. J., Berta, M. & Tomamichel, M. Entropic uncertainty relations and their applications.
Rev. Mod. Phys.
Accepted(2016).
Bannur, V. B. Comments on “Stronger uncertainty relations for all incompatible observables”. arXiv:1502.04853 (2015).
Yao, Y., Xiao, X., Wang, X. & Sun, C. P. Implications and applications of the variance-based uncertainty equalities.
Phys.Rev. A , 062113 (2015). Song, Q. C. & Qiao, C. F. Stronger Shr¨odinger-like uncertainty relations.
Phys. Lett. A , 2925 (2016).
Xiao, Y., Jing, N., Li-Jost, X. & Fei, S. M. Weight uncertainty relations.
Sci. Rep. , 23201 (2016). Zhang, J., Zhang, Y. & Yu, C. S. Stronger uncertainty relations with arbitrarily tight upper and lower bounds.arXiv:1607.08223 (2016).
Huang, Y. Variance-based uncertainty relations.
Phys. Rev. A , 024101 (2012). Li, J. L. & Qiao, C. F. Reformulating the quantum uncertainty relation.
Sci. Rep. , 12708 (2015). Li, J. L. & Qiao, C. F. Equivalence theorem of uncertainty relations. J. Phys. A , 03LT01 (2017). Abbott, A. A., Alzieu, P. L., Hall, M. J. W. & Branciard, C. Tight state-independent uncertainty relations for qubits.
Mathematics , 8 (2016).
Wang, K., et al. Experimental investigation of the stronger uncertainty relations for all incompatible observables.
Phys.Rev. A , 052108 (2016). Wa, W., et al. Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances.
Phys. Rev.Lett. , 160405 (2016).
Baek, S. Y., Kaneda, F., Ozawa, M. & Edamatsu, K. Experimental violation and reformulation of the Heisenberg’serror-disturbance uncertainty relation.
Sci. Rep. , 2221 (2013). Zhou, F., et al. Verifying Heisenberg’s error-disturbance relation using a single trapped ion.
Sci. Adv. , e1600578 (2016). Kechrimparis, S., & Weigert, S. Heisenberg uncertainty relation for three canonical observables.
Phys. Rev. A , 062118(2014). Dammeier, L., Schwonnek, R. & Werner, P. F. Uncertainty relations of angular momentum.
New J. Phys. , 093046(2015). Qiu, H. H., Fei, S. M. & Li-Jost, X. Multi-observable uncertainty relations in product form of variances.
Sci. Rep. , 31192(2016). Xiao, Y. & Jing, N. Mutually exclusive uncertainty relations.
Sci. Rep. , 36616 (2016). Chen, B., Cao, N. P., Fei, S. M. & Long, G. L. Variance-based uncertainty relations for incompatible observables.
QuantumInf. Process , 3909 (2016). Chen, B. & Fei, S. M. Sum uncertainty relations for arbitrary N incompatible observables. Sci. Rep. , 14238 (2015). Acknowledgements
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 NationalNatural Science Foundation of China(NSFC) under the grants 11375200 and 11635009.
Author contributions statement
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.
Additional information