Uncertainty equalities and uncertainty relation in weak measurement
aa r X i v : . [ qu a n t - ph ] M a y Uncertainty equalities and uncertainty relation in weak measurement
Qiu-Cheng Song and Cong-Feng Qiao , ∗ School of Physics, University of Chinese Academy of Sciences - YuQuan Road 19A, Beijing 100049, China CAS Center for Excellence in Particle Physics, Beijing 100049, China
Uncertainty principle is one of the fundamental principles of quantum mechanics. In this work, wederive two uncertainty equalities, which hold for all pairs of incompatible observables. We also obtainan uncertainty relation in weak measurement which captures the limitation on the preparation ofpre- and post-selected ensemble and hold for two non-Hermitian operators corresponding to twonon-commuting observables.
PACS numbers: 03.65.Ta, 42.50.Lc, 03.67.-a
I. INTRODUCTION
Uncertainty principle is one of the basic tenets of quan-tum mechanics. The initial spirit of uncertainty principlewas postulated by Heisenberg [1]. Kennard first mathe-matically derived the Heisenberg uncertainty relation [2].The most famous and popular form is the Heisenberg-Robertson uncertainty relation [3]∆ A ∆ B ≥ | h ψ | [ A, B ] | ψ i| , (1)for any observables A , B , and any state | ψ i , where thevariance of an observable X in state | ψ i is defined as∆ X = h ψ | X | ψ i − h ψ | X | ψ i and the commutator isdefined as [ A, B ] = AB − BA . A stronger extension ofthe Heisenberg-Robertson uncertainty relation (1) wasmade by Schr¨odinger [4], which is generally formulatedas∆ A ∆ B ≥ (cid:12)(cid:12)(cid:12)(cid:12) h [ A, B ] i (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) h{ A, B }i − h A ih B i (cid:12)(cid:12)(cid:12)(cid:12) , (2)where the anticommutator is defined as { A, B } = AB + BA , and h X i is defined as the expectation value h ψ | X | ψ i for any operator X with respect to the normalized state | ψ i .However, the above two uncertainty relations have theproblem that they may be trivial even when A and B areincompatible on the state | ψ i . In order to correct thisproblem, Maccone and Pati [5] presented two strongeruncertainty relations based on the sum of variances. Thefirst one reads∆ A + ∆ B ≥ ± i |h [ A, B ] i| + |h ψ | A ± iB | ψ ⊥ i| , (3)which is valid for arbitrary states | ψ ⊥ i orthogonal to thestate of the system state | ψ i , where the sign should bechosen so that ± i h [ A, B ] i (a real quantity) is positive.The second uncertainty relation is∆ A + ∆ B ≥ |h ψ ⊥ A + B | A + B | ψ i| . (4) ∗ Corresponding author, [email protected]
Here | ψ ⊥ A + B i ∝ ( A + B − h A + B i ) | ψ i is a state orthog-onal to | ψ i . Maccone and Pati also derived an amendedHeisenberg-Robertson uncertainty relation∆ A ∆ B ≥ ± i h [ A, B ] i − |h ψ | A ∆ A ± i B ∆ B | ψ ⊥ i| , (5)which is stronger than the Heisenberg-Robertson uncer-tainty relation (1).Recently, two stronger Schr¨odinger-like uncertainty re-lations have been proved which go beyond the Macconeand Pati’s uncertainty relation [6]. The new relationsprovide stronger bounds whenever the observables areincompatible on the state | ψ i . The first uncertainty re-lation is∆ A + ∆ B ≥|h [ A, B ] i + h{ A, B }i − h A ih B i| + |h ψ | A − e iα B | ψ ⊥ i| , (6)which is valid for arbitrary states | ψ ⊥ i orthogonal tothe state of the system | ψ i and stronger than the Mac-cone and Pati’s uncertainty relation (3). Here α is areal constant, if h{ A, B }i − h A ih B i >
0, then α =arctan − i h [ A,B ] ih{ A,B }i− h A ih B i ; if h{ A, B }i − h A ih B i <
0, then α = π + arctan − i h [ A,B ] ih{ A,B }i− h A ih B i ; and while h{ A, B }i − h A ih B i = 0, the relation (6) will reduce to (3) . Thesecond uncertainty relation is∆ A ∆ B ≥ (cid:12)(cid:12) h [ A, B ] i (cid:12)(cid:12) + (cid:12)(cid:12) h{ A, B }i − h A ih B i (cid:12)(cid:12) (1 − |h ψ | A ∆ A − e iα B ∆ B | ψ ⊥ i| ) (7)which is stronger than the Schr¨odinger uncertainty rela-tion (2).However, these new state dependent uncertainty rela-tions have some problem [7], but some state independentuncertainty relations [8, 9] immune from the drawback.Maccone and Pati’s uncertainty relations [5] still are veryimportant and have some generalizations. Two variance-based uncertainty equalities have been proved recentlyby Yao et al. [10] on the trend of stronger uncertaintyrelations [5], for all pairs of incompatible observables A and B . Meanwhile, two uncertainty relations in weakmeasurement were introduced by Pati et al. [11] for vari-ances of two non-Hermitian operators corresponding totwo non-commuting observables.In this work we derive and proof two uncertainty equal-ities, which hold for all pairs of incompatible observ-ables A and B . We also give an uncertainty relationin weak measurement for two non-Hermitian operatorscorresponding to two non-commuting observables. II. UNCERTAINTY EQUALITIES
In this section, we construct and prove two uncertaintyequalities which imply the uncertainty inequalities (6)and (7).
Uncertainty relation 1 ∆ A + ∆ B = |h [ A, B ] i + h{ A, B }i − h A ih B i| + d − X n =1 |h ψ | A − e iα B | ψ ⊥ n i| , (8) where {| ψ i , | ψ ⊥ n i d − n =1 } comprise an orthonormal completebasis in the d -dimensional Hilbert space.Proof : To prove our uncertainty relation, let us define theoperator Π = I − | ψ ih ψ | , ¯ A = A − h A i I , ¯ B = B − h B i I and the state | φ i = ( ¯ A − e iτ ¯ B ) | ψ i , we have h φ | Π | φ i = h ψ | ( ¯ A − e − iτ ¯ B ) | ( I − | ψ ih ψ | ) | ( ¯ A − e iτ ¯ B ) | ψ i = h ψ | ( ¯ A − e − iτ ¯ B )( ¯ A − e iτ ¯ B ) | ψ i =∆ A + ∆ B − e iτ h ψ | ¯ A ¯ B | ψ i ) . (9)There exists τ = − α , so that e iτ h ψ | ¯ A ¯ B | ψ i is real, and itcan be written as |h ψ | ¯ A ¯ B | ψ i| , we obtain h ψ | ( ¯ A − e iα ¯ B ) | Π | ( ¯ A − e − iα ¯ B ) | ψ i =∆ A + ∆ B − |h ψ | ¯ A ¯ B | ψ i| =∆ A + ∆ B − |h [ A, B ] i + h{ A, B }i − h A ih B i| , (10)Since Π is the orthogonal complement to | ψ ih ψ | , we canchoose an arbitrary orthogonal decomposition of the pro-jector Π Π = d − X n =1 | ψ ⊥ n ih ψ ⊥ n | , (11)where {| ψ i , | ψ ⊥ n i d − n =1 } comprise an orthonormal completebasis in the d -dimensional Hilbert space. Whence, Eq.(10) can be rewritten as d − X n =1 |h ψ | ( ¯ A − e iα ¯ B ) | ψ ⊥ n i| = d − X n =1 |h ψ | A − e iα B | ψ ⊥ n i| =∆ A + ∆ B − |h [ A, B ] i + h{ A, B }i − h A ih B i| , (12)which is equivalent to (8). Uncertainty relation 2 ∆ A ∆ B = (cid:12)(cid:12) h [ A, B ] i (cid:12)(cid:12) + (cid:12)(cid:12) h{ A, B }i − h A ih B i (cid:12)(cid:12) (1 − P d − n =1 |h ψ | A ∆ A − e iα B ∆ B | ψ ⊥ n i| ) , (13) where {| ψ i , | ψ ⊥ n i d − n =1 } comprise an orthonormal completebasis in the d -dimensional Hilbert space.Proof : To prove our uncertainty equality, let us definethe operator Π = I −| ψ ih ψ | , ¯ A = A −h A i I , ¯ B = B −h B i I and the unnormalized state | φ i = ( ¯ A ∆ A − e iτ ¯ B ∆ B ) | ψ i , wehave h φ | Π | φ i = h ψ | ( ¯ A ∆ A − e − iτ ¯ B ∆ B ) | ( I − | ψ ih ψ | ) | ( ¯ A ∆ A − e iτ ¯ B ∆ B ) | ψ i = h ψ | ( ¯ A ∆ A − e − iτ ¯ B ∆ B )( ¯ A ∆ A − e iτ ¯ B ∆ B ) | ψ i =2 − e iτ h ψ | ¯ A ¯ B | ψ i )∆ A ∆ B , (14)There exists τ = − α , so that e iτ h ψ | ¯ A ¯ B | ψ i is real, and itcan be written as |h ψ | ¯ A ¯ B | ψ i| , we obtain h ψ | ( ¯ A ∆ A − e iα ¯ B ∆ B ) | Π | ( ¯ A ∆ A − e − iα ¯ B ∆ B ) | ψ i =2 − |h ψ | ¯ A ¯ B | ψ i| ∆ A ∆ B , (15)Similarly, we choose the projector ΠΠ = d − X n =1 | ψ ⊥ n ih ψ ⊥ n | . (16)Then Eq. (15) can be rewritten as d − X n =1 |h ψ | ( ¯ A ∆ A − e iα ¯ B ∆ B ) | ψ ⊥ n i| = d − X n =1 |h ψ | A ∆ A − e iα B ∆ B | ψ ⊥ n i| =2 − | h [ A, B ] i + h{ A, B }i − h A ih B i| ∆ A ∆ B , (17)which is equivalent to (13).The two uncertainty equalities (8) and (13) are validfor all pairs of incompatible observables. If we retain onlyone term associated with | ψ ⊥ i ∈ {| ψ ⊥ n i d − n =1 } in the sum-mation and discard the others, the uncertainty equalities(8) and (13) reduce to the uncertainty relations (6) and(7), respectively. III. UNCERTAINTY RELATION IN WEAKMEASUREMENT
First introduced by Aharonov, Albert, and Vaidman[12], weak values are complex numbers that one can de-fine the weak value of A using two states: an initial state | ψ i , called the preselection, and a final state | ϕ i , calledthe postselection. the weak value of A has the form h A i w = h ϕ | A | ψ ih ϕ | ψ i . (18)For a givern preselected and post-selected ensemble, de-fine the operator A w as A w = Π ϕ Ap , (19)where Π ϕ = | ϕ ih ϕ | and p = |h ϕ | ψ i| . This has manyproperties please reference [11].Here, we construct an uncertainty relation in weakmeasurement for variances of two non-Hermitian oper-ators A w and B w corresponding to two non-commutingobservables A and B . The uncertainty relation quanti-tatively express the impossibility of jointly sharp prepa-ration of pre- and post-selected (PPS) quantum states | ψ i and | ϕ i for the weak measurement of incompatibleobservables. Uncertainty relation 3 ∆ A w + ∆ B w ≥ | p h ϕ | [ A, B ] | ϕ i + 1 p h ϕ |{ A, B }| ϕ i− h A w ih B w i ∗ | + (cid:12)(cid:12) h ψ | A w − e iα B w | ψ ⊥ i (cid:12)(cid:12) . (20) which is valid for two non-Hermitian operators A w and B w , where p is equivalent to |h ϕ | ψ i| .Proof : To prove this relation we define the variancefor any general (non-Hermitian) operator X in a state | ψ i which can be defined as [13, 14]∆ X = h ψ | ( X − h X i )( X † − h X † i ) | ψ i . (21)The variance of the non-Hermitian operation A w in thequantum | ψ i can be defined as∆ A w = h ψ | ( A w − h A w i )( A † w − h A † w i ) | ψ i , (22)where h A w i = h ψ | A w | ψ i and h A † w i = h ψ | A † w | ψ i = h A w i ∗ ,∆ A w can also be expressed as∆ A w = h ψ | A w A † w i| ψ i − h ψ | A w | ψ ih ψ | A † w | ψ i . (23)Similarly, for Hermitian operator B , we can define theoperator B w = Π ϕ Bp . (24)Then, the uncertainty for B w can also be defined as∆ B w = h ψ | B w B † w i| ψ i − h ψ | B w | ψ ih ψ | B † w | ψ i . (25)To prove our uncertainty relation in weak measurement,we introduce a general inequality k C † | ψ i − e iτ D † | ψ i + k ( | ψ i − | ¯ ψ i ) k ≥ , (26) where C † ≡ A † w − h A † w i and D † ≡ B † w − h B † w i . By ex-panding the square modulus, we have∆ A w + ∆ B w ≥ − λk − βk + π, (27)where λ ≡ − Re[ h ψ | ¯ ψ i ]), π ≡ e iτ h ψ | CD † | ψ i ],and β ≡ h ψ | ( − C + e − iτ D ) | ¯ ψ i ]. We choose the valueof k that maximizes the right-hand-side of (27), namely k = − β/ λ , we get∆ A w + ∆ B w ≥ β λ + π. (28)The above inequality can be rewritten as∆ A w + ∆ B w ≥ Re[ h ψ | ( − C + e − iτ D ) | ¯ ψ i ] − Re[ h ψ | ¯ ψ i ])+ 2Re[ e iτ h ψ | CD † | ψ i ] (29)Suppose | ¯ ψ i = cos θ | ψ i + e iφ sin θ | ψ ⊥ i , where | ψ ⊥ i is or-thogonal to | ψ i , by taking the limit θ →
0, the state | ¯ ψ i reduces to | ψ i and then the above inequality can bereexpressed as∆ A w + ∆ B w ≥ Re[ e iφ h ψ | ( − A w + e − iτ B w ) | ψ ⊥ i ] + 2Re[ e iτ h ψ | CD † | ψ i ] , (30)there exists τ = − α so that e iτ h ψ | CD † | ψ i is real, andit can be written as |h ψ | CD † | ψ i| , and then the secondterm becomes { Re[ e iφ h ψ | − A w + e iα B w | ψ ⊥ i ] } , we canchoose φ so that this term in square brackets is real, sothat this term can be expressed as |h ψ | A w − e iα B w | ψ ⊥ i| .Whence, inequality (30) becomes∆ A w + ∆ B w ≥|h ψ | A w − e iα B w | ψ ⊥ i| + 2 |h ψ | CD † | ψ i| . (31)The last term can be rewritten as2 |h CD † i| = |h CD † + DC † + CD † − DC † i| , (32)where h CD † + DC † i = 1 p h ϕ |{ A, B }| ϕ i − h A w ih B w i ∗ − h A w i ∗ h B w i (33)and h CD † − DC † i = 1 p h ϕ | [ A, B ] | ϕ i − h A w ih B w i ∗ + h A w i ∗ h B w i . (34)We combine Eqs. (33) and (34), Eq. (32) becomes2 |h CD † i| = (cid:12)(cid:12)(cid:12)(cid:12) p h ϕ | [ A, B ] | ϕ i + 1 p h ϕ |{ A, B }| ϕ i − h A w ih B w i ∗ (cid:12)(cid:12)(cid:12)(cid:12) . (35)Combining Eqs. (32) and (35), we obtain the uncertaintyrelation (20). IV. CONCLUSIONS
In this work, we derived two new uncertainty equalitiesfor sum and product of variances of a pair of incompat-ible observables, which hold for all pairs of incompatibleobservables A and B . In fact, one can obtain a seriesof inequalities by retaining 1 to d − {| ψ ⊥ n i d − n =1 } . We also derived an uncertainty relationin weak measurement for two non-Hermitian operators A w and B w corresponding to two non-commuting observ-ables A and B . The uncertainty relation quantitatively expresses the impossibility of jointly sharp preparationof PPS quantum states | ψ i and | ϕ i for measuring incom-patible observables during the weak measurement. Acknowledgments
We are grateful to Junli Li fordiscussion. This work was supported in part by NationalKey Basic Research Program of China under the grant2015CB856700, and by the National Natural ScienceFoundation of China(NSFC) under the grants 11175249and 11375200. [1] W. Heisenberg, Z. Phys. : 172 (1927).[2] E. Kennard, Z. Phys. , 326 (1927).[3] H. P. Robertson, Phys. Rev. , 163 (1929).[4] E. Schr¨odinger, Sitzungsberichte der Preussis-chen Akademie der Wissenschaften, Physikalisch-mathematische Klasse , 296 (1930).[5] L. Maccone and A. K. Pati, Phys. Rev. Lett. , 260401(2014).[6] Qiu-Cheng Song and Cong-Feng Qiao, arXiv:1504.01137.[7] V. M. Bannur, arXiv:1502.04853.[8] Jun-Li Li and Cong-Feng Qiao, arXiv:1502.06292. [9] Y. Huang, Phys. Rev. A , 024101 (2012).[10] Yao Yao, Xing Xiao, Xiaoguang Wang, and C. P. Sun,arXiv:1503.00239.[11] A. K. Pati and J. Wu, arXiv:1411.7218.[12] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev.Lett. , 1351 (1988).[13] J. S. Anandan, Phys. Lett. A147