Uncertainty principle is one of the basic tenets of quantum mechanics. The initial spirit of uncertainty principle was postulated by Heisenberg ^[1]. Kennard^[2] first mathematically derived the Heisenberg uncertainty relation . The most famous and popular form is the Heisenberg-Robertson uncertainty relation ^[3]

$\Delta {{A}^{2}}\Delta {{B}^{2}}\ge {{\left| \frac{1}{2}\langle \psi \left| \left[ A, B \right] \right|\psi \rangle \right|}^{2}}, $

(1)

for any observables A and B and any state |ψ>, where the variance of an observable X in state |ψ> is defined as ΔX²=〈ψ|X²|ψ〉-〈ψ|X|ψ〉² and the commutator is defined as [A, B]=AB-BA. A stronger extension of the Heisenberg-Robertson uncertainty relation (1) was made by Schrödinger^[4], which is generally formulated as

$\begin{align} & \Delta {{A}^{2}}\Delta {{B}^{2}}\ge {{\left| \frac{1}{2}\langle \left[ A, B \right]\rangle \right|}^{2}}+ \\ & {{\left| \frac{1}{2}\langle \left\{ A, B \right\}\rangle -\left\langle A \right\rangle \left\langle B \right\rangle \right|}^{2}}, \\ \end{align}$

(2)

where the anticommutator is defined as {A, B}=AB+BA, and 〈X〉 is defined as the expectation value 〈ψ|X|ψ〉 for any operator X with respect to the normalized state |ψ〉.

However, the above two uncertainty relations have the problem that they may be trivial even when A and B are incompatible on the state |ψ〉. In order to correct this problem, Maccone and Pati ^[5] presented two stronger uncertainty relations based on the sum of variances. The first uncertainty relation reads

$\begin{align} & \Delta {{A}^{2}}+\Delta {{B}^{2}}\ge \pm i\left\langle \left[ A, B \right] \right\rangle + \\ & \left| \langle \psi \right|A\pm iB|{{\psi }^{\bot }}\rangle {{|}^{2}}, \\ \end{align}$

(3)

which is valid for arbitrary states |ψ^⊥〉 orthogonal to the state of the system |ψ〉, where the sign should be chosen so that ±i〈[A, B]〉 (a real quantity) is positive. The second uncertainty relation is

$\Delta {{A}^{2}}+\Delta {{B}^{2}}\ge \frac{1}{2}|\langle \psi _{A+B}^{\bot }\left| A+B \right|\psi \rangle {{|}^{2}}.$

(4)

Here |ψ_A+B^⊥〉∝(A+B-〈A+B〉)|ψ〉 is a state orthogonal to |ψ〉. Maccone and Pati also derived an amended Heisenberg-Robertson uncertainty relation ΔAΔB≥±i12〈[A, B]〉1-12|〈ψ|AΔA±iBΔB|ψ^⊥〉|²,

$\Delta A\Delta B\ge \frac{\pm i\frac{1}{2}\left\langle \left[ A, B \right] \right\rangle }{1-\frac{1}{2}\left| \langle \psi \right|\frac{A}{\Delta A}\pm i\frac{B}{\Delta B}|{{\psi }^{\bot }}\rangle {{|}^{2}}}, $

(5)

which is stronger than the Heisenberg-Robertson uncertainty relation (1).

Recently, two stronger Schrödinger-like uncerta-inty relations^[6] have been proved which go beyond the Maccone and Pati’s uncertainty relation. The new relations provide stronger bounds whenever the observables are incompatible on the state |ψ〉. The first uncertainty relation is

$\begin{align} & \Delta {{A}^{2}}+\Delta {{B}^{2}}\ge \left| \left\langle \left[ A, B \right] \right\rangle +\left\langle \left\{ A, B \right\} \right\rangle -2\left\langle A \right\rangle \left\langle B \right\rangle \right| \\ & +\left| ＜\psi \right|A-{{e}^{i\alpha }}B|{{\psi }^{\bot }}>{{|}^{2}}, \\ \end{align}$

(6)

which is valid for arbitrary states |ψ^⊥〉 orthogonal to the state of the system |ψ〉 and stronger than the Maccone and Pati’s uncertainty relation (3). In(6), α is a real constant. If 〈{A, B}〉-2〈A〉〈B〉>0, then α=arctan-i〈[A, B]〉〈{A, B}〉-2〈A〉〈B〉. If 〈{A, B}〉-2〈A〉〈B〉 ＜0, then α=π+arctan-i〈[A, B]〉〈{A, B}〉-2〈A〉〈B〉. While 〈{A, B}〉-2〈A〉〈B〉=0, the relation (6) reduces to (3) . The second uncertainty relation is

$\begin{align} & \Delta {{A}^{2}}\Delta {{B}^{2}}\ge \\ & \frac{{{\left| \frac{1}{2}\left\langle \left[ A, B \right] \right\rangle \right|}^{2}}+{{\left| \frac{1}{2}\left\langle \left\{ A, B \right\} \right\rangle -\left\langle A \right\rangle \left\langle B \right\rangle \right|}^{2}}}{{{(1-\frac{1}{2}\left| \langle \psi \right|\frac{A}{\Delta A}-{{e}^{i\alpha }}\frac{B}{\Delta B}|{{\psi }^{\bot }}\rangle {{|}^{2}})}^{2}}} \\ \end{align}$

(7)

which is stronger than the Schrödinger uncertainty relation (2).

These new state-dependent uncertainty relations have some problem^[7], but some state-independent uncertainty relations^[8-9] are immune from the drawback. Maccone and Pati’s uncertainty relations^[5] are still very important and have some generalizations. Two variance-based uncertainty equalities were proved recently by Yao et al.^[10] on the trend of stronger uncertainty relations^[5], for all pairs of incompatible observables A and B. Meanwhile, two uncertainty relations in weak measurement were derived by Pati and Wu^[11] for variances of two non-Hermitian operators corresponding to two noncommuting observables.

In this work we derive and prove two uncertainty equalities, which hold for all pairs of incompatible observables A and B. We also give an uncertainty relation in weak measurement for two non-Hermitian operators corresponding to two non-commuting observables.

1 Uncertainty equalities

In this section, we construct and prove two uncertainty equalities, which imply the uncertainty inequalities (6) and (7).

Uncertainty relation 1.

$\begin{align} & \Delta {{A}^{2}}+\Delta {{B}^{2}}= \\ & \left| \left\langle \left[ A, B \right] \right\rangle +\left\langle \left\{ A, B \right\} \right\rangle -2\left\langle A \right\rangle \left\langle B \right\rangle \right| \\ & +\sum\limits_{n=1}^{d-1}{\left| \langle \psi \right|A-{{e}^{i\alpha }}B|{{\psi }^{\bot }}_{n}\rangle {{|}^{2}}}, \\ \end{align}$

(8)

where {|ψ〉, |ψ_n^⊥〉_n=1^d-1} comprise an orthonormal complete basis in the d-dimensional Hilbert space.

Proof To prove our uncertainty relation, let us define the operators Π=I-|ψ〉〈ψ|, A-=A〈A〉I, and B=B〈B〉I and the state |φ〉=(A-e^iτB)|ψ〉. We have 〈φ|Π|φ〉=〈ψ|(A-e^-iτB)|(I-|ψ〉

$\begin{align} & \left\langle \phi \left| \Pi \right|\phi \right\rangle =\left\langle \psi \left| \left( \bar{A}-{{e}^{-i\tau }}\bar{B} \right) \right|\left( I-|\psi \right) \right\rangle \\ & \langle \psi |)\left| \left( \bar{A}-{{e}^{-i\tau }}\bar{B} \right) \right|\psi \rangle \\ & =\langle \psi |\left( \bar{A}-{{e}^{-i\tau }}\bar{B} \right)\left( \bar{A}-{{e}^{-i\tau }}\bar{B} \right)|\psi \rangle \\ & =\Delta {{A}^{2}}+\Delta {{B}^{2}}-2Re({{e}^{i\tau }}\langle \psi \left| \bar{A}\bar{B} \right|\psi \rangle ) \\ \end{align}$

(9)

There exists τ=-α so that e^iτ〈ψ|AB|ψ〉 is real, and it can be written as |〈ψ|A-B-|ψ〉|. we obtain 〈ψ|(A-e^iαB)|Π|(A-e^-iαB)|ψ〉

$\begin{align} & \langle \psi |(\bar{A}-{{e}^{i\alpha }}\bar{B})|\Pi |(\bar{A}-{{e}^{-i\alpha }}\bar{B})|\psi \rangle \\ & =\Delta {{A}^{2}}+\Delta {{B}^{2}}-2\left| \langle \psi \right|\bar{A}\bar{B}\left| \psi \rangle \right| \\ & =\Delta {{A}^{2}}+\Delta {{B}^{2}}-|\left\langle \left[ A, B \right] \right\rangle + \\ & \left\langle \left\{ A, B \right\} \right\rangle -2\left\langle A \right\rangle \left\langle B \right\rangle |. \\ \end{align}$

(10)

Since Π is the orthogonal complement to |ψ〉〈ψ|, we can choose an arbitrary orthogonal decomposition of the projector Π,

$\Pi =\sum\limits_{n=1}^{d-1}{|\psi _{_{n}}^{^{\bot }}}\rangle \langle {{\psi }^{\bot }}_{n}|, $

(11)

where {|ψ>, |ψ_n^⊥>^d-1_n=1} comprise an orthonormal complete basis in the d-dimensional Hilbert space. Whence, Eq. (10) can be rewritten as

$\begin{align} & \sum\limits_{n=1}^{d-1}{\left| \langle \psi \right|}(\bar{A}-{{e}^{i\alpha }}\bar{B})|\psi _{n}^{\bot }>{{|}^{2}} \\ & =\sum\limits_{n=1}^{d-1}{\left| \langle \psi \right|}A-{{e}^{i\alpha }}B|{{\psi }^{\bot }}_{n}\rangle {{|}^{2}} \\ & =\Delta {{A}^{2}}+\Delta {{B}^{2}}-|\langle \left[ A, B \right]\rangle + \\ & \langle \left\{ A, B \right\}\rangle -2\langle A\rangle \langle B\rangle |, \\ \end{align}$

(12)

which is equivalent to (8).

Uncertainty relation 2.

$\begin{align} & \Delta {{A}^{2}}\Delta {{B}^{2}}= \\ & \frac{{{\left| \frac{1}{2}{{\left\langle \left[ A, B \right] \right\rangle }^{2}}+12\left\langle \left\{ A, B \right\} \right\rangle -\left\langle A \right\rangle \left\langle B \right\rangle \right|}^{2}}}{{{(1-\frac{1}{2}\sum\nolimits_{n=1}^{d-1}{\left| \langle \psi \right|}\frac{A}{\Delta A}-{{e}^{i\alpha }}\frac{B}{\Delta B}|\psi _{n}^{\bot }\rangle {{|}^{2}})}^{2}}}, \\ \end{align}$

(13)

where {|ψ〉, |ψ_n^⊥〉^d-1_n=1} comprise an orthonormal complete basis in the d-dimensional Hilbert space.

Proof To prove our uncertainty equality, let us define the operators Π=I-|ψ〉〈ψ|, A=A〈A〉I, and B=B〈B〉I and the unnormalized state $\left| \varphi \rangle =\frac{{\bar{A}}}{\Delta A}-{{e}^{i\tau }}\frac{{\bar{B}}}{\Delta B} \right|\psi \rangle $. We have

$\begin{align} & \langle \phi |\Pi |\phi \rangle \\ & =\langle \psi |(\frac{{\bar{A}}}{\Delta A}-{{e}^{-i\tau }}\frac{{\bar{B}}}{\Delta B})\left| \left( I \right.-\left| \psi \right\rangle \left. \left\langle \psi \right| \right) \right| \\ & (\frac{{\bar{A}}}{\Delta A}-{{e}^{-i\tau }}\frac{{\bar{B}}}{\Delta B})|\psi \\ & =\langle \psi |(\frac{{\bar{A}}}{\Delta A}-{{e}^{-i\tau }}\frac{{\bar{B}}}{\Delta B})(\frac{{\bar{A}}}{\Delta A}-{{e}^{-i\tau }}\frac{{\bar{B}}}{\Delta B})|\psi \rangle \\ & =2-2\frac{Re({{e}^{i\tau }}\langle \psi \left| A-\bar{B} \right|\psi \rangle )}{\Delta A\Delta B}, \\ \end{align}$

(14)

There exists τ=-α so that e^iτ〈ψ|A B|ψ〉 is real, and it can be written as |〈ψ|A B|ψ〉|. We obtain

$\begin{align} & \langle \psi |(\frac{{\bar{A}}}{\Delta A}-{{e}^{i\alpha }}\frac{{\bar{B}}}{\Delta B})|\Pi |(\frac{{\bar{A}}}{\Delta A}-{{e}^{i\alpha }}\frac{{\bar{B}}}{\Delta B})|\psi \rangle \\ & =2-2\frac{\left| \langle \psi \right|\bar{A}\bar{B}\left| \psi \rangle \right|}{\Delta A\Delta B}. \\ \end{align}$

(15)

$\Pi =\sum\limits_{n=1}^{d-1}{\left| \psi _{n}^{\bot } \right\rangle \left\langle \psi _{n}^{\bot } \right|}.$

(16)

Then Eq. (15) can be rewritten as

$\begin{align} & \sum\limits_{n=1}^{d-1}{\left\langle \psi \right|}\left( \frac{{\bar{A}}}{\Delta A}-{{e}^{i\alpha }}\frac{{\bar{B}}}{\Delta B} \right){{\left| \psi _{n}^{\bot } \right\rangle }^{2}} \\ & ={{\sum\limits_{n=1}^{d-1}{\left| \left\langle \psi \right|\frac{A}{\Delta A}-{{e}^{i\alpha }}\frac{B}{\Delta B}\left| \psi _{n}^{\bot } \right\rangle \right|}}^{2}} \\ & =2-2\frac{\left| \frac{1}{2}\left\langle \left[ A, B \right] \right\rangle +12\left\langle \left\{ A, B \right\} \right\rangle -\left\langle A \right\rangle \left\langle B \right\rangle \right|}{\Delta A\Delta B} \\ \end{align}$

(17)

which is equivalent to (13).

The two uncertainty equalities (8) and (13) hold for all pairs of incompatible observables. If we retain only the term associated with |ψ^⊥〉∈{|ψ_n^⊥〉^d-1_n=1} in the summation and discard the rest, the uncertainty equalities (8) and (13) reduce to the uncertainty relations (6) and (7), respectively.

2 Uncertaintyrelationinweak

measurement

First proposed by Aharonov et al. ^[12], weak values are complex numbers so that one can define the weak value of A using two states: an initial state |ψ〉 called the pre-selection and a final state |φ〉 called the post-selection. The weak value of A has the form

${{\left\langle A \right\rangle }_{w}}=\frac{\langle \varphi \left| A \right|\psi \rangle }{\langle \varphi |\psi \rangle }.$

(18)

For a given pre-selected and post-selected ensemble, we define the operator A_w as

${{A}_{w}}=\frac{{{\Pi }_{\varphi }}A}{p}, $

(19)

where Π_φ=|φ〉〈φ| and p=|〈φ|ψ〉|². The non-Hermitian operator has many properties ^[11] and is very useful in duality quantum computer ^[13-14].

Here, we construct an uncertainty relation in weak measurement for variances of two non-Hermitian operators A_w and B_w corresponding to two noncommuting observables A and B. The uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of pre- and post-selected (PPS) quantum states |ψ〉 and |φ〉 for the weak measurement of incompatible observables.

Uncertainty relation 3.

$\begin{align} & \Delta A_{w}^{2}+\Delta B_{w}^{2}\ge \left| \frac{1}{p}\langle \varphi \right|\left[ A, B \right]|\varphi \rangle + \\ & \frac{1}{p}\langle \varphi \left| \left\{ A, B \right\} \right|\varphi \rangle -2\langle {{A}_{w}}\rangle \langle {{B}_{w}}{{\rangle }^{*}}|+ \\ & \langle \psi |{{A}_{w}}-{{e}^{i\alpha }}{{B}_{w}}|{{\psi }^{\bot }}{{\rangle }^{2}}, \\ \end{align}$

(20)

which is valid for two non-Hermitian operators A_w and B_w, where p is equivalent to |〈φ|ψ〉|².

Proof To prove this relation we define the variance for any general (non-Hermitian) operator X in a state |ψ〉 which can be defined as in Refs.^[15-16]

$\Delta {{X}^{2}}=\langle \psi |\left( X-\langle X\rangle \right)({{X}^{\dagger }}-\langle {{X}^{\dagger }}\rangle )|\psi \rangle .$

(21)

The variance of the non-Hermitian operation A_w in the quantum state |ψ> can be defined as

$\Delta A_{w}^{2}=\langle \psi |({{A}_{w}}-\langle {{A}_{w}}\rangle )(A_{w}^{\dagger }-\langle A_{w}^{\dagger }\rangle )|\psi \rangle , $

(22)

where 〈A_w〉=〈ψ|A_w|ψ〉 and 〈A_w^†〉=〈ψ|A_w^†|ψ〉=〈A_w〉^*. ΔA_w² can also be expressed as

$\begin{align} & \Delta A_{w}^{2}=\langle \psi |{{A}_{w}}A_{w}^{\dagger }\left| \psi \rangle -\langle \psi \right|{{A}_{w}}|\psi \rangle \\ & \langle \psi |{{A}^{\dagger }}_{w}|\psi \rangle . \\ \end{align}$

(23)

Similarly, for Hermitian operator B, we can define the operator

${{B}_{w}}=\frac{{{\Pi }_{\varphi }}B}{p}.$

(24)

Then, the uncertainty for B_w can also be defined as

$\begin{align} & \Delta {{B}^{2}}_{w}=\langle \psi |{{B}_{w}}B_{w}^{\dagger }\left| \psi \rangle -\langle \psi \right|{{B}_{w}}|\psi \rangle \\ & \langle \psi |B_{w}^{\dagger }|\psi \rangle . \\ \end{align}$

(25)

To prove our uncertainty relation in weak measurement, we introduce a general inequality

$\|{{C}^{\dagger }}|\psi \rangle -{{e}^{i\tau }}{{D}^{\dagger }}\left| \psi \rangle +k( \right|\psi \rangle -|\bar{\psi }\rangle ){{\|}^{2}}\ge 0, $

(26)

where C^†≡A_w^†-〈A_w^†〉 and D^†≡B_w^†-〈B_w^†〉. By expanding the square modulus, we have

$\Delta A_{w}^{2}+\Delta B_{w}^{2}\ge -\lambda {{k}^{2}}-\beta k+\pi , $

(27)

where λ≡2(1-Re[〈ψ|ψ〉]), π≡2Re[e^iτ〈ψ|CD^†|ψ〉], and β≡2Re[〈ψ|(-C+e^-iτD)|ψ〉]. By choosing the value of k that maximizes the right-hand-side of (27), namely k=-β/2λ, we get

$\Delta A_{w}^{2}+\Delta B_{w}^{2}\ge \frac{{{\beta }^{2}}}{4\lambda }+\pi $

(28)

The above inequality can be rewritten as

$\begin{align} & \Delta A_{w}^{2}+\Delta B_{w}^{2}\ge \frac{Re{{[\langle \psi |(-C+{{e}^{-i\tau }}D)|\bar{\psi }\rangle ]}^{2}}}{2(1-Re\left[ \langle \psi |\bar{\psi }\rangle \right])}+ \\ & 2Re[{{e}^{i\tau }}\langle \psi |C{{D}^{\dagger }}|\psi \rangle ]. \\ \end{align}$

(29)

Suppose |ψ〉=cosθ|ψ〉+e^iφsinθ|ψ^⊥〉, where |ψ^⊥〉 is orthogonal to |ψ〉. By taking the limit θ→0, the state |ψ〉 reduces to |ψ〉 and then the above inequality can be reexpressed as ΔA_w²+ΔB_w²≥

$\begin{align} & \Delta A_{w}^{2}+\Delta B_{w}^{2}\ge \\ & Re{{[{{e}^{i\varphi }}\langle \psi |(-{{A}_{w}}+{{e}^{-i\tau }}{{B}_{w}})|{{\psi }^{\bot }}\rangle ]}^{2}}+ \\ & 2Re[{{e}^{i\tau }}\langle \psi |C{{D}^{\dagger }}|\psi \rangle ]. \\ \end{align}$

(30)

There exists τ=-α so that e^iτ〈ψ|CD^†|ψ〉 is real, and it can be written as |〈ψ|CD^†|ψ〉| . Then the second term becomes {Re[e^iφ〈ψ|-A_w+e^iαB_w|ψ^⊥〉]}². We can choose φ so that the term in square brackets is real and this term can be expressed as |〈ψ|A_w-e^iαB_w|ψ^⊥〉|². Whence, inequality (30) becomes

$\begin{align} & \Delta A_{w}^{2}+\Delta B_{w}^{2}\ge \left| \langle \psi \right|{{A}_{w}}-{{e}^{i\alpha }}{{B}_{w}}|{{\psi }^{\bot }}\rangle {{|}^{2}}+ \\ & 2\left| \langle \psi \right|C{{D}^{\dagger }}\left| \psi \rangle \right|. \\ \end{align}$

(31)

The last term can be rewritten as

$2|\langle C{{D}^{\dagger }}\rangle \left| = \right|\langle C{{D}^{\dagger }}+D{{C}^{\dagger }}+C{{D}^{\dagger }}-D{{C}^{\dagger }}\rangle |, $

(32)

where

$\begin{align} & \langle C{{D}^{\dagger }}+D{{C}^{\dagger }}\rangle = \\ & 1p\langle \varphi \left| \left\{ A, B \right\} \right|\varphi \rangle -\langle {{A}_{w}}\rangle \langle {{B}_{w}}{{\rangle }^{*}}-\langle {{A}_{w}}{{\rangle }^{*}}\langle {{B}_{w}}\rangle , \\ \end{align}$

(33)

and

$\begin{align} & \langle C{{D}^{\dagger }}-D{{C}^{\dagger }}\rangle = \\ & \frac{1}{p}\langle \varphi \left| \left[ A, B \right] \right|\varphi \rangle -\langle {{A}_{w}}\rangle \langle {{B}_{w}}{{\rangle }^{*}}+\langle {{A}_{w}}{{\rangle }^{*}}\langle {{B}_{w}}\rangle . \\ \end{align}$

(34)

We combine Eqs. (33) and (34), Eq. (32) becomes

$\begin{align} & 2|\langle C{{D}^{\dagger }}\rangle |= \\ & \left| \frac{1}{p} \right.\langle \varphi \left| \left[ A, B \right] \right|\varphi \rangle + \\ & \frac{1}{p}\langle \varphi \left| \left\{ A, B \right\} \right|\varphi \rangle -2\langle {{A}_{w}}\rangle \left. \langle {{B}_{w}}{{\rangle }^{*}} \right|. \\ \end{align}$

(35)

Combining Eqs. (32) and (35), we obtain the uncertainty relation (20).

3 Conclusions

In this work, we derived two new uncertainty equalities for the sum and product of variances of a pair of incompatible observables, which hold for all pairs of incompatible observables A and B. In fact, one can obtain a series of inequalities by retaining 1 to (d-2) terms within the set {|ψ^⊥_n〉^d-1_n=1}. We also derived an uncertainty relation in weak measurement for two non-Hermitian operators A_w and B_w corresponding to two non-commuting observables A and B. The uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of PPS quantum states |ψ〉 and |φ〉 for measuring incompatible observables in weak measurement.