The uncertainty relations have played a fundamental role in the development of quantum theory not only in the foundation but also in recent investigations of quantum information and quantum communication, in particular in the areas such as entanglement detection,^[1,2] security analysis of quantum key distribution in quantum cryptography,^[3] quantum metrology, and quantum speed limit.^[4–6] Usually the uncertainty relations are expressed in terms of the product of variances of the measurement results of two incompatible observables. Other forms of the uncertainty relations include entropic uncertainty principle,^[7–9] variance-based sum uncertainty relation,^[10–12] the skew information,^[13,14] the majorization technique,^[15–17] and the recent weighted uncertainty relation.^[18] Recently, several experimental investigations were also performed to check the corresponding relations.^[19,20]

In 1927, Heisenberg^[21] analyzed the observation of an individual electron with photons and obtained the famous uncertainty principle

The inequality (1) shows that the uncertainties in the position and momentum of a quantum particle are inversely proportional to each other: a particle’s position and momentum cannot be known simultaneously. Thus the accuracy of quantum measurement is limited by the uncertainty principle. This principle uncovers a fundamental and peculiar feature in the atomic world, and is considered as one of the cornerstones of quantum mechanics.

Robertson^[22] formulated the uncertainty relation for arbitrary pair of non-commuting observables A and B (with bounded spectra) as follows:

Robertson’s uncertainty relation was further generalized by Schrödinger to be^[3]

The goal of this note is to give a family of generalized Schrödinger uncertainty relations using a stronger Cauchy–Schwarz inequality.

We start with a quantum system in the quantum state ρ = |Ψ⟩⟨Ψ| on the Hilbert space with the inner product ⟨|⟩, where ⟨Ψ|Ψ⟩ = 1, and consider observables A and B. Define the operator associated with a given operator A.

Let {|φ_i⟩} be an orthonormal basis of the Hilbert space and write . , so that , where ⟨α|β⟩ is the usual inner product (linear in the second argument) for the vectors α = (α₁, α₂,...,α_n) and β = (β₁, β₂,...,β_n). We will not distinguish the two inner products as long as it is clear from the context.

The variance of observable A can be expressed as , thus

Until now we have considered only a pure state |Ψ⟩ of the system, this relation can be extended to the case of mixed states ρ = ∑p_j|Ψ_j⟩⟨Ψ_j|, p_j ≥ 0 and ∑_j p_j = 1. Since , , and we just need to consider the terms in the sum for both |α_i|≠ 0 and |β_i| ≠ 0, so we can rewrite inequality (4) as

Since the mean value of observable A can be expressed in the form of trace, ⟨Ψ|A|Ψ⟩ = TrA|Ψ⟩⟨Ψ|, we rewrite the inequality above as

For each state |Ψ_j⟩, we have

We remark that our generalized uncertainty relation is stronger than Schrödinger’s uncertainty relation, as the generalized Cauchy inequality shows that |⟨α|β⟩| is smaller than the right-hand side of our uncertainty relation.

The uncertainty relation (4) can be tightened by optimizing over the sets of complete orthonormal bases. Then we can improve the uncertainty relation by the Callebaut inequality as in the proof of Theorem 1.

Both the Callebaut and Milne inequalities (5) and (12) are stronger than the usual Cauchy–Schwarz inequality, thus our uncertainty relations are tighter than those derived from the Cauchy–Schwaz inequality, for example, the Mondal–Bagchi–Pati’s uncertainty inequality in Ref. [26] about observables

Fig. 1.

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Fig. 1. (color online) The uncertainty relation for observables Lx and Ly at state |Ψ⟩: the blue curve is the lower bound in inequality (4) with the weight λ = 1/10, the red one is with the weight λ = 9/10, the black one stands for Mondal–Bagchi–Patis’ lower bound in inequality (14), and the green line denotes Schrödinger’ lower bound in inequality (3). As observed from the plot, the bound given by inequality (4) is one of the tightest bounds in the literature.

Fig. 2.

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	Fig. 2. (color online) The lower bound of the uncertainty relation (9) for observables Ax, By at mixed state ρiso, it is clear that the lower bound is bigger as the weight λ and parameter p increase.

The uncertainty relations play a central role in the current research in quantum theory and quantum information.^[28–30] We have derived a family of new product forms of variance-based uncertainty relations, which are expected to help further investigate the uncertainty relation.