Improving the Cauchy-Schwarz inequality

We highlight overlap as one of the simplest inequalities in linear space that yields a number of useful results. One obtains the Cauchy-Schwarz inequality as a special case. More importantly, a variant of it is seen to work desirably in certain singular situations where the celebrated inequality appears to be useless. The basic tenet generates a few other interesting relations, including the improvements over certain common uncertainty bounds. Role of projection operators in modifying the Cauchy-Schwarz relation is noted. Selected applications reveal the efficacy.


Introduction
Inequalities are encountered in many areas of theoretical sciences [1,2]. There exist quite a few standard routes [1] to arrive at specific types of inequalities. Our modest aim here is to explore how far one can extract useful results starting from a remarkably simple idea, viz., the overlap. To pursue, we start from the intuitively obvious inequality for two normalized states |ψ N1 and |ψ N2 in a linear space as where S is the overlap. It simply means that the overlap of any two unit-norm states is never greater than unity. Inequality (1) is worst if ψ N1 |ψ N2 = 0, and it is saturated (equality) when |ψ N2 = |ψ N1 . Thus, S is a direct measure of nonorthogonality of the two chosen states. Sometimes, S is also interpreted as distance between two pure states, while its square is termed as fidelity. One may wonder that the message of (1) can be fruitfully employed to obtain the Cauchy-Schwarz inequality (CSI) and related ones, an improved CSI (ICSI), and sometimes a tighter CSI involving projection operators.

A few known results
Appropriate choices of the states in (1) may now be seen to yield certain known results, as outlined below: (i) Choose two arbitrary states |ψ 1 and |ψ 2 such that Inequality (1) then quickly takes the familiar form of the Schwarz inequality [2], viz., If the states |ψ 1 and |ψ 2 are expanded in terms of an orthonormal set of states {|φ k } as then (3) leads to Cauchy's inequality [2] |a k | 2 |b k | 2 ≥ (a * k b k ) .
For real {a k } and {b k }, it further simplifies in terms of average (over N) values as that possesses some relevance to statistics. Indeed, (3) and (5) are equivalent, and hence (3) is often also called the CSI. Needless to mention, the CSI (3) is usually derived from the relation ψ|ψ ≥ 0 where |ψ is an arbitrary normalizable state. Also, while (1) is straightforwardly found from (3), the extension (see (14) below) of (3) rests solely on the spirit of (1) again.
(ii) Another choice in (1), viz., where |φ is any normalized state andÂ is Hermitian, leads to This result primarily connects the various moments of a spatial distribution forÂ =x. However, it may also be useful elsewhere.
(iii) The CSI is usually employed to obtain the uncertainty product inequality (UPI) [3]. A direct application of (1), however, acts with equal facility. Define |ψ 1 and |ψ 2 in terms of two non-commuting Hermitian operatorŝ A andB acting on some arbitrary normalized state |φ as Then, states |ψ N1 and |ψ N2 may be taken in the forms and the inequality (1) shows immediately

Some additional relations
It is now imperative to search for some more relations from (1) to justify its strength and worth further. To achieve this end, we again proceed point-wise:

An improved Cauchy-Schwarz inequality
An important special case in the context of CSI (3) arises when ψ 1 |ψ 2 = 0 (12) so that the right side becomes zero, rendering the celebrated inequality almost useless. Improvements of the CSI along various routes are available (see, e.g., references [4][5][6][7] and those quoted therein). Still, the specific problem with (12) does not seem to have attracted sufficient attention. Nonetheless, we have found it expedient [8] to tackle this problem by rewriting (1) as Essentially, in inequality (13), we employ two normalized given states |ψ N j and two similar auxiliary ones |θ N j , j ∈ {1, 2}. Auxiliary states are otherwise arbitrary, only the integrals at the right of (13) should exist. Now, following (13), we find the desired ICSI that reads as in place of (3). Note that condition (12) cannot cause any harm now, because that vulnerable inner product is avoided in (14). Further, (14) can reduce to (3) for the specific choice However, other possibilities do exist, and they can really bypass (12) to yield a non-zero right side in (14). The ICSI thus justifies its name and generality. A point of secondary interest lies in strengthening the CSI when the overlap S in (1) is known to be much less than unity. Then, (3) will certainly turn out to be a poor inequality. Our relation (14) in such a case possesses the potential to provide better bounds.
Thirdly, while the ICSI (14) is more general than (3), it requires import of two auxiliary states. Elsewhere [8], we have found that this prescription too may be somewhat relaxed in case (12) is exactly valid. This new form reads as In (16), unlike (14), just one auxiliary state is involved.
Finally, let us now have a look at the UPI (11). It may be beset with similar trouble as outlined above under the condition (cf. (9)) A common finite-dimensional example of (17) involves the angular momentum operatorsÂ =Ĵ x ,B =Ĵ y in (9), the special state being |φ = |φ( j, µ = 0) . An infinitedimensional case [8] might refer toÂ =x 2 ,B =p acting on the ground state of the one-dimensional harmonic oscillator. Our bypass route in tackling such cases will be similar. For example, one obtains by using ICSI (14) the following result in place of (11). The disaster (cf. condition (17)) is thus avoided. This is indeed a modified UPI. Moreover, in view of (16), one may add an extra step to arrive at a simpler version of (18), viz., if (17) is obeyed. The embedded auxiliary states in (14) or (18), or the single auxiliary state in (16) or (19), may be chosen at will, so much so that saturation can occur [8] in either case.

A refined uncertainty sum inequality
Kinship of the CSI with another inequality, viz., is well-known [9]. We employ (20) with the choice (cf. definition (9)) to obtain, for example, the result On the other hand, by replacing −|ψ 2 for |ψ 2 in (20), the same definition (21) yields The left side of (22) or (23) involves an uncertainty sum and hence we call such a relation an uncertainty sum inequality (USI). Indeed, a direct route to (23) with potential usefulness is available in the literature [10]. Anyway, in the present context, we combine (22) and (23) to get another powerful relation, viz., While inequalities like (22), (23) and (24) may be otherwise useful, they all follow from (20). Therefore, three weaknesses of (20) should be pointed out here: (i) It becomes trivial when |ψ 2 = |ψ 1 .
(iii) The choice (21), coupled with condition (17), leads one to an obvious result like Notably, under this situation (e.g., (17) and (21)), the right side of (25) also replaces the same of either of inequalities (22) to (24). Thus, (25) essentially relates two kinds of USI, one based on standard deviation and the other on variance. We next notice how (1) (or (13)) can save us from such inconsequentialities. Let us recall the way of construction of the ICSI (14). In the present context, we put it as This relation improves (20). None of the aforesaid weaknesses of (20) (see, e.g., the discussion below (24)) now prevail. Further, using (21), an improved USI (IUSI) is found from (26), viz., that avoids again any direct inner product between states |φ A and |φ B . It is thus stronger than (24) and can even saturate, in contrast to (25). Only under a specific situation, when we choose |θ N1 = |θ N2 = |θ N in (27) and express it (remembering (17)) as a linear combination of |φ A and |φ B , the best choice leads one to (25). However, more general choices exist and they do attest the generality of IUSI (27).

Role of projection operators
Use of a suitable projection operator may sometime increase the tightness of the CSI by appropriately redressing the overlap. For simplicity, we rearrange inequality (3) in the form To tighten this inequality, let us keep the |ψ 2 part as such, but incorporate an auxiliary state |ψ 3 in the |ψ 1 part of (28) that is known to naturally satisfy ψ 2 |ψ 3 = 0. Such a choice renders the left side unaltered, though form (28) changes to Thus, value of the right side changes. A little algebra shows that the tightest situation, i.e., minimum value for the right side in (29), is attained at an optimum α to yield from (29) It implies, we can reorganize (29) in the tightest situation as In (31),P 3 refers to the projector for |ψ 3 and is defined byP Thus, (28) admits modification to (31) when a state |ψ 3 is known a priori to be manifestly orthogonal to |ψ 2 . In our form (14) or (16), on the contrary, the only restriction would be |θ N2 , |θ N |ψ 3 . Of course, when the whole of |ψ 1 is orthogonal to |ψ 2 , the left side in (31) needs to be rectified, and there appears our prescription, the ICSI (14) or (16), as a remedy.
In the context of UPI (recall (9)), this projector issue is nicely met in the standard route. By construction, we have φ B |φ = 0, and so we choose |φ A such that φ A |φ = 0. Indeed, this is in-built in the definition. Had we chosen instead, e.g., |φ A =Â|φ , we would have been led to a weaker inequality (as in (28)), keeping aside the fact that this option does not involve the standard deviation of operatorÂ. However, by following (31), one regains the usual form, as found in (9). Therefore, while applying (1), we shall continue with the above wisdom in studies on the complementary Eckart bound [11,12] and survival probability [13][14][15][16][17] to follow.

Results and applications
Let us quickly turn to certain results that will demonstrate the advantage of the present endeavor.

CSI vs. ICSI
We take the following states for a first-hand experience: One obtains the results shown in Figure 1 for varying x-values. Here, the left side is fixed at unity. The right sides stand for lower bounds to the same. In Figure 1,  Figure 1: Plots of the right sides of inequalities (3), (14) and (16) (14) with both auxiliary states taken as |θ N2 [dark cyan : 5]. Note that the CSI performs nicely for large |x|. But, overlap is small in regions around x = 0 (S = 0 at x = 0), and that is the primary focus of the present study. We witness here varying performances of the choices (cf. (33)) based on (14). Particularly interesting ones are curves 2 and 4. Both fare well around the x = 0 region. Curves 3 and 5 also perform better than curve 1 within |x| = 1. The advantage of using auxiliary states in (14) should now be clear. While none of the curves reach the exact value, we happily note that (16) applies at x = 0, and curve 4 shows its ability to saturate at this point. Our next example concerns two lowest normalized energy eigenstates of the particle-in-a-box problem in (0, π) for which the overlap is zero, and hence the CSI (3) [1 ≥ 0] is of no use. In this situation, we see how the ICSI (16) performs, aided by just one auxiliary state |θ N taken in functional form as N x 2 (π − x), and zero otherwise, where N stands for the normalization constant. Our recipe betters the bound from 1 ≥ 0 to 1 ≥ 0.6553. The tightness achieved is again noteworthy.
In both the above cases, however, auxiliary states of our choice are employed. One may wonder whether betterment can be accomplished at all by using only the parent states (e.g., |ψ N1 , |ψ N2 ). To explore, we proceed by choosing different linear combinations of these parent states, viz., Let us also take all the states as real, with real positive combining coefficients and overlap. The prefactors N 1 and N 2 in (34) represent the normalization constants. Figure 2 attests that the whole accessible region at S = 0 is better in ICSI (14). Even, saturation is possible here. Generality of the ICSI may now be appreciated.

A complementary Eckart bound
The Eckart inequality [11,12] stands as one of the earliest measures for the goodness of an approximate normalized eigenstate of energy |φ 1 by providing a lower bound to its overlap with the unknown exact ground stationary state |φ 1 . Writing the energy eigenvalue equation aŝ H|φ j = ε j |φ j , φ j |φ j = 1, and defining the average ground-state energy asε 1 , one finds [11,12] Coupled with the upper bound S 1 ≤ 1, the above result actually reflects the closeness of |φ 1 with the unknown |φ 1 . However, an improved upper bound may be obtained by defining and subsequently using (1). The outcome is This is complementary to the standard Eckart bound. Moreover, unlike (35), (37) applies to any n-th state, not just the ground state, and it does not require any information about ε 2 . Our preliminary checks reveal that (37) furnishes far better bound than the primitive one, viz., S n ≤ 1, primarily due to the denominator at the left. For any finite right side in (37), S n has to be less than unity. Had we chosen |ψ 1 =Â|φ n in (36) instead, we would not reach this stronger form.

Decay probability
Turning to quantum dynamics, we now consider the problem of survival probability P(t) [13][14][15][16][17] or, more specifically, the decay probability Q(t). To proceed, we first define a state |ψ(t) whose evolution is governed by a conservative HamiltonianĤ as Next, we identify below a specific projection operatorÂ whose average in state |ψ(t) yields P(t).
The states in (1) are now chosen in the forms (cf. (9) and (10)) Putting these in (1), we find after a little algebra the inequality But, a direct application of the CSI (3) for the right side of (41) leads us to a weaker bound, viz., Notice, the multiplying factor Q(t) [Q(t) ≤ 1] at the left of (41) does not appear at the right side of (42), and so the latter loses the time dependence. Indeed, this becomes decisive in tightening the bound (41) compared to (42), chiefly at short times when the starting state changes little, and hence Q(t) 1 follows. At t = 0, however, both the left and right sides of (41) are zero; hence, there is no paradox. Thus, we again witness the worth of (1). Role of the projector is also evident in the choice (40). The next task would naturally be to put (41) to test. In the short-time regime, one obtains from (41) a general result of the form However, depending on the eigenvalue spectrum ofĤ in (38), two situations should now be distinguished. If the spectrum is continuous, the state gradually decays. In contrast, one observes decay and revival in succession whenĤ has a purely discrete spectrum. This quantum recurrence [18][19][20] is important in various areas [21]. We consider the efficacy of (41) in both these cases.
Concentrating first on pure decay [15], an upper bound to the decay probability Q(t) (see Eq. (10) in Ref. [15]) may be found as In (41), however, we have arrived at a complementary bound. What is more, while (44) is valid only over short times, our present lower bound (41) does not impose any such restriction on time. As an example, let us pay attention to a solvable problem, viz., the decay of a Gaussian packet [9,15] in field-free space. Implementing (41), we observe after some algebra that the standard energy form factor [15] finally yields We have checked that this relation is valid over the entire region of time. But, as already stated, inequality (41) works better at smaller t. Thus, at a time when ∆Et = 1 4 , one finds from (45) √ Q(t) ≥ 0.23, whereas (44) gives √ Q(t) ≤ 0.25. The exact result [15] yields √ Q(t) ≈ 0.24, justifying the tightness of either bound.
(48) A few remarks are now in order. First, recurrences begin with decay, and we shall consider this primary decay part below. Secondly, (46) or (48) shows a certain symmetry with respect to exchange of r 1 and r 2 for both Q(t) and ∆E, something that is lacking in the average energy Ĥ . It reveals, Ĥ is presumably not a very significant observable [26] in the current context. Thirdly, from (43) and (44), we see that a t − √ Q(t) plot is initially linear. Indeed, one obtains Here, t is the dynamical time. Relation (49) ties the energy uncertainty with the decay probability and time, and applies to both decay and recurrence problems. Fourthly, (48) shows that the ϕ = π 4 case (equiprobable) decays most rapidly to Q(t) = 1 (the orthogonal state) at t = π ω 21 . Calling this time as τ, we obtain τ = h 4∆E using (48). This is the Fleming bound [14,15] for quantum speed [22][23][24][25].
If ω 1 = 0, one also finds the Margolus-Levitin bound [26], viz., τ = h 4 Ĥ from (46) and (48). Fifthly, the maximum decay for any general, non-equiprobable situation is given by √ Q(t) = sin (2ϕ) , ϕ π 4 . This is also reached at a minimum time t = π ω 21 = τ min g (say). On the other hand, along the fastest decay route (equiprobable), the result √ Q(t) = sin (2ϕ) is achieved at a time τ min e = 4ϕ ω 21 . These results along with their corresponding ∆E-values are summarized below: One observes now that the relative time and the relative energy spread obey For small enough ϕ, (51) shows that τ rel becomes exceedingly large. The dependence of the relative decay time on the initial state of a qubit should now be transparent. However, there exists a nice relation in this regime connecting τ rel and ∆E rel , viz., In (52), h does not appear at the right just because each term at the left is dimensionless. Figure 3 displays how the maximum of Q(t) reduces with decreasing contribution of one of the two states, denoted here by ϕ. A concomitant increase in the minimum time to attain some pre-assigned value of Q(t) is also clear. In these situations, the minimum-time bounds set by the fastest decay route could be quite useless. One may instead concentrate on (51) or (52). The figure clarifies the critical role of the initial state, as emphasized elsewhere [27].
In fine, we also note that the Fleming bound applies to both pure decay and recurrence problems, but the Margolus-Levitin bound concerns the latter situation only.

Conclusions
To summarize, we sketched here how the overlap inequality (1) can be exploited to obtain the CSI (3). More importantly, in case the CSI fails to work (e.g., under condition (12)), a form of the overlap inequality (13) leads to an ICSI (14) that applies to any arbitrary situation. A further simplification to ICSI (16) is a positive addition. The UPI (18) or (19) reveals the gains in suitable contexts. For the USI case, a similar extension of (14) to (26) yields an IUSI (27) that is more general than the prevalent form.
The ICSI (14) and (16) are applied to a few pathological situations in Figures 1 and 2. These are general linear-space problems. Auxiliary states are imported in Figure 1 to specifically study the x → 0 limit. However, in Figure 2, such states are avoided. The benefit of a projector is highlighted in certain areas of quantum mechanics. These studies include a complementary Eckart bound (relation (37)), a bound to the decay probability Q(t) (cf. (41)) and a few other characteristics (cf. (43), (45), (49) and (52)) of it in decay and recurrence problems, particularly for qubits.