Theory of Compression Channels for Postselected Quantum Metrology

Postselected quantum metrological scheme is especially advantageous when the final measurements are either very noisy or expensive in practical experiments. In this work, we put forward a general theory on the compression channels in postselected quantum metrology. We define the basic notions characterizing the compression quality and illuminate the underlying structure of lossless compression channels. Previous experiments on Postselected optical phase estimation and weak-value amplification are shown to be particular cases of this general theory. Furthermore, for two categories of bipartite systems, we show that the compression loss can be made arbitrarily small even when the compression channel acts only on one subsystem. These findings can be employed to distribute quantum measurements so that the measurement noise and cost are dramatically reduced.

Quantum metrology utilizes quantum coherence and entanglement to boost the measurement precision quantified by the quantum Fisher information (QFI) [1][2][3][4].In standard quantum metrology, given an ensemble of metrological samples, quantum mechanics allows one to optimize the quantum measurements so that the information about the signal is maximally extracted.Yet, another metrological scheme, called Postselected quantum metrology arises in the context of weak-value amplification (WVA) [5][6][7][8][9][10][11], where a post-selection measurement is performed to select a sub-ensemble of the samples before the information-extracting measurement.Comparing to standard metrology, though the QFI encoded in the subensemble averaged over its post-selection probability cannot be larger than the QFI in standard metrology [12][13][14][15][16][17], there are several advantages due to post-selection when the cost of the post-selection measurement becomes cheap: (i) WVA outperforms the standard metrology in the presence of certain types of technical noise [18].(ii) WVA can be viewed as a filter to reduce the number of detected samples in standard metrology without losing the precision significantly.As such, in Hamiltonian learning [19,20], post-selection can be employed to reduce the sample complexity [21], i.e., the number of samples to achieve a given precision.Practically speaking, when the final information-extracting measurements are subjected to various kinds of imperfections, such as detector saturation, limited memory and computational power etc, Postselected quantum metrology is provably outperforms the standard one [22][23][24][25][26].
Recently post-selection has been applied in a broad context in quantum metrology beyond the setup of WVA [27].The advantage of post-selection as a filter or compression channel persists in this broad context as demonstrated in the experiment of optical phase estimation [28].While postselection can be also applied to classical metrology, previous works [27,29] show that the non-classicality can further boost the precision.However, despite these advances, a comprehensive theory for designing the lossless post-selection measurement channels in the most general setups beyond WVA remains uncharted.In standard quantum metrology, for arbitrary parameter-dependent quantum states, the optimal measurements saturating the quantum Cramér-Rao bound were studied by Braunstein and Caves [30] and recently applied Figure 1.The protocol of Postselected quantum metrology."A" denotes the ancilla.The unitary operation combined with the following projective measurement on the ancilla implement the post-selection channel K ω .After post-selection, measurements can be further performed (not shown), as in standard metrology, on these Postselected states to extract information about the estimation parameter.
to the case of noisy detection [31].Analogously, in Postselected metrology similar fundamental question has not been addressed.
In this work, we answer this question by proposing a theory that unifies weak-value metrology, Postselected metrology, as well as standard metrology.The crucial observation is that standard metrology only makes use of the measurement statistics and discards the post-measurement states completely while Postselected metrology utilizes a specific set of post-measurement states and discards the rest.As such, we generalize the optimal measurement condition from standard metrology to Postselected metrology.By keeping track of these conditions, we identify the generic structure of the lossless post-selection channel for pure states.Previous setup on Postselected metrology [21,27,28,32] and WVA [17] are special cases of this general theory.Finally, for bipartite entangled states, when the compression channel is restricted to one subsystem, we construct two categories of examples, which can be compressed substantially with only a negligible amount of loss.
Generalized optimal measurement condition.-We consider a pure quantum state of a quantum sensor described by ρ x = |ψ x ⟩ ⟨ψ x |, where x is the estimation parameter.The QFI associated with this state is , and E ω is a positiveoperator-valued measure (POVM) operator, satisfying E ω ≥ 0 and ω E ω = I [35].In Postselected quantum metrology, a post-selection measurement channel denoted as {K ω } is per-formed on the system, where ω ∈ Ω and Ω is the set of all measurement outcomes.As shown in Fig. 1, such a generalized measurement can be implemented by a unitary operation entangling the system and ancilla, followed by a projective measurement on the ancilla.After performing the post-selection channel, but before post-selection is made, the joint state of the system and the ancilla becomes σ [36].Throughout this work, states and operators that do not act on the system only will be specified through the superscript, "A", "SA" etc.The QFI corresponding to σ SA x is [13,37], , where and The physical meaning of Eq. ( 1) is clear: The QFI for each measurement outcome ω consists of two parts, the classical QFI and the average QFI for the post-measurement states.Clearly, the post-selection channel cannot increase the QFI, i.e., I(σ SA x ) ≤ I(ρ x ).A more refined statement is the following [37]: If Eq. ( 2) is saturated for all measurement outcomes, then To compress the number of samples without sacrificing the precision, we demand the discarded set contains no information.Therefore, a precondition to reaching this goal is that Eq. ( 2) must be saturated even before the selection process is made.For a regular POVM measurement, where ⟨ψ x E ω ψ x ⟩ 0 (see [37] and Ref. [33] for an elaborated definition), the necessary and sufficient condition to saturate the inequality (2) is given by Eq. (T1) in Table I.For a null POVM measurement where ⟨ψ x E ω ψ x ⟩ = 0, I cl ω p(ω x) = I ω (ρ x ) and I(σ x|ω ) = 0, see Ref. [33] for details.Thus no information is left in the post-measurement state σ x|ω .As a consequence, if σ x|ω is the state one would like to retain, one should avoid designing E ω as a null POVM measurement operator.
In standard quantum metrology, the post-measurement states are all discarded and only the measurement statistics is retained.In this case, one would like to saturate the inequality, I cl ω p(ω x) ≤ I ω (ρ x ), which was studied by the classic work of Braunstein and Caves [30] and the recent work Ref. [31], see also Eq. (T5).In Postselected metrology, we require the average QFI of the retained post-measurement state saturates the quantum limit, i.e., p(ω x)I(σ x|ω ) ≤ I ω (ρ x ).Finally, for the measurement outcome or the corresponding post-measurement state to be discarded, we would like them to carry no information, i.e., I cl ω p(ω x) = 0 or I(σ x|ω ) = 0.The saturation conditions of these bounds are given in Table I.We shall call Eqs.(T1-T5) as the generalized optimal measurement conditions, as they generalize the results by Braunstein and Caves [30] to account for the case where the information about the parameter is losslessly encoded either in the measurement statistics or the post-measurement states.As we will see, they play a fundamental role in the theory of compression channels.
Lossless Compression Channel.-In the standard metrology, the discarded set, as indicated by the red trash bin in Fig. 1, is = {σ x|ω } ω∈Ω and no QFI is left in the postmeasurement states.In Postselected metrology, we require all the QFI is transferred to the desired post-measurement states and the discarded set contains no information.We use ω ∈ ✓ and ω ∈ × to indicate the desired and undesired outcomes, respectively.Throughout this work, we consider a minimum retained set {σ x|ω } ω∈✓ , corresponding to = Ω ∪ {σ x|ω } ω∈× [38].When ω∈✓ p(ω|x) < 1, we can view the post-selection as a compression channel.It is worth noting that even if ω∈✓ p(ω|x) is small, resulting in a small number of metrological samples in the Postselected ensemble in each round, the experiment is assumed to be repeated for a sufficiently large number of rounds so that the classical Cramér-Rao bound is saturated.
Let us first introduce the essential notions for the theory of compression channels.The loss of the QFI per input sample can be expressed as γ ≡ 1 − ω∈✓ p(ω|x)I ω (σ x|ω )/I(ρ x ).We define c ≡ 1/ ω∈✓ p(ω|x) as the compression capacity for a post-selection channel, characterizing the ability of a designed post-selection measurement to reduce the number of samples.If γ = 0 and c > 1 then a post-selection measurement is called an lossless compression channel (LCC).We shall restrict our attention to efficient post-selection channels where c ∈ (1, ∞) [39].We further define the compression gain, η ≡ ω∈✓ I ω (σ x|ω )/I(ρ x ), as the ratio between the Postselected QFI and the one standard metrology, characterizing information gain per detected sample.This characterizes the advantage of the former over the latter when the cost of final detection dominates over the cost of post-selection [27].
For generic quantum systems, we find LCC must satisfy the following theorem: Theorem 1.For a pure state |ψ x ⟩, the POVM operators in an efficient LCC must satisfy ⟨ψ ⊥ x ω∈✓ with p(ω|x) > 0 for ω ∈ ✓ and ω∈✓ p(ω|x) < 1, where The proof is straightforward with the following intuition: Eq. ( 3) guarantees that in the undesired outcome, measurements statistics and post-measurement states contain no QFI, i.e., I ω (ρ x ) = 0 for ω ∈ ×; Eq. ( 4) ensures that for the desired outcome where ω ∈ ✓, the measurements statistics again contains no QFI and retained states reach the quantum limit given by Eq. (T4).As such, the QFI is fully preserved after the postselection.Alternatively, the following theorem illuminates the underlying structure of an LCC: Theorem 2. For a pure state |ψ x ⟩, the POVM operators in the retained set of an LCC can be expressed as follows: where x | and Λ ω is the gauge operator, which does not contributes to the QFI and can be chosen in many ways as long as it satisfies and ⟨ψ x Λ ω ψ x ⟩ = λ ω ∈ (0, 1).The compression capacity and gain are c = 1/ ω∈✓ λ ω and η = ω∈✓ q ω /λ ω , respectively.
Theorem 1 and 2 are our second main results.Practically, the LCC (5) depends on the true value of x, so in general adaptive estimation is required [40][41][42][43][44][45][46].If we assume full accessibility of the post-selection measurements on the whole Hilbert space, i.e., Eq. ( 5) is always implementable regardless of the choice of Λ ω , then λ ω can be tuned arbitrarily small, say λ ω = ε.Then we find η = Lc = 1/ε, where L is the number of desired outcomes.
When η > 1, we know the QFI per detected sample in an LCC is amplified, though it will be counter-balanced when the post-selection probability is accounted for [12,13,27].While Ref. [27] relates such an amplification to the noncommutativity of observables, thanks to Theorem 1 and 2 , we can directly compute the enhancement of the parametric sensitivity of the Postselected state to the estimation parameter.For example, apart from an irrelevant unitary rotation, one can take While the post-measurement state is still |ψ x ⟩, the parameter derivative of the Postselected state becomes [37]: ) Finally, it is worth to note that if λ ω = 0, the LCC (5) degenerates into a null measurement operator q ω ρ ⊥ x .Using our prior knowledge about the estimation parameter denoted as x * , the projector ρ ⊥ x * can approach to quantum limit asymptotically as x * → x [33].
In two-level systems, the gauge operator Λ ✓ is forced to take the form , where ∆ is a known constant and {|0⟩, |1⟩} is a parameter-independent basis.This is the example investigated in the experiment in Ref. [28].Our theory predicts that the Postselected POVM measurement operator in the basis of {|0⟩ , |1⟩} is It should be noted that unlike Ref. [28] which assumes small values of x∆, Eq. ( 8) is the exact LCC for any values of x.Of course, when x∆ << 1, to the zeroth order of x∆, it recovers the LCC in Ref. [28], i.e., E ✓ = λ ✓ |0⟩ ⟨0| + |1⟩ ⟨1|.
Restricted post-selections.-When|ψ x ⟩ is a bipartite entangled state between two subsystems A and B, the post-selection measurement on only A generically leads to loss i.e. γ > 0. This is because the LCC, according to Theorem 2, in general acts globally on both the system and the environment.Nevertheless, we demonstrate the existence of approximate LCC for two classes of examples, where the loss is very tiny.To proceed, let us define Consider a post-selection channel on the subsystem A only, i.e., E (A)  ω ⊗ I (B) .Then Theorem 1 becomes Tr ϱ ⊥A The first category of examples is the weak-entanglement limit, which includes WVA as a special case.We consider a separable pure initial state . By a judicious choice of the initial states, one can always make In local estimation theory, x is usually considered to be very small.In the limit x → 0, we find similar with Eq. ( 5), one can construct where ω∈✓ q ω = 1, ε is arbitrarily small.In this case η = Lc = 1/ε as before, i.e., we can achieve arbitrarily large compression capacity and gain without loss.On the other hand, if One can simply take In this case, the compression capacity is c which is the ratio between the second and first order moments of the energy of the subsystem A. The compression gain is η = Lc.
Another category of examples with negligible loss is when the energy fluctuation of the Postselected subsystems dominates over the other if they are non-interacting but the initial state is entangled.We consider the Hamiltonian of the two systems is We then split the Hilbert space into several orthogonal and disjoint subspaces spanned by the energy eigenstates, i.e., for k l so that one can construct a set of mutually orthogonal states with the same average energy, i.e., ⟨ϕ k is a superposition of energy eigenstates to ensure non-vanishing QFI.We consider entangled initial state and δh B ≪ δh A is assumed.While an LCC exists for arbitrary value of x [37], as before, we focus on the local estimation for small x and consider the following LCC where ε is an arbitrarily small positive number, ω∈✓ r ωk = 1, ) is any projector to the support of the reduced density matrix 11) generalizes Eq. ( 5) beautifully while preserving the similar structure.The scaling of loss, capacity and gain in this case are For example, consider A and B consists of two qubits and one qubit respectively with the Hamiltonian H A = ω 0 (σ (1)  z + σ (2)  z ) and where |ϕ θ ⟩ is defined previously.We consider binary post-selection and employ the LCC and |ϕ ⊥A 2 ⟩ = 0 so it does not appear.The performance of this compression channel is numerically calculated in Fig. 2.
Conclusion.-We propose a unified theory, which implies that quantum measurements can be viewed as either information-extracting apparatus as in the standard quantum metrology, or information filters as in the Postselected quantum metrology.It can be employed to distribute the optimal measurements through post-selections so that the cost of the final detections are dramatically reduced, in synergy with recent efforts on distributed quantum sensing (see e.g.[51][52][53][54]).As a result, we anticipate our results will find applications in quantum sensing technologies, such as optical imaging and interferometry [49,50,55], magnetometry [56], frequency estimation [57], etc.Many problems are open for future exploration, including the compression of mixed states, multiparameter states [4,58], multipartite-entangled states, etc.

Supplemental Material
In this Supplemental Material, we present the expression of the quantum Fisher information (QFI) after the post-selection measurement, the saturation conditions for the various bounds of the QFI, proof of Theorem 2 in the main text, calculations on the sensitivity of the Postselected state, weak value amplifications and the example of restricted compression, and expressions of and the QFI associated with σ SA x is where and I cl ω p(ω x) ≡ ∂ x p(ω x) 2 /p(ω x).We denote It is straightforward to calculate Subtracting above two equations yields For later mathematical convenience, let us express all the quantities in terms of |ψ x ⟩ and its orthogonal vector |∂ ⊥ x ψ x ⟩.It can be readily checked that As a result, we find Therefore where Let us note the QFIs must be non-negative.Secondly, according to Eq. (S16), it is clear that Summing over all the measurement outcome leads to I(σ SA x ) ≤ I(ρ x ).Clearly, given Eq. ( 1) in the main text, we also have and In standard metrology, only the post-selection measurement statistics matters as the post-measurement states σ x|ω will be discarded.That is, only I cl ω p(ω x) matters.We can calculate Summing over all the measurement outcome leads to ω I cl ω p(ω x) ≤ I(ρ x ).As one can see from Eqs. (S14, S15, S16), the expressions of the QFI all involve the probability of the corresponding measurement outcome ⟨ψ x E ω ψ x ⟩ in the denominator, one should distinguish the case where ⟨ψ x E ω ψ x ⟩ 0 and the case where ⟨ψ x E ω ψ x ⟩ = 0, which was known as the regular and null POVM operator, respectively, according to Ref. [33].Equivalently, one should distinguish whether √ E ω |ψ x ⟩ vanishes or not .Finally, let us note a trivial case where √ E ω |∂ ⊥ x ψ x ⟩ = 0.In this case I ω (ρ x ) = 0, which dictates I cl ω p(ω x) = I(σ x|ω ) = 0, according to the non-negativity of the QFI.We shall exclude this trivial case in the following discussion and the main text.
A. Regular POVM operator with In this case, If we perform a spectral decomposition of E ω = µ λ ωµ |e ωµ ⟩ ⟨e ωµ |, the it indicates at there exists at least one vector such that ⟨e ωµ ψ x ⟩ 0. According to Eq. (S16), the inequality (S18) is saturated if Eq. (T1) in the main text is satisfied.
To saturate Eq. (S19), we would like to saturate Eq. (S18) but also ensure I cl ω p(ω x) vanishes at the same time.This leads to Eq. (T4) in the main text.In this case, the measurement statistics contains no information, but the maximum amount of information all goes into the selective post-selection state σ x|ω .
Finally, the saturation of Eq. (S21) requires, in addition to Eq. (T1) in the main text, These two condition leads to Eq. (T5) in the main text, which was addressed in the paper by Braunstein and Caves [30] and more recently in Ref. [33] and Ref. [31].Indeed, this condition includes Eq. (T2) in the main text as a special case and therefore I(σ S x|ω ) = 0.That is, no information is left after the optimal measurements in standard metrology, which physically makes sense!
In this case, we know that is, |e ωµ ⟩ lies in the kernel of the density matrix ρ x .Using L'hospital rule, Ref. [33] prove that for single-parameter estimation.This condition implies that when our prior knowledge of the estimation parameter denoted x * is very close to x, the channel E ω (x * ) leaves almost no information in the selective post-measurement state σ x|ω .Indeed, using L'hospital rule, one can further confirm that I(σ x|ω ) vanishes if E ω is a null POVM operator.

)
II. LOW AND UPPER BOUNDS OF I ω AND I clω p(ω x)

Table I .
The necessary and sufficient conditions for the saturation of the bounds of various QFIs corresponding to a regular POVM operator E ω with √ E ω |ψ x ⟩ 0. See Supplemental Material for details[37].