No-go result for quantum postselection measurements of rank-$m$ degenerate subspace

Introduction.— Theoretical description of quantum measurements with a postselection protocol is fundamental and of practical interest Aharonov et al. (1988); Zhu et al. (2011); Wagner et al. (2021); Monroe et al. (2021); Purves and Short (2021). It is a sequential measurements of a generalized (POVM) measurement followed by another projective measurement Aharonov et al. (1988). The postselection process alters the statistical results of the measured observable and let to an extraordinary amplification effect: the expectation value obtained by the postselection can go far beyond the conventional eigenvalues of the measured observable Aharonov et al. (1988); Torres and Salazar-Serrano (2016); Ren et al. (2020). Beyond the fundamental interest Dressel et al. (2014); Qin et al. (2016); Vaidman (2017); Kunjwal et al. (2019); Ogawa et al. (2020); Ho and Imoto (2018, 2016, 2017), postselection measurements have attracted tremendous research interest in multiple fields, including testing of quantum paradoxes and nonlocality Lundeen and Steinberg (2009); Yokota et al. (2009); Das and Sen (2021); Calderón-Losada et al. (2020); Xu et al. (2020); Aharonov et al. (2013); Denkmayr et al. (2014); Kim et al. (2021); Aharonov et al. (2016); Waegell et al. (2017), measurement uncertainty Budiyono and Dipojono (2021), weak value amplification Hosten and Kwiat (2008); Dixon et al. (2009); Modak et al. (2021); Krafczyk et al. (2021); Modak et al. (2021); Zhou et al. (2021); Fang et al. (2021); Liu et al. (2021); Huang et al. (2021), quantum-enhanced metrology Arvidsson-Shukur et al. (2020); Pati et al. (2020); Yin et al. (2021); Ho and Kondo (2021, 2019); Harris et al. (2017); Pang and Brun (2015), direct quantum state measurement Lundeen et al. (2011); Lundeen and Bamber (2012); Calderaro et al. (2018); Vallone and Dequal (2016); Maccone and Rusconi (2014); Chen et al. (2018); Pan et al. (2019); Turek (2020); Ho (2020); Tuan et al. (2021), among others.

Consider a prepared state $\rho$ and a measured observable $A$ represented by a self-adjoint operator $\bm{A}$. Assume the spectral decomposition of $\bm{A}$ has purely discrete spectrum as $\bm{A}=\sum_{k}r_{k}\bm{P}_{k}$, where $r_{k}$ is the eigenvalues, and projection operators $\bm{P}_{k}=|r_{k}\rangle\langle r_{k}|$ satisfy $\bm{P}_{j}\bm{P}_{k}=\delta_{jk}\bm{P}_{k};\sum_{k}\bm{P}_{k}=\bm{I}$. Following the projection postulate P. Busch (2009); Nielsen and Chuang (2010), the expectation value gives $\langle\bm{A}\rangle_{\rho}=\sum_{k}r_{k}P(r_{k}|\rho)$, where $P(r_{k}|\rho)={\rm Tr}[\bm{P}_{k}\rho]$ is the probability upon obtaining outcome $r_{k}$. The state transforms to, following the Lüders rule P. Busch (2009); Lüders (2006), $\rho^{\prime}=\bm{P}_{k}\rho\bm{P}_{k}/{\rm Tr}[\bm{P}_{k}\rho]$. After the $\bm{A}$-measurement, a subsequent projection measurement is carried out, i.e., using $\bm{\Pi}_{\phi}=|\phi\rangle\langle\phi|$, such that we postselect the system onto a final state $|\phi\rangle$. The expectation value of $\bm{A}$ now conditions on the postselected state $|\phi\rangle$ and reads ${}_{\phi}\langle\bm{A}\rangle_{\rho}=\sum_{k}r_{k}P(r_{k}|\phi,\rho)$, where $P(r_{k}|\phi,\rho)={\rm Tr}[\bm{\Pi}_{\phi}\bm{P}_{k}\rho\bm{P}_{k}]/\sum_{k^{\prime}}{\rm Tr}[\bm{\Pi}_{\phi}\bm{P}_{k^{\prime}}\rho\bm{P}_{k^{\prime}}]$ is the conditional probability following the Aharonov-Bergmann-Lebowitz (ABL) rule Aharonov et al. (1964). The conditional expectation value becomes the weak value when the system weakly couples to the device.

Even the effect of postsections on measurement results is significant, it is not always so. For example, with a full degenerated spectrum, i.e., $r_{k}=r\ \forall k$, or with a projective observable $\bm{A}=|r\rangle\langle r|$, we have $\langle\bm{A}\rangle_{\rho}=\ _{\phi}\langle\bm{A}\rangle_{\rho}$. Previously, Vaidman et al. Vaidman et al. (2017) have claimed that the nature of weak values is the same as the eigenvalues for an infinitesimally small interaction strength. Besides, weak values become conventional expectation values in the enlarged Hilbert space Ho and Imoto (2018).

In this paper, we generalize these intuitive claims by presenting a “no-go” theorem, where the postselection does not affect the measurement results. We first extend the projection postulate to a composite system, such as a measured system and its apparatus (device). [A composite system also induces subsystem-subsystem and system-environment interaction models.] Whenever a joint system-device observable has rank-$m$ degenerate subspace, with $m$ is the device’s dimension, the measured observable’s results will not be affected by the postselection measurement. This is the main statement of the theorem. Afterward, we illustrate the no-go theorem in the error and disturbance of quantum measurements. Following Ozawa’s interpretation Ozawa (2003, 2004), the error is a root-mean-square of the noise operator formed by the device’s operator after the interaction and the system operator before the interaction, and the disturbance in a root-mean-square of the disturbance operator formed by the system’s observables after and before the interaction.

Conditional expectation values.— In the von Neumann mechanism von Neumann (1955), we consider a measured system $\mathcal{S}$ and a device $\mathcal{M}$, initially prepared in uncorrelated state $|\Psi\rangle=|\psi\rangle\otimes|\xi\rangle$. The interaction is given by a unitary $\mathbfcal U={\rm exp}(-it\bm{H}_{\mathcal{S}}\otimes\bm{H}_{\mathcal{M}})$, where $\bm{H}_{\mathcal{S}}$ and $\bm{H}_{\mathcal{M}}$ are Hamiltonians over the system’s and device’s complex Hilbert spaces $\mathcal{H}_{\mathcal{S}}$ and $\mathcal{H}_{\mathcal{M}}$, respectively. Given any initial joint observable $\mathbfcal O_{0}=\bm{S}_{0}\otimes\bm{M}_{0}$ in the joint $\mathcal{SM}$ Hilbert space, following the Heisenberg picture, it evolves to $\mathbfcal O_{t}=\mathbfcal U^{\dagger}\mathbfcal O_{0}\mathbfcal U$ after interaction. Let $\mathbfcal O$ be a joint measured operator after interaction, which is a function of $\mathbfcal O_{t}$ and satiflies $\mathbfcal{O}\equiv f(\mathbfcal{O}_{t})=\sum_{k}\bm{S}_{k}\otimes\bm{M}_{k}$ ¹¹1 Using the Baker-Campbell-Hausdorff formula Hall (2015), we expand $\mathbfcal U^{\dagger}\mathbfcal O_{0}\ \mathbfcal U=\mathbfcal O_{0}+it\big{[}\bm{H}_{\mathcal{S}}\otimes\bm{H}_{\mathcal{M}},\mathbfcal O_{0}\big{]}+\frac{(it)^{2}}{2!}\Big{[}\bm{H}_{\mathcal{S}}\otimes\bm{H}_{\mathcal{M}},\big{[}\bm{H}_{\mathcal{S}}\otimes\bm{H}_{\mathcal{M}},\mathbfcal O_{0}\big{]}\Big{]}+\cdots$. For small $t$ (or weak interaction), it yields $\mathbfcal U^{\dagger}\mathbfcal O_{0}\ \mathbfcal U\approx\mathbfcal O_{0}+it\big{[}\bm{H}_{\mathcal{S}}\otimes\bm{H}_{\mathcal{M}},\mathbfcal O_{0}\big{]}=\sum_{k}\bm{S}_{k}\otimes\bm{M}_{k}$. (e.g., $\mathbfcal{O}=\mathbfcal{O}_{t}-\mathbfcal{O}_{0}$, which pertains to the error and disturbance operators discussed later.) To calculate the expectation value of $\mathbfcal O$, we start with an element $\mathbfcal O_{k}\equiv\bm{S}_{k}\otimes\bm{M}_{k}$ which is an $(n.m\times n.m)$-matrix, where $n(m)$ is the system(device) dimension. Let $|u_{i}\rangle,|v_{j}\rangle$ with $i\leq n$ and $j\leq m$ are eigenvectors associated to the eigenvalues $u_{i}$ and $v_{j}$ of $\bm{S}_{k}$ and $\bm{M}_{k}$, respectively, then $\mathbfcal O_{k}$ can be expressed in the spectral representation as $\mathbfcal O_{k}=\sum_{i=1}^{n}\sum_{j=1}^{m}r_{ij}^{(k)}\bm{P}_{ij}$, where $r_{ij}^{(k)}=u_{i}v_{j}$ are eigenvalues, and $\bm{P}_{ij}=|u_{i}v_{j}\rangle\langle u_{i}v_{j}|$ satisfies the orthonormality relations $\bm{P}_{ij}\bm{P}_{i^{\prime}j^{\prime}}=\delta_{i,i^{\prime}}\delta_{j,j^{\prime}}\bm{P}_{i^{\prime}j^{\prime}}$ and a completeness relation $\sum\bm{P}_{ij}=\bm{I}$. Following von Neumann, the probability to obtain the outcome $r_{ij}^{(k)}$ is given by Lüders (2006)

where we set $\rho=|\Psi\rangle\langle\Psi|$. The expectation value of $\mathbfcal O_{k}$ is given by

After the projection measurement $\bm{P}_{ij}$, the joint state transforms to a conditional (not normalized) $\rho^{\prime}_{ij}=\bm{P}_{ij}\rho\bm{P}_{ij}$. We then postselect system $\mathcal{S}$ onto a final state $|\phi\rangle$, represented by a projection operator $\bm{\Pi}_{\phi}=|\phi\rangle\langle\phi|\otimes\bm{I}$. The joint probability to obtain $r_{ij}^{(k)}$ and postselection is

Using the Bayesian theorem, the conditional probability to obtain $r_{ij}^{(k)}$ for given pre and postselected states is

Then, the conditional expectation value of $\mathbfcal O_{k}$ yields

We present the following theorem:
Theorem (no-go postselection theorem).— For any given joint state $\rho=|\Psi\rangle\langle\Psi|$ and post-selected state $|\phi\rangle$, the following rank-$m$ degenerate for every joint operator $\mathbfcal{O}_{k}$

must lead to a “no-go” for postselection measurement

where $\langle\mathbfcal O\rangle=\sum_{k}\langle\mathbfcal O_{k}\rangle$.

Proof of Theorem.— Let $\{|e_{i}\rangle\otimes|g_{j}\rangle\}$ be the canonical basis of the joint space, in which the joint state $|\Psi\rangle$ can be expressed as

where $\psi_{i}=\langle e_{i}|\psi\rangle$ and $\xi_{j}=\langle g_{j}|\xi\rangle$. Let the eigenvalues $|r_{ij}^{(k)}\rangle\equiv|u_{i}v_{j}\rangle=\sum_{i^{\prime},j^{\prime}}a_{i^{\prime}j^{\prime},ij}|e_{i^{\prime}}g_{j^{\prime}}\rangle$ is the eigenbasis, and $\bm{T}=(|r_{11}^{(k)}\rangle,\cdots,|r_{nm}^{(k)}\rangle)$ is the transformation matrix that formed by the ket vector of all eigenvalues. In the eigenbases, the joint state is expressed as $|\Psi\rangle=\sum_{i,j}\psi^{\prime}_{i}\xi^{\prime}_{j}|u_{i}v_{j}\rangle$, where $\psi^{\prime}_{i}=\langle u_{i}|\psi\rangle$ and $\xi^{\prime}_{j}=\langle v_{j}|\xi\rangle$, whose obey $\sum_{i}|\psi^{\prime}_{i}|^{2}=1$ and $\sum_{j}|\xi^{\prime}_{j}|^{2}=1$. We then obtain $\langle r_{ij}^{(k)}|\Psi\rangle=\psi_{i}^{\prime}\xi_{j}^{\prime}$.

In the postseleted projection operator $\bm{\Pi}_{\phi}=|\phi\rangle\langle\phi|\otimes\bm{I}$, let $|\phi\rangle=\sum_{i}\phi_{i}|e_{i}\rangle$, where $\phi_{i}=\langle e_{i}|\phi\rangle$ a complex amplitude, then we obtain $(n.m\times n.m)$-matrix:

where the off-diagonal elements are omitted as they do not include in the canonical basis, see more details in App. A. Here, each box is an $(m\times m)$-matrix, with totally $n$ boxes. In the eigenbasis, the postselected state is expressed as $\bm{\Pi}^{\prime}_{\phi}=\bm{T}^{\dagger}\bm{\Pi}_{\phi}\bm{T}$. An extra requirement (for the transformation matrix) that the diagonal elements of $\bm{\Pi}^{\prime}_{\phi}$ admit the rank-$m$ degenerated subspace similar as $\bm{\Pi}_{\phi}$, i.e., ${\rm diag}(\bm{\Pi}^{\prime}_{\phi})=(|\phi^{\prime}_{1}|^{2},\cdots,|\phi^{\prime}_{1}|^{2},\cdots,|\phi^{\prime}_{n}|^{2},\cdots,|\phi^{\prime}_{n}|^{2})$. We emphasize that this requirement always satisfies when the eigenbasis is the canonical basis as given in Eq. (9).

Now, from Eq. (5), we have the denominator

and the numerator

where we applied condition (6). Equation (5) is recast as

Similarly, we have $\langle\mathbfcal O_{k}\rangle_{\rho}=\sum_{j=1}^{m}{\widetilde{r}}_{j}^{(k)}|\xi^{\prime}_{j}|^{2}$, then ${}_{\phi}\langle\mathbfcal O_{k}\rangle_{\rho}=\langle\mathbfcal O_{k}\rangle_{\rho}$ (See detailed proof in App. A .) The proof for second term in (7) is straightforward since all $\mathbfcal{O}_{k}$ satisfy condition (6) $\;\;\Box$

Corollary 1.— For any eigenbasis $\{|r_{ij}\rangle\}$ is the canonical basis, the no-go theorem states

The proof for this Corollary is the same as above.

Remark.— Different from conditional expectation values we are considering here, the weak value of an observable $\bm{A}$ in system $\mathcal{S}$ generally depends on the postselected state, as it is $\langle\bm{A}\rangle_{\rm w}={\langle\phi|\bm{A}|\psi\rangle}/{\langle\phi|\psi\rangle}$, except for some certain conditions wherein it can be an expectation value Ho and Imoto (2018) or an eigenvalue Vaidman et al. (2017). However, this is not a consequential result from our theorem here. Instead, a consequence from the no-go theorem can state as follows:

Corollary 2.— Weak values can reduce to eigenvalues if the measured observable $\bm{A}$ has full-rank degenerate subspace. To proof this Corollary, let’s say $\bm{A}=\sum_{k}a_{k}|k\rangle\langle k|$ with $a_{k}=a$ for all $k=1,\cdots,n$, then, $\langle\bm{A}\rangle_{\rm w}=\langle\bm{A}\rangle=a$.

Observation.— Any joint observable, i.e., $\mathbfcal O=\bm{S}\otimes\bm{M}$ with $\bm{S}=\bm{I}$ always satisfies the no-go theorem. The proof for this Observation is given directly by noting that the eigenvalues of $\bm{I}$ are all one. Thus eigenvalues of $\mathbfcal O$ satisfies (6), and thus satisfies the theorem.

No-go theorem in the error and disturbance.— Error and disturbance are essential quantities for determining measurements’ uncertainties Ozawa (2003, 2004, 2005). We consider the error of an $\bm{A}$-measurement in system $\mathcal{S}$ through an $\bm{M}$-measurement in device $\mathcal{M}$ and the disturbance of a $\bm{B}$-measurement in system $\mathcal{S}$. In the joint $\mathcal{SM}$ system, we denote $\mathbfcal A_{0}=\bm{A}\otimes\bm{I},\text{ and }\mathbfcal B_{0}=\bm{B}\otimes\bm{I}$, where $\bm{A}$ and $\bm{B}$ are the observables to be measured in system $\mathcal{S}$. We also define a device observable $\bm{M}$ in the device space, such that it becomes $\mathbfcal M_{0}=\bm{I}\otimes\bm{M}$ in the joint $\mathcal{SM}$ space Ozawa (2003, 2004, 2005). The interaction is switched on during a short time $t$, where the joint $\mathcal{SM}$ system evolves under the unitary transformation $\mathbfcal U$. These operators transform to $\mathbfcal M_{t}=\mathbfcal U^{\dagger}\mathbfcal M_{0}\mathbfcal U$ and $\mathbfcal B_{t}=\mathbfcal U^{\dagger}\mathbfcal B_{0}\mathbfcal U$. According to Ozawa Ozawa (2003, 2004, 2005), the error and disturbance operators are defined by $\mathbfcal N_{\bm{A}}=\mathbfcal M_{t}-\mathbfcal A_{0},$ and $\mathbfcal D_{\bm{B}}=\mathbfcal B_{t}-\mathbfcal B_{0}$, respectively. Then, the mean square error and the disturbance are given by

where the bra-ket symbol $\langle...\rangle_{\Psi}$ means $\langle\Psi|...|\Psi\rangle$ throughout this paper.

In the following, we illustrate that such the error and disturbance satisfy condition (6) in the Theorem, and thus it makes no sence for postselection measurements of the error and disturbance. In other words, the error and disturbance will not be affected under the postselection.

Concretely, let us consider a CNOT-type measurement, where both system $\mathcal{S}$ and device $\mathcal{M}$ are qubits, initially prepared in $|\psi\rangle=|i^{+}\rangle$ and $|\xi\rangle=\sqrt{\frac{1+s}{2}}|0\rangle+\sqrt{\frac{1-s}{2}}|1\rangle$, where $|0\rangle$ and $|1\rangle$ are two eigenstates of Pauli matrix $\bm{Z}$, and $|i^{+}\rangle=\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$; $s$ is the measurement strength ranging from 0 (weak measurement) to 1 (strong measurement). The measured observables $\bm{A}$ and $\bm{B}$ in system $\mathcal{S}$ are chosen t o be Pauli matrices $\bm{Z}$ and $\bm{X}$, respectively, and the device observable $\bm{M}$ is also $\bm{Z}$. The interaction is CNOT-gate, i.e., $\mathbfcal U=|0\rangle\langle 0|\otimes\bm{I}+|1\rangle\langle 1|\otimes\bm{X}$. The square error and disturbance operators give

Fortunately, CNOT is a typical interaction, which leads to simplify measured operators (square error and disturbance). The eigenvalues of the square error are (0, 4, 0, 4), and the same for the square disturbance, which all satisfy the rank-2 degenerate. As a result, ${}_{\phi}\langle\mathbfcal N^{2}_{\bm{Z}}\rangle_{\Psi}=\langle\mathbfcal N^{2}_{\bm{Z}}\rangle_{\Psi}=2(1-s),$ and ${}_{\phi}\langle\mathbfcal D^{2}_{\bm{X}}\rangle_{\Psi}=\langle\mathbfcal D^{2}_{\bm{X}}\rangle_{\Psi}=2(1-\sqrt{1-s^{2}}),$ for any postselected state $|\phi\rangle=\cos\theta|0\rangle+e^{-i\varphi}\sin\theta|1\rangle$. These results imply that postselection measurements affect neither the error nor the disturbance. (See detailed calculation in App. B.)

The observations in the example can be explained as follows. The error is determined via the device after the first measurement, and thus, it is not affected by postselection measurements as long as the rank-$m$ degenerate in the device holds. For the same reason, the disturbance is affected by the backaction caused by the device while unpolluted with postselection measurements.

Conclusion.— We introduced a no-go theorem for postselection measurements, where the obtained conditional expectation value is not affected by postselection measurements and is equal to a conventional expectation value. This can happen when the joint observable has a rank-$m$ degenerate subspace, where $m$ is the dimension of the device space. As a consequence, the error and disturbance in quantum measurements immune with postselection measurements.

Acknowledgments.— We thanks H.C. Nguyen for the fruitful discussion.