Quantum instruments as a foundation for both states and observables

We demonstrate that quantum instruments can provide a unified operational foundation for quantum theory. Since these instruments directly correspond to laboratory devices, this foundation provides an alternate, more experimentally grounded, perspective from which to understand the elements of the traditional approach. We first show that in principle all measurable probabilities and correlations can be expressed entirely in terms of quantum instruments without the need for conventional quantum states or observables. We then show how these states and observables reappear as derived quantities by conditioning joint detection probabilities on the first or last measurement in a sequence as a preparation or a postselection. Both predictive and retrodictive versions of states and observables appear in this manner, as well as more exotic bidirectional and interdictive states and observables that cannot be easily expressed using the traditional approach. We also revisit the conceptual meaning of the Heisenberg and Schrödinger pictures of time evolution as applied to the various derived quantities, illustrate how detector loss can be included naturally, and discuss how the instrumental approach fully generalizes the time-symmetric two-vector approach of Aharonov et al. to any realistic laboratory situation.

Figure 1

(a) Graphical depiction of the Hilbert-Schmidt inner product $\langle \check{B},\widehat{A}\rangle$. The scalar value of $\text{Tr}\left({\check{B}}^{†}\widehat{A}\right)$ (hexagon) is conceptually separated into complementary halves $\check{B}$ (trapezoid) and $\widehat{A}$ (inverse trapezoid). The choice of hat on the operators indicates this distinction that is induced by the inner product. (b) The inner product being used to show the normalization of a quantum state $\widehat{\rho }$.

Figure 2

(a) The quantum instrument $\mathcal{A}$ representing a detector is a transformation-valued functional that produces quantum operations $\mathcal{A}\left[{\chi }_S\right]$ corresponding to each set of detector outcomes $S$, as in Eq. (5). Here ${\chi }_S$ is a suitable indicator function for the set $S$. Computing the modified norm using the inner product yields the probability $p_S$ for detecting the outcomes $S$. (b) More generally, by assigning an appropriate set of values $\alpha$ to each detector outcome, the same quantum instrument can be used to compute any expectation value $\langle \alpha \rangle$ that can be measured by the detector.

Figure 3

The measurable correlation $\langle \alpha \beta \rangle$ as in Eq. (6) between the outcomes of two detectors $\mathcal{A}$ and $\mathcal{B}$ arranged in a sequence. We show this correlation represented three different ways as in Eq. (7) using the adjoint instruments ${\mathcal{A}}^{*}$ and ${\mathcal{B}}^{*}$. Each diagram corresponds to a different choice of temporal reference point, indicated by the black dot, which conceptually separates future detections from past detections in accordance with the time line on the right.

Figure 4

Predictive observable operators. (a) The adjoint quantum instrument ${\mathcal{A}}^{*}$ for the final measured detector generates predictive probability operators $\check{A}\left[{\chi }_S\right]$ in a POM from indicator functions ${\chi }_S$ for sets $S$ of detector outcomes, as in Eq. (14). (b) Similarly, weighting the adjoint instrument outcomes with real contextual values $\alpha$ produces traditional predictive (Hermitian) observable operators $\check{A}\left[\alpha \right]$ that can be indirectly measured by the final detector, as in Eq. (16).

Figure 5

Stateless reformulation. (a) The unnormalized correlation $c_{{\alpha }_1,...,{\alpha }_k}$ between the contextual values assigned to $k$ detectors $({\mathcal{A}}^{\left(1\right)},...,{\mathcal{A}}^{\left(k\right)})$ in sequence, as in Eq. (20). The quantum instruments for the detectors are entirely sufficient for computing this correlation, provided that no prior or posterior detectors that are not included in the sequence influence the computed correlation. (b) The normalization constant $N$ is computed with the nonselective measurements for the same detector sequence, and produces normalized correlation functions as $\langle {\alpha }_1\cdots {\alpha }_k\rangle =c_{{\alpha }_1,...,{\alpha }_k}/N$. Joint probabilities are special cases when the contextual values are chosen to be indicator functions.

Figure 6

Retrodictive observable operators. (a) The quantum instrument $\mathcal{A}$ for the first measured detector generates retrodictive probability operators $\widehat{A}\left[{\chi }_S\right]$ in a retrodictive POM from indicator functions ${\chi }_S$ for sets $S$ of detector outcomes, as in Eq. (21). (b) Similarly, weighting the instrument outcomes with real contextual values $\alpha$ produces retrodictive observable operators $\widehat{A}\left[\alpha \right]$ that can be indirectly measured by the first detector, as in Eq. (22).

Figure 7

Three measurement examples. (a) The unnormalized joint probability $n_{a,b,c}$ as in Eqs. (25) and (28) between the contextual values assigned to the sequence of three detectors $(\mathcal{A},\mathcal{B},\mathcal{C})$ defined in Eqs. (24). The correlation is shown both with quantum instruments and with the associated retrodictive and predictive probability observables for the first and last measurements, respectively. (b) The normalization constant $N$ for the same detector sequence, producing normalized joint probabilities as $p_{a,b,c}=n_{a,b,c}/N$.

Figure 8

Predictive state. Conditioning on the first measurement and associating the updated normalization constant $N_a=\langle \check{1},\widehat{A}\left[{\chi }_a\right]\rangle$ with the retrodictive PO for the first measurement produces the standard predictive quantum state as in Eq. (30), provided that the the detectors $\mathcal{B}$ and $\mathcal{C}$ are predictively complete.

Figure 9

Retrodictive state. Conditioning on the final measurement and associating the updated normalization constant $N_c=\langle \check{C}\left[{\chi }_c\right],\widehat{1}\rangle$ with the PO for the final measurement produces the retrodictive quantum state as in Eq. (34), provided that the detectors $\mathcal{A}$ and $\mathcal{B}$ are retrodictively complete.

Figure 10

Interdictive state. Conditioning on the intermediate measurement and associating the updated normalization constant $N_b=\langle \check{B}\left[{\chi }_b\right],\widehat{1}\rangle =\langle \check{1},\widehat{B}\left[{\chi }_b\right]\rangle$ with the QO for the intermediate detector produces an interdictive quantum state as in Eq. (41), provided that the detectors $\mathcal{A}$ and $\mathcal{C}$ are appropriately complete. Unlike the predictive and retrodictive states, the interdictive state is an operation.

Figure 11

Bidirectional state. (a) The unnormalized conditional probability $n_{b|a,c}$ as in Eq. (47). The predictive and retrodictive PO can be independently normalized by convention to produce the pair of states $({\widehat{\rho }}_a,{\check{\rho }}_c)$ that encodes the boundary condition information. (b) The normalization constant $N_{a,c}$ that produces normalized conditional probabilities as $p_{b|a,c}=n_{b|a,c}/N_{a,c}$.