Quasi-Quantization: Classical Statistical Theories with an Epistemic Restriction

Abstract

A significant part of quantum theory can be obtained from a single innovation relative to classical theories, namely, that there is a fundamental restriction on the sorts of statistical distributions over physical states that can be prepared. This is termed an “epistemic restriction” because it implies a fundamental limit on the amount of knowledge that any observer can have about the physical state of a classical system. This article provides an overview of epistricted theories, that is, theories that start from a classical statistical theory and apply an epistemic restriction. We consider both continuous and discrete degrees of freedom, and show that a particular epistemic restriction called classical complementarity provides the beginning of a unification of all known epistricted theories. This restriction appeals to the symplectic structure of the underlying classical theory and consequently can be applied to an arbitrary classical degree of freedom. As such, it can be considered as a kind of quasi-quantization scheme; “quasi” because it generally only yields a theory describing a subset of the preparations, transformations and measurements allowed in the full quantum theory for that degree of freedom, and because in some cases, such as for binary variables, it yields a theory that is a distortion of such a subset. Finally, we propose to classify quantum phenomena as weakly or strongly nonclassical by whether or not they can arise in an epistricted theory.

Appendix A: Quadrature Quantum Subtheories and the Stabilizer Formalism

In quantum information theory, there has been a great deal of work on a particular quantum subtheory for discrete systems of prime dimension (qubits and qutrits in particular) which is known as the stabilizer formalism [6, 53].

A stabilizer state is defined as a joint eigenstate of a set of commuting Weyl operators. By Eq. (53), two Weyl operators commute if and only if the corresponding phase-space displacement vectors have vanishing symplectic inner product,

$$ [ \hat{W}(\mathbf{a}) , \hat{W}(\mathbf{a}') ]=0 \;\text { if and only if }\; \langle \mathbf{a}, \mathbf{a}'\rangle =0. $$

Consequently, the sets of commuting Weyl operators, and therefore the stabilizer states, are parametrized by the isotropic subspaces of $\Omega$. Specifically, for each isotropic subspace M of $\Omega$ and each vector $\mathbf{v}\in {\textit{JM}}\equiv \{ J\mathbf{u} : \mathbf{u} \in M \}$, we can define a stabilizer state $\rho ^{(\mathrm stab)}_{M,\mathbf{v}}$ as the projector onto the joint eigenspace of $\{ \hat{W}(\mathbf{a}): \mathbf{a} \in M\}$ where $\hat{W}(\mathbf{a})$ has eigenvalue $\chi (\langle \mathbf{v}, \mathbf{a} \rangle )$.

We will show here that the set of stabilizer states is precisely equivalent to the set of quadrature states.

To describe the connection, it is convenient to introduce some additional notions from symplectic geometry. The symplectic complement of a subspace V, which we will denote as $V^C$, is the set of vectors that have vanishing symplectic inner product with every vector in V,

$$ V^{C}\equiv \{ \mathbf{m}' \in \Omega : \mathbf{m'}^T J \mathbf{m} =0, \; \forall \mathbf{m} \in V \}, $$

where J is the symplectic form, defined in Eq. (15). This is not equivalent to the Euclidean complement of a subspace V, which is the set of vectors that have vanishing Euclidean inner product with every vector in V,

$$ V^{\perp }\equiv \{ \mathbf{m}' \in \Omega : \mathbf{m'}^T \mathbf{m} =0, \; \forall \mathbf{m} \in V \}, $$

The composition of the two complements will be relevant in what follows. It turns out that the latter is related to V by an isomorphism; it is simply the image of V under left-multiplication by the symplectic form J,

$$ (V^{\perp })^{C}\equiv {\textit{JV}} = \{ J\mathbf{u} : \mathbf{u} \in V \}. $$

Note that if V is isotropic, then $(V^{\perp })^{C}$ is as well.

Proposition 1

Consider the quadrature state $\rho _{V,\mathbf{v}}$, with V an isotropic subspace of $\Omega$ and $\mathbf{v}\in V$ a valuation vector, which is the joint eigenstate of the commuting set of quadrature observables $\{ \mathcal {O}_{f} : \mathbf{f}\in V \}$, where the eigenvalue of $\mathcal {O}_f$ is $f(\mathbf{v})$. This is equivalent to the stabilizer state $\rho ^{(\mathrm{stab})}_{M,\mathbf{v}}$, which is the joint eigenstate of the commuting set of Weyl operators $\{ \hat{W}(\mathbf{a}) : \mathbf{a} \in M\}$ where $M \equiv (V^{\perp })^C$ is the isotropic subspace that is the symplectic complement of the Euclidean complement of V, and where the eigenvalue of $\hat{W}(\mathbf{a})$ is $\chi (\langle \mathbf{v},\mathbf{a} \rangle )$.

Proof

Consider first a single degree of freedom. Every quadrature observable $\mathcal {O}_{f}$ can be expressed in terms of the position observable $\mathcal {O}_q$ as follows: if $S_f$ is the symplectic matrix such that $\mathbf{f} = S_f \mathbf{q}$, then $\mathcal {O}_{f} = \hat{V}(S_f) \mathcal {O}_{q} \hat{V}(S_f)^{\dag }$. Now note that the position basis can equally well be characterized as the eigenstates of the boost operators. Specifically, $\hat{B}(\mathtt{p}) |\mathtt{q}\rangle _q = \chi (\mathtt{q p}) |\mathtt{q}\rangle _q$, that is, an element $|\mathtt{q}\rangle _q$ of the position basis is an eigenstate of the set of operators $\{ \hat{B}(\mathtt{p}): \mathtt{p}\in \mathbb {R}/\mathbb {Z}_d \}$ where the eigenvalue of $\hat{B}(\mathtt{p})$ is $\chi (\mathtt{qp})$. The element $|\mathtt{f}\rangle _f$ of the basis associated to the quadrature operator $\mathcal {O}_f$ is defined as $|\mathtt{f}\rangle _f \equiv \hat{V}(S_f)|\mathtt{f}\rangle _q$ and consequently can be characterized as an eigenstate of the set of operators $\{ \hat{V}(S_f) \hat{B}(\mathtt{g}) \hat{V}(S_f)^{\dag }: \mathtt{g}\in \mathbb {R}/\mathbb {Z}_d \}$ where the eigenvalue of $\hat{V}(S_f) \hat{B}(\mathtt{g}) \hat{V}(S_f)^{\dag }$ is $\chi (\mathtt{fg})$. This can be stated equivalently as follows: the element $|\mathtt{f}\rangle _f$ of the basis associated to the quadrature operator $\mathcal {O}_f$ is the eigenstate of the set of Weyl operators $\{ \hat{W}(\mathbf{a}): \mathbf{a}\in \mathrm{span}(S_f\mathbf{p}) \}$ where the eigenvalue of $\hat{W}(\mathbf{a})$ is $\chi ( \mathtt{f}\langle \mathbf{f},\mathbf{a}\rangle )$. Noting that

$$\begin{aligned} \mathrm{span}(S_f\mathbf{p})&=\mathrm{span}(S_f J\mathbf{q}),\nonumber \\&= \mathrm{span}({\textit{JS}}_f \mathbf{q}),\nonumber \\&= \mathrm{span}(J \mathbf{f}), \end{aligned}$$

(91)

we can just as well characterize $\hat{\Pi }_f(\mathtt{f})$ as the projector onto the joint eigenspace of the Weyl operators $\{ \hat{W}(\mathbf{a}): \mathbf{a}\in \mathrm{span}(J \mathbf{f}) \}$.

Now consider n degrees of freedom. The quadrature state associated with $(V,\mathbf{v})$ has the form

$$\begin{aligned} \rho _{V,\mathbf{v}} = \frac{1}{\mathcal {N}} \prod _{\mathbf{f}^{(i)}: \mathrm{span}(\mathbf{f}^{(i)})=V} \hat{\Pi }_{f^{(i)}}\left( f^{(i)}(\mathbf{v})\right) . \end{aligned}$$

(92)

By an argument similar to that used for a single degree of freedom, this is an eigenstate of the Weyl operators $\{ \hat{W}(\mathbf{a}): \mathbf{a}\in \mathrm{span}(J \mathbf{f}^{(i)}) \}$ where the eigenvalue of $\hat{W}(\mathbf{a})$ is $\chi ( \langle \mathbf{v}, \mathbf{a} \rangle )$. Noting that $\mathrm{span}(J \mathbf{f}^{(i)}) = {\textit{JV}} = (V^{\perp })^C$, we have our desired isomorphism.$\blacksquare$

The stabilizer formalism allows all and only the Clifford superoperators as reversible transformations. The sharp measurements that are included in the stabilizer formalism are the ones associated with PVMs corresponding to the joint eigenspaces of a set of commuting Weyl operators, which, by Proposition 1, are precisely those corresponding to the joint eigenspaces of a set of commuting quadrature observables. It follows that the stabilizer formalism coincides precisely with the quadrature subtheory.

Gross has argued that the discrete analogue of the Gaussian quantum subtheory for continuous variable systems is the stabilizer formalism [6]. Our results show that the connection between the discrete and continuous variable cases is a bit more subtle than this. In the continuous variable case, there is a distinction between the Gaussian subtheory and the quadrature subtheory, with the latter being contained within the former. In the discrete case, there is no distinction, so the stabilizer formalism can be usefully viewed as either the discrete analogue of the Gaussian subtheory or as the discrete analogue of the quadrature subtheory. While Gross’s work showed that the stabilizer formalism for discrete systems could be defined similarly to how one defines Gaussian quantum mechanics, our work has shown that it can also be defined in the same way that one defines quadrature quantum mechanics.

To our knowledge, quadrature quantum mechanics has not previously received much attention. However, given that it is a natural continuous variable analogue of the stabilizer formalism for discrete systems, it may provide an interesting paradigm for exploring quantum information processing with continuous variable systems.