1 Introduction↩︎

A central goal of quantum many-body physics is to classify and understand the phases of quantum matter [1]–[14]. Traditionally, phases of matter have been studied in thermal equilibrium, at zero or nonzero temperature. However, it has more recently been appreciated that the concept of phases of matter is also applicable to more general non-equilibrium states of quantum many-body systems [15]–[28]. These states could occur, for example, in open systems in which the system undergoes some noise process or interaction with its environment. “Phase transitions” in this context can correspond to sharp transitions in quantum information-theoretic properties of the quantum many-body system such as decodability.

When classifying phases of matter, symmetries play an essential role. For mixed (i.e.non-pure) quantum states (which non-equilibrium states will generally be), an important distinction has emerged between so-called weak and strong symmetries. Strong symmetries are generally the appropriate concept when a system interacts with a bath but does not exchange charge with it. Moreover, in the presence of strong symmetry, it has been argued that there can be phases of matter characterized by strong-to-weak spontaneous symmetry breaking (SW-SSB) [17], [23], [24], [29]–[32].

Nevertheless, compared to conventional spontaneous symmetry-breaking, in some ways SW-SSB remains a mysterious and elusive concept. The diagnostics of SW-SSB that have been proposed are not easy to interpret physically, and moreover are such that implementing them experimentally could in general could require a number of measurements that scales in a bad way (e.g.exponentially) with the system size. This had led to arguments that SW-SSB might in fact be impossible to diagnose in an efficient way as the system size is increased [32].

In this paper, we show that these issues can be overcome by a “back-to-basics” approach to quantum many-body phases of matter. Our central contention is that phases of matter should ideally be defined in terms of the expectation values of local operators in the thermodynamic limit. We will show that this leads to a more robust and mathematically rigorous way to distinguish and diagnose phases of matter that have previously been described in terms of “SW-SSB”, while also providing clearer physical interpretations.

Our key technical tool will be the formulation of several equivalent definitions that generalize the concept of “strong symmetry” to infinite systems. Our work is based on the framework of local quantum physics¹ [33]–[37]. In this approach, the algebra of (quasi-)local observables is taken as the primary object of study.

We will also show that by invoking our definition of strong symmetries, it is possible to prove generalizations of the Lieb-Schultz-Mattis theorem [38]–[67] (in its modern formulation as a constraint from anomalous symmetries – see also Refs. [63], [68]–[72]) that are applicable to mixed states with strong symmetries. Finally, we will show that in infinite systems, there is a kind of symmetry which is intermediate between strong and weak symmetries, which we refer to as “von Neumann symmetry”.

1.1 Preview: local operators, strong symmetries and “strong-to-weak symmetry breaking”↩︎

The key physical assumption that we want to make is that it should always be possible to distinguish different phases of matter via the expectation values of local operators². Indeed, in pure state phases of matter this appears to always be the case. For example, topologically ordered states on an infinite plane (or disk) can be studied in terms of local operators; see, e.g., Ref. [36], [73], [74].

Focusing on local operators has a practical advantage in that it might be much easier in experiment (or even numerics) to measure local operators than non-local operators. However, it also has a significant theoretical advantage, as follows. The concept of a “phase of matter” is strictly speaking only defined in the thermodynamic limit. In finite systems, phase transitions always become smooth crossovers. If one demands that all systems that we consider are strictly finite, then it is necessary to carefully consider various finite-size effects in order to define phase of matter. Therefore, it is conceptually more elegant (and certainly much easier to make mathematically rigorous) to define phase of matter first in a strictly infinite system, and address finite-size effects afterwards. This is indeed the approach we will take in this paper.

Later in Sec. 2.1, we will review some basic notions in the standard mathematical formalism of infinite quantum many-body systems [34]–[36]. In this formalism, states are generally defined through the expectation values they assign to local operators (Def. 1). Therefore, if we wish to formulate a notion of phase of matter that applies naturally to infinite systems, the definition must refer only to local operators.

As we will see shortly, the existing formulations of strong symmetry and strong-to-weak symmetry breaking (SWSSB) cannot be expressed purely in terms of local operators and therefore cannot be directly applied to infinite systems. In this paper we aim to close this gap. Doing so will lead to a clearer conceptual foundation for strong symmetries and strong-to-weak symmetry breaking that does not rely on subtle and hard-to-measure global properties.

One of the lessons that will emerge is that formulating the problem purely in terms of local operators requires a reorganization of how we think about strong symmetries and “strong-to-weak symmetry breaking” . To illustrate this point, consider a finite (but large) spin-\(1/2\) chain with a \(\mathbb{Z}_2\) symmetry generated by the unitary operator \[X = \prod_i \sigma_i^x .\] Now consider the following two states: \[\label{eq:example95states} \rho_1 = \mathbb{I}, \qquad \rho_2 = \frac{1}{2}(\mathbb{I} + X).\tag{1}\] The first state \(\rho_1\) is the maximally mixed state. It has the weak symmetry \(X\) (since \(X\rho_1 X^\dagger=\rho_1\)) but no strong symmetry. The second state \(\rho_2\), on the other hand, has strong symmetry \(X\) in the usual sense since \(\rho_2 X = X \rho_2 = \rho_2\).

From the perspective of local operators, however, \(\rho_1\) and \(\rho_2\) are completely indistinguishable. Indeed, they produce the same state upon tracing out even a single spin. In this paper we therefore take the viewpoint that after we take the thermodynamic limit, both \(\rho_1\) and \(\rho_2\) should be regarded as lacking strong symmetry. Formulating a definition of strong symmetry in the thermodynamic limit in terms of local operators will be one of the main goals of this paper, see Def. 5.

By contrast, an example of a state that does possess strong symmetry in the thermodynamic limit, according to our definition, is the pure product state \[| \Psi \rangle = | + \rangle^{\otimes N},\] where \(| \pm \rangle\) is the \(\pm 1\) eigenstate of \(\sigma^x\). For another example that is not pure, we can group the spins into pairs and then form the product state \[\rho_3 = \left(p | ++ \rangle \langle ++| + (1-p) | -- -- \rangle \langle -- --|\right)^{\otimes N/2}\]

What is the physical distinction between \(\rho_1,\rho_2\) on the one hand, and \(| \Psi \rangle\) and \(\rho_3\) on the other hand? Intuitively, the difference is clear: \(\rho_1\) and \(\rho_2\) exhibit incoherent local fluctuations of \(\mathbb{Z}_2\) charge, whereas \(| \Psi \rangle\) and \(\rho_3\) do not. The definition of strong symmetry that we develop in this paper will make this intuition precise.

This example also reveals that, from the local perspective, the terminology of SWSSB must be reconsidered. Indeed, the state \(\rho_2\) has been presented as the prototypical example of a state exhibiting SWSSB [24], yet from the local viewpoint it possesses only weak symmetry in the thermodynamic limit. Thus, strictly speaking, it should not be regarded as displaying SWSSB. This is primarily a matter of terminology. Phases of matter that, in the conventional terminology, are distinguished by the presence or absence of SWSSB, will from our perspective instead be distinguished by the presence or absence of strong symmetry in the infinite system limit.

This does not mean that the notion of “SWSSB” is irrelevant for us. Rather, in our view SWSSB is more naturally interpreted as a property of dynamical processes than of states. For example, ordinary spontaneous symmetry breaking refers to the situation in which a Hamiltonian has a symmetry, while its ground states or thermal equilibrium states do not. By analogy, SWSSB can be understood as the situation where a quantum channel or Lindbladian possesses a strong symmetry, but the steady state exhibits only weak symmetry. This appears to be the notion of SWSSB implicit in Ref. [75], where it was shown that in the presence of SWSSB of a \(\mathrm{U}(1)\) symmetry, the hydrodynamic mode associated with the \(\mathrm{U}(1)\) charge can be interpreted as the “Goldstone mode” of the spontaneous symmetry breaking.

Finally, let us highlight one key feature of our definition of strong symmetry. For infinite-size systems, we will show in Sec. 8 that a strongly symmetric channel can convert a state with strong symmetry into one in which the symmetry becomes weak, but the reverse process can never occur (Prop. 4). In other words, once local charge coherence is destroyed, it cannot be recovered. This reflects a form of macroscopic irreversibility—an emergent “arrow of time” that appears only in the thermodynamic limit.

2 Basic setup and summary of main results↩︎

2.1 Algebra and states↩︎

In this section, we introduce some basic notions in local quantum physics, which are necessary to formulate our main results. Details can be found in Ref. [34]–[36], [66], [76]. For convenience, we summarize our notations in a list placed after the references. Readers familiar with operator algebras may safely skip this section and consult it as needed.

In the present work, we work on an infinite lattice \(\Lambda \subseteq \mathbb{R}^d\), where on each lattice site \(i\in\Lambda\) is assigned with a Hilbert space \(\mathcal{H}_{i}\) with \(\dim_{\mathbb{C}}(\mathcal{H}_{i})={n_i}\) for some \(2\leqslant n_i\in\mathbb{N}\). We will work with general spatial dimension \(d\geqslant 1\) except when otherwise stated.

It is tempting to declare that this system is described by a Hilbert space \(\mathcal{H}\stackrel{?}{=}\bigotimes_{i\in\Lambda}\mathcal{H}_{i}\). However, as noted in [77], such infinite tensor product of Hilbert spaces is ill-defined. Concretely, the inner product between \(|\xi\rangle:=\bigotimes_{i\in\Lambda}|\xi_{i}\rangle\) and \(|\eta\rangle:=\bigotimes_{i\in\Lambda}|\eta_{i}\rangle\), \[\langle\xi|\eta\rangle\stackrel{?}{=}\prod_{i\in\Lambda}\langle\xi_{i}|\eta_{i}\rangle\] typically diverges and there is no obvious way to regularize it. Thus, the Hilbert-space formalism is not an appropriate mathematical description of infinite systems. Nevertheless, the Heisenberg picture of quantum mechanics, which focuses on operators rather than states, is still applicable even in the presence of infinitely many degrees of freedom, such as spin systems on infinite lattices.

Specifically, for each site \(i\in\Lambda\), we associate it with an algebra \(\mathscr{A}_{i}:=\mathrm{M}_{n_i}(\mathbb{C})\) (the \(n_{i}\times n_{i}\)-matrix algebra over complex numbers). For each finite subset \(\Gamma\subseteq\Lambda\), local operators supported on \(\Gamma\) are defined as \[\begin{align} \mathscr{A}_{\Gamma}:=\bigotimes_{i\in\Gamma}\mathscr{A}_{i} \end{align}\] We then define the algebra of local operators \(\mathscr{A}^\ell\) to contain all operators supported on any finite subset \(\Gamma\). Formally, we can write \[\begin{align} \mathscr{A}^{\ell}:=(\bigcup_{\Gamma\subseteq\Lambda}\mathscr{A}_{\Gamma})/\cong, \end{align}\] where we have identified \(a\cong a\otimes 1_{\Gamma'\setminus\Gamma}\) if \(\Gamma\subseteq\Gamma'\).

The matrix adjoint map \(a \mapsto a^\dagger\) on \(\mathscr{A}_\Gamma\) defines an involutive anti-linear anti-automorphism on \(\mathscr{A}^\ell\), i.e. an anti-linear map \(* : \mathscr{A}^\ell \to \mathscr{A}^\ell, a \mapsto a^*\), such that \[\begin{align} \label{eq:42-op} \begin{aligned} (ab)^*&= b^{*}a^{*},\\ (a^*)^*&=a. \end{aligned} \end{align}\tag{2}\] Any algebra with \(*\)-operation satisfying Eq. 2 is called a \(*\)-algebra.

A state \(\psi\) associated to the \(*\)-algebra \(\mathscr{A}^{\ell}\) is defined as follows:

It is readily checked that \(\mathcal{S}(\mathscr{A}^{\ell})\) is a convex set, i.e., a convex combination of states is again a state. This leads to the purity of states

We will be interested in purification, or more generally, the extension of a mixed state \(\psi\) on \(\mathscr{A}^{\ell}\).

We note that given a state \(\psi\), its purification may not exist at all once the locality is imposed. Indeed, Lemma 9 shows that any pure state in the sense of Def. 2 must satisfy the clustering property (i.e., decay of connected correlation functions with the spatial separation; see Lemma 9 for the precise definition). If a state fails to be clustering, there is no way to purify it while preserving locality.

2.2 The symmetry actions↩︎

Let us now discuss how to formulate symmetry actions in the local language. Specifically, a physically relevant symmetry should map local operators to local operators. Thus, a symmetry operation will correspond to an automorphism \(\alpha \in \mathrm{Aut}(A^\ell)\). This means that \(\alpha\) is a linear map \(\alpha : \mathscr{A}^\ell \to \mathscr{A}^\ell\) such that \(\alpha(ab) = \alpha(a) \alpha(b)\) and \(\alpha(a^*) = \alpha(a)^*\), and \(\alpha\) has an inverse \(\alpha^{-1}\) satisfying the same conditions. For example, one could consider an “on-site” symmetry where one chooses a unitary action \(u_i\) on each site and then conjugates operators supported on \(\Gamma \subseteq \Lambda\) by \(\bigotimes_{i \in \Gamma} u_i\). Non-on site symmetries such as translation symmetry also correspond to automorphisms of \(\mathscr{A}^\ell\).

For some of our results (Theorem 1), it will be helpful to impose a stricter notion of locality [78], [79]:

Given a symmetry group \(G\), by slightly abusing the notations³, the symmetry action can be represented by a group homomorphism \(\alpha: G\to \mathscr{G}^{\mathrm{QCA}}\) [or more generally \(\alpha : G \to \mathrm{Aut}(A^\ell)\)]. The image of \(g\in G\) under \(\alpha\) is written as \(\alpha_{g}\). This symmetry may contain internal and/or translation symmetry, and the internal symmetry, which acts as a finite-depth quantum circuits. This general type of symmetry actions covers many physically relevant cases.

Let us mention that for 1d quantum spin chains, given \(\alpha:G\to\mathscr{G}^{\mathrm{QCA}}\), one can define its anomaly index \(\omega\in\mathrm{H}^{3}(G;\mathrm{U}(1))\) [63], [68], which we review in Appendix 11. In fact, this index can be defined in a similar manner for more general symmetry actions with tails). Anomaly indices in higher dimensions have also been studied in [80], [81] but we will not need such generalizations in the present paper.

2.3 Taking thermodynamic limit↩︎

In this paper we work directly in the infinite-volume setting. One might question the physical relevance of this, since any realistic system will have finite size. The point is that, in order to study phases of matter, it is at the very least necessary to consider the limiting behavior as the system size approaches infinity. The infinite-system framework can be interpreted simply as a convenient way to package this limit.

Specifically, suppose that for each \(m\) we have a finite-size system that occupies a finite subset \(\Gamma_m \subseteq \Lambda\) of the infinite-system lattice \(\Lambda \subseteq \mathbb{R}^d\). We demand the following properties of the sequence \(\{\Gamma_m \}_{m \in \mathbb{N}^*}\):

1. it is increasing: \(\Gamma_{m}\subseteq\Gamma_{m+1}\), \(\forall\,m\in\mathbb{N}^*\).

2. it is exhausting: \(\bigcup_{m=1}^{\infty}\Gamma_{m}=\Lambda\).

Let \(\mathscr{A}_{\Gamma_{m}}\) be the algebra of operators on \(\Gamma_{m}\), consider and consider a sequence of states \(\{\psi_{m}\}_{m\in\mathbb{N}^*}\), where each \(\psi_{m}\) is a state of \(\mathscr{A}_{\Gamma_{m}}\). We say that this sequence of states converges to a limit state \(\psi\) as \(m\to\infty\) if ⁴ \[\begin{align} \lim_{m\to\infty}\psi_{m}(a)=\psi(a),\quad\forall\,a\in\mathscr{A}^{\ell} \end{align}\] Therefore, when we talk about a state in the infinite system framework, the reader is always free, if they choose, to interpret it as a limit of finite-size system states.

Similarly, if we have a sequence of automorphisms \(\{\alpha^{(m)}\}_{m\in\mathbb{N}^*}\) where each \(\alpha^{(m)}\in\mathrm{Aut}(\mathscr{A}_{\Gamma_{m}})\), then we say they converge to a limit automorphism \(\alpha\in\mathrm{Aut}(\mathscr{A}^{\ell})\) if \[\begin{align} \label{eq:strong95limit} \lim_{m\to\infty}\alpha^{(m)}(a)=\alpha(a),\quad\forall\,a\in\mathscr{A}^{\ell}. \end{align}\tag{3}\] Mathematically, this amounts to saying that we work with the strong topology of automorphisms, and 3 describes “strong convergence” of the sequence \(\{\alpha^{(m)}\}_{m\in\mathbb{N}}\) to \(\alpha\). It will be easy to check that

2.4 Summary of main results↩︎

In this section, we summarize our main results in terms of the framework described above. Let us begin with defining strong symmetries in the infinite-volume systems. Recall that, in finite systems, a strong symmetry of a state described by a density matrix \(\rho\) is defined by the condition \(U\rho\propto\rho\). In infinite systems, we instead make the following definition:

Recall the definition of “extension” from Section 2.1. It is straightforward to show that in finite systems, this agrees with the usual definition. For example, the canonical purification of a mixed state \(\rho\) is symmetric under \(\mathbb{I} \otimes U\) if and only if \(\rho\) is strongly symmetric under \(U\).

Next, we discuss how this definition is related to notions of “charge coherence”. Let \(\alpha \colon G \to \mathrm{Aut}(\mathscr{A}^{\ell})\) be a group homomorphism of a compact group⁵ \(G\) into automorphisms of the algebra \(\mathscr{A}^{\ell}\). Note that we do not have to assume \(G\) is represented by QCA’s here.

An operator \(0\not=O \in \mathscr{A}^{\ell}\) is called a charged local operator if it satisfies \[\begin{align} \label{eq:ord95para} \int_G \alpha_g(O)\, \mathrm{d}g = 0 \end{align}\tag{4}\] where \(\mathrm{d}g\) denotes the normalized Haar measure on \(G\). An equivalent way to formulate this is that if one decomposes \(O\) into a sum over irreps of \(G\), there is no weight on the trivial irrep. For notational simplicity, we will henceforth refer to such operator \(O\) simply as a charged operator.

Let us recall that in quantum mechanics, the fidelity between two density operators \(\rho,\sigma\) is defined as \[F(\rho,\sigma):=\!\left({\rm tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^{2},\]

which generalizes the (squared) inner product to mixed states. As shown in [82], the notion of fidelity can be extended to positive linear functionals in infinite-size systems (see Def. 17). One can concretely compute this fidelity between two states \(\psi,\psi'\) on an infinite system according to \[F(\psi,\psi') = \lim_{n \to \infty} F(\rho_{A_n}, \rho_{A_n}'),\] where \(\rho_{A_n}\) and \(\rho_{A_n}'\) are the density matrices describing the reduced state of \(\psi\) and \(\psi'\) respectively on the finite subsystem \(A_n\), and \(\{ A_n\}\) is a sequence of subsystems of increasing size that eventually grow to encompass the entire system (i.e. they satisfy the “increasing” and “exhausting” conditions from Section 2.3). We then propose the following criterion for diagnosing the strong-symmetry breaking based on fidelity.

We note that the parenthetical “(locally)” is arguably redundant, since in this paper we only ever consider (quasi-)local operators; therefore, we will omit it in the rest of this paper. However, we included it here in order to emphasize that we are not referring to global charge coherence, which is not even well-defined in infinite systems anyway. In words, the charge-coherence condition is saying that if we act on the state with an operator that locally creates or destroys charge, then the resulting state has exactly zero overlap with the original state. We note that this “fidelity correlator” was proposed as a diagnostic of SW-SSB in Ref. [83].

It turns out that this charge-coherence condition is equivalent to strong symmetries.

::: {#thm:strong95sym61charge95coherence95main .theorem} Theorem 1 (Theorem 6). A state \(\psi\) is strongly symmetric under a compact symmetry group \(G\) if and only if it is charge-coherent.* :::

It is useful to know whether the property of having a strong symmetry is invariant under dynamical processes. To this end, we introduce a class of unital completely positive (UCP) maps, called bath evolutions, which describe general dynamical processes in quantum many-body systems and generalize finite-depth quantum channels. We then analyze their symmetry properties, showing in particular that a strongly symmetric bath evolution can never map a weakly symmetric state to a strongly symmetric one; see Proposition 4.

Finally, we will also discuss anomaly constraints for strong symmetries, as well as their mixed strong–weak variants. The generalization to strong-weak mixed anomalies is necessary because, as we show in Proposition 1, if a state is clustering⁶ and strongly symmetric under translation, then it must be pure. Therefore, for mixed states, imposing strong translation symmetry is often too restrictive.

Let us now spell out our version of anomaly constraints. Recall that a state \(\psi\) on 1d spin chain is said to satisfy the area law of mutual information if \(I(\Gamma;\Gamma^{c}):=S(\psi\|\psi_{\Gamma}\otimes \psi_{\Gamma^{c}})<{\rm const}\) for any finite interval \(\Gamma\subseteq\Lambda=\mathbb{Z}\), where \(S(\cdot\|\cdot)\) stands for the relative entropy [84]–[86] and \(\psi_{\Gamma}\) means the restriction of \(\psi\) on \(\Gamma\) (a.k.a the reduced density matrix).

Later, we will generalize above theorem to a more general symmetry conditions, called the von Neumann symmetry Def. 16.

2.5 Relation with previous definitions of SW-SSB↩︎

In this paper, we have argued that in infinite systems, the property of a state that was previously referred to as SW-SSB should simply be interpreted as the absence of strong symmetry (see Section 1). Nevertheless, this still raises the question of how the diagnostics for SW-SSB that have previously been proposed [24], [87] relate to our criteria for having strong symmetry in the thermodynamic limit.

An important point is that the previously proposed diagnostics cannot be directly applied in infinite systems. To see this, note that when applied in finite systems, they turn out to be very sensitive to global data that is not reflected in the expectation values of local operators. For example, both the fidelity correlators and Renyi 2-correlators of Ref. [24] give completely different results when applied to the states \(\rho_1\) and \(\rho_2\) of 1 , despite their local indistinguishability. Thus, they cannot correspond to any well defined quantity in the thermodynamic limit.

Nevertheless, we will argue that diagnostics such as those of Refs. [24], [87] do correspond to some interesting aspects of how the the thermodynamic limit is achieved starting from finite systems, as we will now describe. The discussion will also clarify how the “SW-SSB” defined via such diagnostics is related to the absence of strong symmetry in the thermodynamic limit.

Let us first recall how the thermodynamic limit works for regular spontaneous symmetry breaking (not SW-SSB). We can begin with spin chains of size \(m\), and then approach the thermodynamic limit by sending \(m \to \infty\) (Recall our precise characterization of this limit as described in Section 2.3). Let \(G\) be a finite group, and consider a representation \(\alpha^{(m)}\) of \(G\) on the system of size \(m\), which we assume converges to a homomorphism \(\alpha : G \to \mathrm{Aut}(\mathscr{A}^\ell)\) as \(m \to \infty\) (in the sense described in Section 2.3). Now consider a family of local Hamiltonians \(H_{m}\) on the systems of size \(m\) that are symmetric under \(\alpha^{(m)}\), and consider the Gibbs state described the density matrix \[\rho_m= \frac{1}{Z} e^{-\beta H_m}.\] Under suitable assumptions, one expects that \(\rho_m\) will converge (in the sense described in Section 2.3) to an infinite-volume state \(\psi\) that is symmetric under \(\alpha\). However, when we have SSB, one finds that \(\psi\) has long-range order – that is, it has two-point correlation functions that do not decay with distance (in other words, it fails to satisfy the “clustering property”). However, generally we can write \[\psi = \frac{1}{|G|} \sum_{g \in G} \psi_{\mathrm{SSB} }\circ \alpha_{g}\] where \(\psi_{\mathrm{SSB}}\) is a state satisfying the clustering property that, however, fails to be invariant under \(\alpha\).

Now we turn to the analogous statements for SW-SSB. Consider a family of states \(\psi_{m}\) with strong symmetry \(\alpha^{(m)}\). Furthermore, let us assume that \(\psi_m\) converges to some state \(\omega\) as \(m \to \infty\). We assume this time that there is no regular SSB, in which case it is reasonable to assume that \(\psi\) satisfies the clustering property.

We can also consider the canonical purifications \(\widetilde{\psi}_{m}\) of \(\psi_{m}\), which by the assumption of strong symmetry are invariant under \(\mathrm{id}\otimes \alpha^{(m)}\). We expect under suitable conditions that \(\widetilde{\psi}_m\) will converge to a state \(\widetilde{\psi}'\) on the doubled version of the infinite chain⁷, in which case \(\widetilde{\psi}'\) must be invariant under \(\mathrm{id}\otimes \alpha\). However, in general \(\widetilde{\psi}'\) might not obey the clustering property. Thus, by analogy to the SSB case, one might expect that it can be decomposed as a sum \[\widetilde{\psi}' = \frac{1}{|G|} \sum_{g \in G} \widetilde{\psi}_{\mathrm{SWSSB}} \circ (\mathrm{id}\otimes \alpha_{g}), \label{eq:swssb95decomposition}\tag{5}\] where the state \(\widetilde{\psi}_{\mathrm{SWSSB}}\) obeys the clustering property but might not be invariant under \(\mathrm{id}\otimes \alpha\). Note that by restricting to the original system, 5 implies that \[\label{eq:swssb95restricted95sum} \psi = \frac{1}{|G|} \sum_{g \in G} (\widetilde{\psi}_{\mathrm{SWSSB}})_R \circ \alpha_{g},\tag{6}\] where \((\widetilde{\psi}_{\mathrm{SWSSB}})_R\) is the restriction to the original system. In order for \(\psi\) to obey the clustering property, it must be the case that all the states appearing in the sum 6 are equal⁸, and hence that \((\widetilde{\psi}_{\mathrm{SWSSB}})_R\) is invariant under \(\alpha\) and \((\widetilde{\psi}_{\mathrm{SWSSB}})_R = \psi\). Thus, \(\widetilde{\psi}_{\mathrm{SWSSB}}\) is an extension of \(\psi\). Therefore, invoking the results on extensions outlined in Section 2.4, we conclude that if \(\psi\) has the strong symmetry, then \(\widetilde{\psi}_{\mathrm{SWSSB}}\) is in fact invariant under \(\mathrm{id}\otimes \alpha\), and therefore \(\widetilde{\psi}' = \widetilde{\psi}_{\mathrm{SWSSB}}\) obeys the clustering property. Therefore, the violation of the clustering property for \(\widetilde{\psi}'\) is a signature of the absence of strong symmetry in the state \(\psi\). Indeed, this is precisely the diagnostic for “SW-SSB” proposed in Ref. [87]. (We remark that Ref. [87] proved that their criterion for SW-SSB is also equivalent to the original one proposed in Ref. [24] in terms of a fidelity correlator.) Thus we see how the “SW-SSB” of the previous literature is related to the absence of strong symmetry in the thermodynamic limit.

Note that we have not given an argument for the converse statement, i.e.that the strong symmetry being absent in \(\psi\) necessarily implies failure of clustering for \(\widetilde{\psi}'\). This converse statement seems to hold in the most physically reasonable cases, but it is false in general, if one allows the finite-size states to be sufficiently “wild”, as the following counterexample demonstrates. Suppose that we consider an on-site symmetry, and for each \(L\) we choose a “typical” pure state (i.e.choosen from a Haar-random distribution) within the subspace of the Hilbert space comprising the +1 eigenstates of \(U^{(m)}(g)\) (where \(U^{(m)}(g)\) is the unitary action of the symmetry on the Hilbert space). Such typical states look like the maximally mixed state with respect to local operators, so in this case \(\psi_m\) will converge to the maximally mixed state \(\psi\), On the other hand, since each \(\psi_m\) is pure, its canonical purification is simply \(\psi_m \otimes \psi_m^*\) (where \(\psi_m^*\) is the complex conjugation of \(\psi_m\)), which converges to \(\widetilde{\psi}' = \psi \otimes \psi^*\). This state indeed obeys the clustering property, despite the fact that \(\psi\) does not have the strong symmetry.

This example demonstrates the limitations of the previous diagnostics of SW-SSB, since in some cases they fail to correctly identify the lack of strong symmetry in the infinite system limit. Thus, in general it is better to use the new criteria put forward in the present paper.

Finally, let us remark that a similar example to the counter-example discussed above was previously considered in Ref. [32]. However, they drew quite different conclusions from it. They considered a family of states for increasing system sizes \(m\) that fails to satisfy SW-SSB according to the diagnostics of Ref. [24], and therefore characterized it as being “in a different phase” compared to another family of states that does satisfy SW-SSB. However, in our view this is not the best interpretation. Rather, both families in fact converge to the same infinite-volume state in the thermodynamic limit, and therefore should be viewed as corresponding to the same phase, characterized by the absence of strong symmetry in the thermodynamic limit. With this interpretation, the conclusions of Ref. [32] regarding the computational difficulty of distinguishing between phases of matter are no longer applicable. We feel this example also serves to illustrate the danger of focusing on non-local quantities that do not have any well-defined extension to the thermodynamic limit, as it tends to lead to making distinctions that arguably have very little practical relevance (as illustrated by the results of Ref. [32] regarding the computational difficulty of distinguishing between the two families of states).

3 More backgrounds on operator algebras↩︎

3.1 Quasi-local operators↩︎

We will often need to work with operators with tails. To this end, we need to introduce the algebra of quasi-local operators. This algebra is defined to be the completion of \(\mathscr{A}^{\ell}\) with respect to the operator norm: \[\begin{align} \label{eq:quasi95local95algebra} \mathscr{A}:=\overline{\mathscr{A}^{\ell}}^{\|\cdot\|} \end{align}\tag{7}\] The completion procedure is similar to how one obtains \(\mathbb{R}\) from \(\mathbb{Q}\) by adding the limits of Cauchy sequences (see e.g., Ref. [88]). We recall that the operator-norm satisfies: \[\begin{align} \label{eq:Banach-inequality} \| ab\|\leqslant \|a\|\cdot \|b\| \end{align}\tag{8}\] Any norm-complete algebra satisfying Eq. 8 is called a Banach algebra.

Moreover, \(\mathscr{A}\) inherits the \(*\)-operation from \(\mathscr{A}^{\ell}\), and this \(*\)-operation is compatible with the operator norm: \[\begin{align} \label{eq:C42-identity} \|a^{*}a\|=\|a\|^{2}. \end{align}\tag{9}\] In general, a Banach \(*\)-algebra satisfying Eq. 9 is called a \(C^*\)-algebra. The algebra \(\mathscr{A}\) provides an example of such a \(C^*\)-algebra.

It is important to note that there are many physically relevant objects that are not in \(\mathscr{A}\). A prominent example is a finite-range Hamiltonian, which is given by the following formal sum: \[\begin{align} H=\sum_{i\in\Lambda}h_{i} \end{align}\] where each \(h_{i}\) is supported on a neighborhood \(N_{i}\) of \(i\), with diameter \(\mathrm{diam}(N_{i})<r\) for some constant \(r\). Surely \(H\) does not converge in the operator norm. However, we note that \(H\) makes sense as a derivation on local operators: \[\begin{align} \label{eq:derivation} \delta_{H}(a)=\sum_{i\in\Lambda}[h_{i},a],\quad\forall a\in\mathscr{A}^{l} \end{align}\tag{10}\] With the help of Lieb-Robinson bound, it is shown that \(\delta_{H}\) can be exponentiated into a \(*\)-automorphism \(\alpha_{t}\in\mathrm{Aut}(\mathscr{A})\) (called the time evolution of \(\delta_{H}\)), see e.g., Ref. [89] for details. In general, \(\alpha_{t}\) is an outer automorphism of \(\mathscr{A}\), meaning that \(\alpha_{t}\not=\mathrm{Ad}_{u}\) for any unitary \(u\in\mathscr{A}\). Similarly, finite-depth quantum circuits (FDQC) are also outer automorphisms of \(\mathscr{A}\).

States on \(\mathscr{A}\), as well as pure states, are defined in the same way as in Def. 1, so we do not repeat the definition here.

Let us also briefly clarify our convention for tensor products of \(C^*\)-algebras. In general, there are several inequivalent \(C^*\)-tensor products, depending on the choice of completion procedure (see, e.g., Chap. 6 of [90]). In the present paper, however, we work exclusively with spin systems, whose quasi-local algebras are uniformly hyperfinite (UHF). For UHF algebras, the \(C^*\)-tensor product is unique; see Theorem 6.3.11 of [90]. To be specific, throughout this paper, the notation \(\otimes\) always refers to the spatial tensor product, both for \(C^*\)-algebras and for von Neumann algebras (to be introduced in Sec. 3.2 below).

3.2 GNS representations and von Neumann algebras↩︎

In Sec. 3.1, we have defined the quasi-local algebra \(\mathscr{A}\). We also defined the states as positive linear functionals (see Def. 1) as well as their classical mixture. However, we are still missing one of the most important ingredients of quantum mechanics, i.e., the coherent superposition. This can be cured by introducing the so-called Gelfand-Naimark-Segal (GNS) construction.

Recall that in perturbative quantum field theory, the Hilbert space is constructed by applying all creation operators to a reference state (i.e.the vacuum of the theory). In the operator-algebraic setting, the GNS construction replaces the role of creation and annihilation operators with elements of the \(C^*\)-algebra \(\mathscr{A}\).

We first define the GNS ideal associated with a state \(\psi\) on \(\mathscr{A}\): \[\begin{align} N_{\psi} := \{ a \in \mathscr{A}\mid \psi(a^{*}a) = 0 \}. \end{align}\] Elements of \(N_{\psi}\) may be regarded as “annihilation operators” for \(\psi\). Accordingly, vector states in the GNS representation can be identified with equivalence classes in the quotient space \(\mathscr{A}/ N_{\psi}\). If \(a\in\mathscr{A}\), we write \([a]\) for its equivalence class.

The quotient \(\mathscr{A}/ N_{\psi}\) carries a natural inner product given by \[\begin{align} \label{eq:GNS95inner95product} \langle [a], [b] \rangle := \psi(a^{*} b). \end{align}\tag{11}\] One can readily verify that this inner product is well defined. However, \(\mathscr{A}/ N_{\psi}\) is not complete with respect to the norm induced by 11 ; hence, it is only a pre-Hilbert space. The GNS Hilbert space \(\mathcal{H}_{\psi}\) is obtained by completing \(\mathscr{A}/N_{\psi}\) with respect to the inner product Eq. 11 . Besides, we have a representation \(\pi_{\psi}\) of \(\mathscr{A}\) defined by: \[\begin{align} \label{eq:inner95prod} \pi_{\psi}(a)([b]):=[ab], \quad\forall\,a,b\in\mathscr{A} \end{align}\tag{12}\] Lastly, \(|\psi\rangle:=[1_{\mathscr{A}}]\) represents \(\psi\) in \(\mathcal{H}_{\psi}\): \[\begin{align} \langle\psi|\pi_{\psi}(a)|\psi\rangle= \psi(a),\quad\forall\,a\in\mathscr{A} \end{align}\]

Sometimes, it will be convenient to talk about the GNS representation of a general positive linear functional \(\rho\), i.e., it may not be normalized. Note that we always have⁹ \(\rho(1)>0\) as long as \(\rho\not=0\). The GNS triple of \(\rho\) is defined to be the GNS triple of \(\rho(1_{\mathscr{A}})^{-1}\rho\).

The vector \(|\psi\rangle\) is distinguished in the above construction because \(\pi_{\psi}(\mathscr{A})|\psi\rangle\) is dense in \(\mathcal{H}_{\psi}\). Equivalently, \(\mathcal{H}_{\psi}\) is the completion of \(\pi_{\psi}(\mathscr{A})|\psi\rangle\) with respect to the inner product in Eq. 12 : \[\begin{align} \mathcal{H}_{\psi}=\overline{\pi_{\psi}(\mathscr{A})|\psi\rangle}^{\langle\cdot\,|\,\cdot\rangle}. \end{align}\] When no confusion can arise, we suppress explicit reference to the inner product from now on. Such a vector is called cyclic, and a representation admitting a cyclic vector is called a cyclic representation. More generally:

This uniqueness of GNS representations plays an essential role in the discussion of the symmetries of states. For instance, let \(\alpha \in \mathrm{Aut}(\mathscr{A})\) be a \(*\)-automorphism and let \(\psi\) be a state satisfying \(\psi \circ \alpha = \psi\). A natural question is how \(\alpha\) acts on the GNS Hilbert space \(\mathcal{H}_{\psi}\). By the uniqueness of the GNS triple, there exists a unique unitary operator \(U_{\alpha} \in \mathcal{B}(\mathcal{H}_{\psi})\) implementing \(\alpha\), i.e., \[\begin{align} \begin{aligned} U_{\alpha} |\psi\rangle&= |\psi\rangle, \\ \pi_{\psi} \circ \alpha (a) &= U_{\alpha} \pi_{\psi}(a) U_{\alpha}^{\dagger}, \qquad \forall\, a \in \mathscr{A}. \end{aligned} \end{align}\]

Given a representation, it is natural to ask if it is irreducible. The reducibility is closely related to the notion of invariant subspace and subrepresentations, which we will define now.

We have the following classical result

We remark that \(\pi_{\psi}(\mathscr{A})'\) in this lemma is called the commutant of \(\pi_{\psi}(\mathscr{A})\) in \(\mathcal{B}(\mathcal{H}_{\psi})\). In Remark 2, we observed that \(U_{\alpha}\notin \pi_{\psi}(\mathscr{A})\) in general. If \(\psi\) is a pure state, then the double commutant satisfies ¹⁰: \[\pi_{\psi}(\mathscr{A})''=\mathcal{B}(\mathcal{H}_{\psi}).\] Consequently, as long as \(\psi\) is pure, we always have \(U_{\alpha}\in \pi_{\psi}(\mathscr{A})''\).

This suggests that the double commutant \(\pi_{\psi}(\mathscr{A})''\) may be an interesting object in its own right. Indeed, as discussed before, the time evolution \(\alpha_t\) is in general an outer automorphism of \(\mathscr{A}\). However, for a ground state¹¹ \(\psi\), one can show that \(\alpha_t\) can always be implemented by a unitary \(U_t\in \pi_{\psi}(\mathscr{A})''\), even when \(\psi\) is mixed. See Proposition 5.3.19 of Ref. [35] for more details.

Based on above discussion,

vN algebras play an essential role in the discussion of symmetries of mixed-state phases, as we will see later.

Before ending this subsection, we mention a powerful result in the theory of von Neumann algebras, called Kaplansky’s density theorem. To this end, let us define some useful topologies on \(\mathcal{B}(\mathcal{H})\).

To relate von Neumann algebras and these two topologies, we have the following famous result, due to von Neumann himself.

See e.g., theorem I.9.1.1 of Ref. [76] for a proof. This result is remarkably convenient because the double commutant is defined in a purely algebraic way.

The power of SOT and WOT can be seen from the following two results.

Although SOT and WOT are very powerful, one has to be careful when dealing with them.

The next lemma is known as Kaplansky’s density theorem. It will be used later when we discuss the mutual information area law and the split property; see Prop. 2.

This lemma allows us to approximate bounded, self-adjoint or unitary elements in \(\mathscr{M}_{\psi}\) by elements in \(\pi_{\psi}(\mathscr{A})\) in weak-operator topology.

3.3 Equivalences↩︎

Given two representations of \(\mathscr{A}\), it is useful to know if and when they are unitarily equivalent. The following lemma can be helpful in this situation.

See Corollary 2.6.11 of Ref. [34] and Theorem 10.2.6 of Ref. [92] for a proof.

For two pure states \(\psi_{1}\not\simeq\psi_{2}\), we say that they are disjoint or in different superselection sectors. The following result characterizes disjoint states.

Note that \(\|\psi_{1}-\psi_{2}\|\) is nothing but twice of the trace-norm distance in the usual quantum mechanics. Thus Lemma 6 simply says states from different superselection sectors are “orthogonal” to each other.

The unitary equivalence and Lemma 5 are useful for pure states. In order to study mixed states, a slightly weaker equivalence, called quasi-equivalence can be more helpful. In order to define quasi-equivalences, let us define the notion of subrepresentations.

For pure states, quasi-equivalence is nothing but unitary equivalence defined as in Lemma 5. Thus quasi-equivalence indeed generalizes unitary equivalence between pure states. To give a nontrivial example of quasi-equivalent states which are not unitarily equivalent, consider two equivalent pure states \(\omega_{1}\simeq\omega_{2}\), then we have quasi-equivalence \(\omega:=\frac{1}{2}(\omega_{1}+\omega_{2})\sim\omega_{1}\sim\omega_{2}\); See Example. 1 below.

There are other useful characterization for quasi-equivalence between states:

In order to detect subrepresentations, the following lemma can be quite convenient.

It is natural to ask whether a GNS representation can be decomposed into simpler constituents (e.g., , irreducible representations). We emphasize that, in general, such a naive decomposition is impossible. Nevertheless, there exists a closely related structural result, which we describe below.

To this end, we need the notion of “simple” vN algebras, i.e., the factors.

The physical importance of factor states can be seen from

This is one of the most surprising results in the algebraic approach to quantum many-body physics. There is a generalization of Lemma 5 for factor states.

There is a type classification for vN factors, see e.g., Ref. [76]. We briefly recall it below.

From the last example above, we see that the type of a factor can serve as a measure of entanglement. This is indeed the case; see e.g., Ref. [93]–[95] for related discussions. We will return to this point when we discuss the so-called split property.

Mathematically, the relationship between factor states is particularly simple as indicated by the following lemmas.

We are ready for the decomposition theorem of von Neumann algebras.

As a corollary,

4 Strong symmetries and von Neumann (vN) symmetries of States↩︎

In this section, we show how anomalous symmetries can constrain the mixed states with such symmetries. Given a (not necessarily pure or factor) state \(\psi:\mathscr{A}\to\mathbb{C}\), consider \(\alpha\in\mathrm{Aut}(\mathscr{A})\) that leaves \(\psi\) invariant. By the uniqueness of GNS representation (Remark. 1), there exists a unitary operator \(U_{\alpha}\in\mathcal{B}(\mathcal{H}_{\psi})\) such that \[\begin{align} U_{\alpha}\pi_{\psi}(a)U_{\alpha}^{\dagger} = \pi_{\psi}(\alpha(a)),\quad\forall\,a\in\mathscr{A} \end{align}\] This \(U_{\alpha}\) is unique once we impose \(U_{\alpha}|\psi\rangle=|\psi\rangle\) and we say \(U_{\alpha}\) implements \(\alpha\) on \(\mathcal{H}_{\psi}\).

Below we give several equivalent definitions for strong symmetries.

This definition is quite general and concise, but it is not physically intuitive. Indeed, it is not obvious why Def. 15 is equivalent to Def. 5 at all. This equivalence is the content of Theorem 3.

Below, we provide another (equivalent) characterization of strong symmetries based on the purification map introduced by Woronowicz in [96]. To this end, we introduce \(\overline{\mathscr{A}}\), the complex conjugation of \(\mathscr{A}\). Note that there is a canonical anti-linear automorphism \(j\) on \(\overline{\mathscr{A}}\otimes\mathscr{A}\) defined as: \[\begin{align} \label{eq:jpositive} j(\overline{b}\otimes a):=\overline{a}\otimes b \end{align}\tag{13}\] A state \(\tilde{\psi}\) on \(\overline{\mathscr{A}}\otimes\mathscr{A}\) is called \(j\)-positive if \[\begin{align} \tilde{\psi}(\overline{a}\otimes a)\geqslant0,\quad\forall\,a\in\mathscr{A} \end{align}\] It is easy to show that any \(j\)-positive state is \(j\)-invariant. Besides, \(\tilde{\psi}\) is called exact if \[\begin{align} \label{eq:exact} \tilde{\pi}(\overline{1}\otimes \mathscr{A})' = \tilde{\pi}(\overline{\mathscr{A}}\otimes 1)'' \end{align}\tag{14}\] where \(\tilde{\pi}\) is the GNS representation of \(\tilde{\psi}\) and the commutants are taken inside \(\mathcal{B}(\mathcal{H}_{\tilde{\psi}})\).

The canonical purification is one example of purification as is defined in Def. 3 with \(\mathscr{B}= \overline{\mathscr{A}}\).

Let us turn to general extensions.

Below we show the equivalence of definitions for strong symmetries in infinite-volume systems.

Later, we will also be interested in a different symmetry condition that is intermediate between weak and strong symmetries, called the von Neumann symmetry (or vN symmetry for short).

A reasonable symmetry condition is that the symmetry acts separately on the system (i.e., \(\mathscr{M}_{\psi}\)) and the environment (i.e., \(\mathscr{M}_{\psi}'\)). This leads to the following definition:

Let us first compare vN symmetries with strong symmetries as defined in Def. 15. Obviously, any strong symmetry is automatically a vN symmetry with \(v_{\alpha}=1\). Therefore, the notion of vN symmetry is a generalization of strong symmetries. We will discuss the Lieb-Schultz-Mattis type constraints for vN symmetries and its possible physical interpretation in later sections, see Sec. 7.

In fact, vN symmetries beyond strong symmetries are quite common. For example, we have the following:

This follows from the fact that all \(*\)-automorphisms are inner for \(\mathscr{M}_{\psi}\) if \(\psi\) is a type \(I\) factor (see Corollary 9.3.5 of Ref. [92]). In contrast, it is straightforward to construct type \(I\) factors that admit no nontrivial strong symmetries.

In finite systems, every symmetry of a state is automatically a vN symmetry. Indeed, for any finite-dimensional \(C^*\)-algebra \(\mathscr{C}\), every automorphism is inner, i.e. of the form \(\mathrm{Ad}_{u}\) for some unitary \(u\in\mathscr{C}\), which trivially extends to an inner automorphism on \(\mathscr{M}_{\psi}\) for any state \(\psi\).

In fact, we will show the following result for lattice translation symmetry:

As an easy corollary, we have:

This corollary can also be seen from canonical purification, i.e. applying \(\mathrm{id}\otimes \tau^{n}\) to \(\tilde{\psi}\) for sufficiently large \(n\) so one can show \(\tilde{\psi}\) fails to be clustering. Note that all symmetries of a type \(I\) factor is automatically a vN symmetry, from Prop. 1, we deduce that

Below we illustrate the generality of vN symmetries with a simple example drawn from \((1+1)\)-dimensional symmetry-protected topological (SPT) phases. Consider an on-site symmetry action \[\alpha : G \longrightarrow \mathscr{G}^{\mathrm{QCA}}.\] By definition, the action is on-site if \[\alpha_g = \prod_{i \in \mathbb{Z}} \mathrm{Ad}_{\rho(g)}, \qquad \forall\, g \in G,\] where \(\rho\) is a (finite-dimensional) representation of \(G\) acting on each local Hilbert space.

Let \(\psi\) be a pure short-range entangled (SRE) state, meaning that it can be transformed into a pure product state by a finite-time evolution generated by (almost-)local Hamiltonians. Assume that \(\psi\) is invariant under \(\alpha_g\) for all \(g \in G\). Then \(\psi\) defines a \((1+1)\)-dimensional symmetry protected topological (SPT) phase with symmetry \(G\) [2], [9], [12].

Let \(\psi_R := \psi|_R\) denote the restriction of \(\psi\) to the right half-chain, i.e.the reduced state obtained by tracing out the left half-chain. It follows immediately that \(\psi_R\) is invariant under \(\alpha_g^R\) for all \(g \in G\), where \(\alpha^R\) denotes the truncation of \(\alpha\) to the right half-chain.

We now ask the following:

We answer this question with the following theorem:

We remark that the converse of the second assertion does not hold in general: the triviality of the SPT phase of \(\psi\) does not imply that \(\alpha^{R}\) acts as a strong symmetry on \(\psi_{R}\).

To illustrate this, consider \(G=\mathbb{Z}_{2}=\{0,1\}_{+}\), with symmetry action generated by the Pauli operator \(\rho(0)=\mathrm{id},\,\rho(1)=\sigma^{z}\). Since \(\mathrm{H}^{2}(\mathbb{Z}_{2};\mathrm{U}(1))=0\), all \(\mathbb{Z}_{2}\) SPT phases are trivial in this case.

As a concrete counterexample, let \(\psi\) be the product state with spin-up at every site except at \(i=-1,0\), where the two spins form a singlet. In this situation, the restricted state \(\psi_{R}\) fails to be strongly symmetric under \(\alpha^{R}\).

More generally, in Eq. ?? , the unitaries \(u_{\alpha}\) and \(v_{\alpha}\) are defined only up to a phase whenever \(\psi\) is a factor state. Consequently, even if the unitary implementation \(\{U_{g}\}_{g\in G}\) forms a genuine (non-projective) representation of \(G\), the corresponding unitaries \(u_{g}\in \mathscr{M}_{\psi}\) define a projective representation of \(G\).

The associated degree-2 group cohomology class of this projective representation coincides with the SPT index of \(\psi\) introduced in Ref. [9] if \(\psi\) splits.

5 Charge coherence↩︎

In this section, we explain how our definition of strong symmetry is related to a “charge coherence” condition.

First, let us briefly introduce the definition of fidelity between states in the context of operator algebras [82].

In order to see why it reduces to the usual fidelity, we have the following generalized version of Uhlmann’s theorem for fidelity.

Thus, Def. 17 reduces to usual fidelity for finite dimensional \(C^*\) algebras.

The following lemma tells us how the fidelity on the infinite system \(\mathscr{A}\) can be approximated by finite subsystems. Write \(\{\Gamma_{L}\}\) for increasing (i.e., \(\Gamma_{1}\subseteq\Gamma_{2}\dots\)) and exhausting (i.e., \(\Lambda=\bigcup_{L}\Gamma_{L}\)) collection of finite subsets of \(\Lambda\), indexed by their “size” \(L\). We have

Next, we turn to the charge coherence condition and charged operators introduced in Def. 6. We briefly recall the relevant definition below.

Let \(G\) be a finite-dimensional compact Lie group, let \(\mathscr{A}\) be a \(C^*\) algebra, and let \(\alpha : G \to \mathrm{Aut}(A)\) be a continuous group homomorphism¹⁷. Consider a state \(\psi\) on \(\mathscr{A}\) which is weakly symmetric under \(\alpha_{g}\) for each \(g \in G\). We say that \(\psi\) is charge-coherent with respect to \(\alpha\) if \[F(\psi, \psi_{O}) = 0\] for any charged operator \(O \in \mathscr{A}\) such that \(\int_{g \in G} \alpha_g(a)\mathrm{d}g = 0\). Here \(\psi_{O}\) is the (un-normalized) state defined by \[\psi_{O}(b) = \psi(O^{*} b \,O).\]

As a corollary, by lemma 19, we have

6 Split property, separability and mutual information↩︎

In this section, we introduce certain notions to characterize the entanglement behavior of a state. We specialize to 1d quantum spin chains in this section. We stress that we work with 1d spin chains in this and the following section.

Note that if \(\psi\) is a factor state, \(\psi_{L}\) and \(\psi_{R}\) are automatically factor states. The following lemma is useful for characterizing the split property.

We noticed that a similar discussion has already appeared in [102].

We provide another version of the split property.

For a pure state \(\psi\), this condition is equivalent to the Schmidt decomposition between the left and right half-chains. Such a decomposition requires the GNS Hilbert space to factorize as \[\mathcal{H}_{\psi} \simeq \mathcal{H}_{L}\otimes \mathcal{H}_{R}\] As shown in Ref. [95], this factorization holds precisely when a pure state \(\psi\) satisfies the split property or, equivalently, when the half-chain restriction of \(\psi\) generates a type \(I\) factor.

We prove a mixed analog of this observation below.

Now, we relate the split property to correlations and mutual information.

To show above proposition, we need several lemmas. The first lemma below is a generalization of Lemma 4 for projections in \(C^*\)-algebra which has real rank¹⁸ 0, which will be defined below. Write \(\mathscr{C}_{sa}\) for the set of self-adjoint elements of \(\mathscr{C}\).

In particular, our quasi-local algebra \(\mathscr{A}\) can be shown to have real rank 0, see Ref. [76] for more details.

This lemma should be well-known to experts (see e.g. [95]), but we include a proof here since we are unaware of a proof in the literature.

7 Lieb-Schultz-Mattis (LSM) constraints in 1d↩︎

In this section, we specialize to one-dimensional quantum spin chains. We begin with the LSM constraints for strong symmetries, which are technically simpler, and then extend the argument to the more subtle case of vN symmetries. In the main text, we restricted attention to symmetry actions implemented by QCAs. Here, however, we allow the more general class of symmetry actions implemented by locality-preserving automorphisms \(\mathscr{G}^{lp}\)

see Def. 29 for the precise definition.

Let us first recall the pure version of the LSM constraint in Ref. [63], [69].

It is straightforward to generalize it to strong symmetries.

Similarly, we can deduce a generalization of the strong-weak mixed version of LSM, reported in Ref. [105].

Below we prove the LSM type constraints for anomalous vN symmetries.

We need the following lemma:

As a corollary, we have

Let us give another perspective from lifting theory below; this approach will be useful in establishing further generalization of Theorem 8 as in Corollary 7.

For any linear (unnecessarily vN!) symmetry action \(\alpha:G\to \mathscr{G}^{lp}\), as before, we cancel GNVW indices and one always has unitary \(U_{g}\) on \(\mathcal{H}_{R}\) such that \(\mathrm{Ad}_{U_{g}}\) implements \(\alpha_{g}^{R}\). Let us claim that \(\mathrm{Ad}_{U_{g}}\) gives a group homomorphism \(G\to \mathrm{Out(\mathscr{M}_{R}})\). To this end, we note \[\begin{align} \begin{aligned} \mathrm{Ad}_{U_{g}}\circ\mathrm{Ad}_{U_{h}}(\mathscr{M}_{R})&= \mathrm{Ad}_{\pi_{\psi}(V_{g,h})}\circ\mathrm{Ad}_{U_{gh}}\circ\mathrm{Ad}_{\eta_{g,h}}(\mathscr{M}_{R}) \end{aligned} \end{align}\] Note that \(\pi_{\psi}(V_{g,h})\in\mathscr{M}_{R}\) is an inner-automorphism of \(\mathscr{M}_{R}\) and \(\eta_{g,h}\in\mathscr{M}_{R}'\) (hence its conjugation is trivial on \(\mathscr{M}_{R}\)). Therefore, \(g\to \mathrm{Ad}_{U_{g}}\) gives a well-defined map from \(G\) to \(\mathrm{Out}(\mathscr{M}_{R})\). We are interested in the following lifting problem: \[\begin{tikzcd} & {\mathrm{Aut}(\mathscr{M}_{R})} \\ G & {\mathrm{Out}(\mathscr{M}_{R})} \arrow[from=1-2, to=2-2] \arrow[dashed, from=2-1, to=1-2] \arrow[from=2-1, to=2-2] \end{tikzcd}\]

We note that closely related lifting problems have already been studied in Refs. [106], [107]. A classical result states that, if such a lifting exists, then the associated obstruction class in \(\mathrm{H}^{3}_{\sigma}(G; Z(\mathscr{U}_{R}))\) must vanish, where \(\mathscr{U}_{R}\) is the group of unitary operators in \(\mathscr{M}_{R}\), \(Z(\mathscr{U}_{R})\) denotes its centralizer, \(\sigma\) denotes the induced \(G\)-action on \(Z(\mathscr{U}_{R}) := \mathscr{U}_{R} \cap \mathscr{M}_{R}'\). See Theorem 7.1.2 of [108].

The new content of our lemma is that, when \(\psi\) is a factor state (i.e. \(Z(\mathscr{U}_{R})=\mathrm{U}(1)\)), the above obstruction can be identified with the previously defined anomaly index \(\omega\in\mathrm{H}^{3}(G;\mathrm{U}(1))\); see Eq. 31 . A priori, these two obstruction classes need not coincide.

The merit of this reformulation is to simplify the proof to the following corollary, which generalizes Corollary 5.

This corollary is useful since Prop. 1 shows translation can never be a vN symmetry unless the factor state \(\psi\) is actually pure. In application, \(G\) is often taken to be the lattice translation symmetry while \(H\) is some internal symmetry that is vN.

8 Bath evolution and locally generated channels↩︎

In quantum mechanics, a general dynamical process is not necessarily given by a unitary evolution, but more generally by a quantum channel [109]; see that reference for the precise definition. The Stinespring theorem states that every quantum channel on a finite system can be realized as a unitary evolution after enlarging the system by ancillary degrees of freedom.

In infinite systems, the analog of quantum channels are unital, completely positive (UCP) maps on the quasi-local algebra [109]. However, general UCP maps do not necessarily have a clear physical interpretation as physically achievable dynamical processes, and will be too general for our purposes. Instead we will consider a subset of UCP maps, which we refer to as “bath evolutions”.

Physically, a bath evolution describes process in which the system is coupled to a “bath”, initially in the state \(\Phi_\mathscr{B}\), then the two systems evolve unitarily together, and finally the bath is “traced out”. Observe that the Stinespring dilation theorem implies that every UCP map on a finite system is a bath evolution; however the same result is not expected to hold in infinite systems.

The following proposition shows that the composition of bath evolutions is again a bath evolution.

Next, we discuss the symmetry property, especially the strong symmetry of a bath evolution.

Physically, a strongly symmetric bath evolution describes a process in which the system does not exchange charge with the bath.

The following proposition describes how the symmetry properties of a state transform under a strongly symmetric bath evolution.

In other words, a strongly symmetric bath evolution cannot take a state with weak symmetry to one with strong symmetry. However, applying a strongly symmetric bath evolution \(\mathscr{E}\) to a strongly symmetric state \(\psi\), the resulting state \(\psi\circ\mathscr{E}\) can be weakly symmetric.

As an example of this phenomenon, consider the case of a 1-D infinite chain of spin-1/2’s, which initially in the state where each spin is in the +1 eigenstate of the Pauli \(\sigma^x\). Now we consider a bath evolution where the “bath” \(\mathscr{B}\) is also a 1-D infinite chain of spin-1/2’s, offset by half a lattice spacing, as shown in Figure [fig:bath95evolution], and is also initially taken to be in the state where each eigenstate is in the +1 eigenstate of \(\sigma^x\). The system-bath coupling involves applying a controlled-Z gate between each system spin and its two nearest neighbors in the bath, as also shown in Figure [fig:bath95evolution]. The combined state of the system and bath is now a 1-D cluster state, and tracing out the bath gives (with respect to local operators) the maximally mixed state on the system. The bath evolution and the initial state of the system are both strongly symmetric under the \(\mathbb{Z}_2\) symmetry generated by \(\prod_i \sigma_x^i\), but the bath evolution takes this initial state to one with only weak symmetry. This can be viewed as a special case of the strongly symmetric dephasing channel discussed in Ref. [24].

We remark that the 1-D cluster state shared between system and bath serves an example of a purification of the maximally mixed state which is different from the canonical one. In fact, a bath evolution which sends a strongly symmetric state to a state without the strong symmetry necessarily requires that any purification of the state of the combined system-bath just before tracing out the bath be a non-canonical purification of the final state of the system (with the bath traced out). In finite systems, all purifications are unitarily equivalent to the canonical purification, so it follows that in finite systems, unlike infinite systems, strongly symmetric channels always preserve strong symmetry.

As an easy but important corollary of Proposition 4:

For many practical purposes, such as the definition of mixed-state phases, bath evolutions are still too general. To obtain better control over locality, we introduce the notion of locally generated channels. To this end, we first recall the concept of locally generated automorphisms (LGAs), introduced in Refs. [12], which are used in the definition of gapped phases for pure states.

In Ref. [110], it was shown that any continuous path \(F:[0,1]\to \frak{D}^{al}\), namely a continuously time-dependent almost-local Hamiltonian, can be exponentiated, using the Lieb-Robinson bound, to a strongly continuous family of \(*\)-automorphisms \(\{\gamma_{s}\}_{s\in[0,1]}\), which describes the corresponding time evolution. This leads to the notion of locally generated automorphisms (LGA), as well as its counterpart in open quantum systems, namely locally generated channels (LGC):

One major application of LGC is defining phases of mixed-states.

Below, we show that, in 1d spin chains, if a state \(\psi\) has some anomalous symmetry \(\alpha:G\to\tilde{\mathscr{G}}^{lp}\) as its vN symmetry, then \(\psi\) cannot be locally generated from any product state. In particular, such a state cannot fall into the trivial phase.

9 Discussion↩︎

What is a phase of matter? In this paper we have put forward the view that it should always be characterized via the expectation values of local operators in the thermodynamic limit. In particular, we investigated the symmetry properties of mixed states from the perspective of (quasi-)local operators. This perspective is advantageous both practically, since local operators are more accessible experimentally and numerically, and theoretically, since phases of matter are properly defined only in the thermodynamic limit. In particular, the charge coherence condition (Def. 6) yields a more robust and mathematically rigorous formulation for distinguishing and diagnosing phases of matter previously described in terms of “SW-SSB”.

Note that, while we chose to focus our presentation on spin systems for concreteness, many of our results are in fact much more general. The definition of strong symmetry, as well as most of the core results (such as the relation with charge coherence, and the constraints on strongly symmetric bath evolutions) in fact hold in any quantum system in which states can be characterized as positive linear functionals on a (separable) \(C^*\)-algebra. This can include, for example: classical systems (which correspond to commutative \(C^*\)-algebras); quantum field theories, when formulated in the framework of algebraic quantum field theory (AQFT) [33]; constrained spin systems where the operator algebra fails to be a tensor product over sites (e.g.lattice gauge theories); lattice systems where the Hilbert space on each site is infinite (e.g. lattices of harmonic oscillators); and systems with non-local interactions such as the Sachdev-Ye-Kitaev (SYK) model (as long as there is still a well-defined thermodynamic limit).

In addition, mutual information plays an important role in our derivation of Lieb–Schultz–Mattis-type constraints for mixed states. It would therefore be interesting to explore how other information-theoretic quantities, such as conditional mutual information (CMI) can be used to characterize mixed states in the thermodynamic limit [111]–[113].

Note added.– in the course of preparing this work, we became aware of the parallel work Ref. [83]. While several of our results and definitions of strong symmetry were arrived at independently of Ref. [83], we learned of the “fidelity correlator” discussed in Sections 2.4 and 5 from the authors of Ref. [83], after which we were able to prove equivalence with our other definitions.

Acknowledgments↩︎

We thank Chong Wang, Leonardo A. Lessa and Francisco Divi for helpful discussions. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges, Universities and Research Excellence. RL is also supported by the Simons Collaboration on Global Categorical Symmetries through Simons Foundation grant 888996. JY is also supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

10 A review on group cohomology↩︎

In this section, we review the basics of group cohomology and differentiable group cohomology. For group cohomology, there are many materials in the literature [2], [114]–[117]. See also appendix A.1 of Ref. [63] and Ref. [118] for the version for Lie groups. We will only cover the motivations and basics here.

10.1 Projective representations in quantum mechanics↩︎

To motivate group cohomology, we start with projective representations in quantum mechanics. Suppose we have a symmetry group \(G\) (assumed to be unitary and discrete for simplicity) acting on a Hilbert space \(\mathcal{H}\). Usually this symmetry action is given by a homomorphism \(\rho:G\to U(\mathcal{H})\), i.e., a unitary representation of \(\mathcal{H}\). More explicitly, for each \(g\in G\), we assign a unitary operator \(\rho(g)\) such that \[\begin{align} \rho(g)\rho(h)=\rho(gh),\quad\forall\,g,h\in G \end{align}\] However, in quantum mechanics, states are not really a vector in \(\mathcal{H}\), but a ray. That means a state \(|\psi\rangle\) is the same as \(e^{i\theta}|\psi\rangle\) as a quantum state. Thus, the space of states is not literally \(\mathcal{H}\), but the projective space \(P(\mathcal{H})\). This for allows more general symmetry actions as \[\begin{align} \rho(g)\rho(h)=\omega(g,h)\rho(gh) \end{align}\] where²² \(\omega(g,h)\in \mathrm{U}(1)\). This \(\rho\) is a representation up to a phase \(\omega\) and is called a projective representation. Moreover, the matrix multiplication is associative, so \[\begin{align} (\rho(g)\rho(h))\rho(k)=\rho(g)(\rho(h)\rho(k)) \end{align}\] This imposes the following constraint on \(\omega\), \[\begin{align} \label{eq:2-cocycle95condition} \omega(g,h)\omega(gh,k)=\omega(g,hk)\omega(h,k) \end{align}\tag{26}\] Any function \(G\times G\to \mathrm{U}(1)\) satisfying Eq. 26 is called a 2-cocycle. Furthermore, one can redefine the phase of \(\rho(g)\to \tilde{\rho}(g)=\rho(g)\eta(g),\eta(g)\in\mathrm{U}(1)\) (we do not require \(\eta:G\to \mathrm{U}(1)\) to be a homomorphism), and the resulting 2-cocycle is \[\begin{align} \label{eq:shift952-coboundary} \tilde{\omega}(g,h)=\omega(g,h)\eta(g)\eta(h)\eta(gh)^{-1} \end{align}\tag{27}\] One can easily check that \(\tilde{\omega}\) again satisfies the 2-cocycle condition, Eq. 26 . If there exists \(\eta(g)\) such that \(\tilde{\omega}(g,h)=1\) for all \(g,h\in G\), then we say that \(\omega\) is a 2-coboundary or trivial. Any two 2-cocycles \(\omega\) and \(\tilde{\omega}\) related by Eq. 27 are viewed as equivalent, since they differ only by the artificial choice of phase factors \(\eta(g)\) of representation matrix \(\rho(g)\). We write \(\omega\sim \tilde{\omega}\) if \(\omega\) and \(\tilde{\omega}\) are equivalent. The space of 2-cocycles modulo this equivalence \(\sim\) is the so-called the degree 2 group cohomology of \(G\), denoted by \(\mathrm{H}^{2}(G;\mathrm{U}(1))\).

A projective representation provides the following constraint on quantum states.

As a corollary, consider a \(G\)-symmetric Hamiltonian \(H\) which has a \(G\) symmetry that acts projectively. We have

This can be viewed as \((0+1)d\) version of anomaly constraints.

10.2 Group cohomology↩︎

Now we present the definition of group cohomology in general. Let \(G\) be a discrete group, one defines a space \(BG\) which is a collection of spaces \(\{G^{n}\}_{n=1,2,...}\) equipped with a collection of maps \(d_{k}:G^{n}\to G^{n-1},\,k=0,1,...,n\) (called face maps). Explicitly, \[\begin{align} \label{eq:face95maps} d_{k}(g_{1},g_{2},...,g_{n})=\begin{cases} (g_2,...,g_n),\,k=0\\ (g_1,...,g_{k}g_{k+1},...,g_{n}),\,0<k<n\\ (g_1,...,g_{n-1}),k=n \end{cases} \end{align}\tag{28}\] One can check that if \(d=\sum_{k=0}^{n}(-1)^{k}d_{k}\), then \(d^{2}=0\). Let \(A\) be an Abelian group (with discrete topology). For example \(A\) can be \(\mathbb{Z}_{2}\), \(\mathbb{Z}\), \(\mathbb{R}\) or \(\mathrm{U}(1)\). We denote all \(A\)-valued functions on \(BG\) as \(C^{\bullet}(BG,A)\). For example, one writes \(\omega\in C^{2}(BG,A)\) if \(\omega:G^{2}\to A\). Consider an \(A\)-valued function \(\omega\) on \(G^{n-1}\). The maps \(d_{k}:G^{n}\to G^{n-1}\) induces a pullback of \(\omega\), i.e., \(d_{k}^{*}\omega:=\omega\circ d_{k}\) on \(G^{n}\). We denote \(\delta=d^{*}\) (it follows that \(\delta^{2}=0\)), thus \(C^{\bullet}(BG,A)\) together with \(\delta\) becomes a cochain complex.

Group cohomology of higher degrees are used to classify ’t Hooft anomalies in physics. We will explain this in some more details in Sec. 2.2.

11 Defining anomaly indices↩︎

In this section, we outline the construction of anomaly index of a symmetry action \(\alpha:G\to\mathscr{G}^{\mathrm{QCA}}\) in spin chains therefore we exclusively work with \(\Lambda\simeq\mathbb{Z}\) in this appendix.

Given a symmetry group \(G\), by slightly abusing the notations²⁴, the symmetry action can be represented by a group homomorphism \(\alpha: G\to \mathscr{G}^{\mathrm{QCA}}\). This symmetry may contain internal and/or translation symmetry, and the internal symmetry, which acts as a finite-depth quantum circuit, may be discrete or continuous, on-site or non-on-site. This general type of symmetry actions covers many physically relevant cases.

First, suppose \(\alpha\) is an internal symmetry action (i.e., it contains no translation). For an arbitrary site, say, the origin, it can be shown that \(\alpha\) can be decomposed as \[\begin{align} \label{eq:32decomposition} \alpha=\alpha^{L} \, \alpha_0 \, \alpha^{R} \end{align}\tag{29}\] where \(\alpha^{R}\) (resp. \(\alpha^{L}\)) is an operation of local operators supported on \([0, \infty)\) (resp. \((-\infty, 0)\)), and \(\alpha_0\) is the conjugation by a local unitary. Although \(\alpha\) is a group homomorphism, in general \(\alpha^{R}\) is not. In fact, for any \(g, h\in G\), \[\begin{align} \label{eq:composition95half95chain} \alpha^{R}_{g} \, \alpha^{R}_{h}=\mathrm{Ad}_{V_{g,h}} \, \alpha^{R}_{gh} \end{align}\tag{30}\] where \(V:G\times G\to \mathscr{U}^{\ell}\) with \(\mathscr{U}^\ell\) the group of local unitaries is not necessarily a homomorphism, and \(\mathrm{Ad}_V(a):=VaV^{*}\) for any \(a\in\mathscr{A}\). The associativity of \(\alpha^{R}\), i.e., \(\left( \alpha^{R}_{g} \, \alpha^{R}_{h} \right) \, \alpha^{R}_{k} = \alpha^{R}_{g} \, \left( \alpha^{R}_{h} \, \alpha^{R}_{k} \right)\) with \(g, h, k\in G\), puts further constraints on \(V\) \[\begin{align} \label{eq:anomaly95index} \mathrm{Ad}_{\omega_{g,h,k}}=1,\quad \omega_{g,h,k} :=V_{g,h}V_{gh,k}V_{g,hk}^{-1}\alpha^{R}_{g}(V_{h,k})^{-1} \end{align}\tag{31}\] This means the above \(\omega\) is actually a phase since it commutes with all local operators. It can be checked that \(\omega\) satisfies the 3-cocycle condition, and multiplying \(V_{g,h}\) by a phase \(\rho_{g,h}\in \mathrm{U}(1)\) shifts \(\omega\) by a 3-coboundary \(\delta\rho\). Therefore, \(\omega\) specifies an element in \(\mathrm{H}^3(G, \mathrm{U}(1))\), and this element is defined as the anomaly index associated with the symmetry action \(\alpha\), see appendix 10 for a review of group cohomology. See also Ref. [68] for more detailed discussions.

If \(\alpha\) contains translation, one can stack the system with another copy on which the translation acts oppositely [63]. The symmetry action on this composite system (denoted by \(\alpha_{\otimes}\)) contains no translation, and the anomaly index of \(\alpha\) is defined to be the anomaly index of \(\alpha_{\otimes}\).

We comment on some possible generalizations of the anomaly index. Firstly, one can allow tails by considering the locality-preserving automorphisms introduced in Ref. [63], [122]. In more details,

Clearly, QCA’s are special cases of LPA’s, corresponding to the strictly local situation in which the tail function satisfies \(f_{\alpha}(r)=0\), \(\text{for all } r>r_{\alpha}\). For this reason, we work exclusively with LPAs in the remainder of this paper.

When \(G\) is a Lie group, it is natural to require the symmetry action to be smooth while \(\mathscr{G}^{lp}\) does not carry a smooth structure. This requirement can be met by restricting to almost-local LPAs, namely those whose tail functions satisfy \(f_{\alpha}(r)=O(r^{-\infty})\). The subgroup of almost local LPA can be equipped with a smooth structure, see Ref. [63] for a detailed discussion. Throughout this work, whenever \(G\) is a Lie group, we implicitly assume we are using almost local LPA without further comment.

In all of the above generalizations, the anomaly indices are defined in a similar manner.

References↩︎

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1. More precisely, we work with its lattice version.↩︎

2. By local operator, we mean more precisely that the size of the support of the operator does not need to scale with the system size.↩︎

3. Previously the notation \(\alpha\) is used to represent an operation acting on operators, but here we use it to represent a map from the symmetry group \(G\) to all possible QCA operations \(\mathscr{G}^{\mathrm{QCA}}\).↩︎

4. Mathematically, we work with weak-* topology of states.↩︎

5. This includes finite groups as special cases.↩︎

6. That is, it exhibits no long-range order; see Lemma 9 for a precise definition.↩︎

7. We denote this limiting state \(\widetilde{\psi}'\) rather than \(\widetilde{\psi}\) to distinguish it from the canonical purification of \(\psi\) (using the infinite-system version of canonical purification defined in later sections), with which it does not necessarily coincide.↩︎

8. Technically there could be another possibility, which is that the states appearing in the sum 6 could differ in a manner that goes to zero at spatial infinity. However, this would not be compatible with translation invariance, for example.↩︎

9. To see this, note \(a^{*}a\leqslant \|a\|^{2}\times 1_{\mathscr{A}}\), if \(\rho(1_{\mathscr{A}})=0\) we would have \(\rho(a^{*}a)\leqslant \|a\|^{2}\rho(1_{\mathscr{A}})=0\) hence \(\rho\) is identically 0.↩︎

10. Note that \(\pi_{\psi}(\mathscr{A})'''=\pi_{\psi}(\mathscr{A})'\) automatically holds, so there will be nothing new by taking further commutants.↩︎

11. Given a local Hamiltonian \(H\) (viewed as a derivation), a state \(\psi\) is called a ground state if \(-i\,\psi(a^*\delta_H(a))\ge 0\) for all \(a\) in the domain of \(\delta_H\). Such a state can be mixed in general.↩︎

12. A trace \({\rm tr}\) on \(\mathscr{M}_{\psi}\) is a positive linear functional on \(\mathscr{M}_{\psi}\) whose value can diverge, satisfying \({\rm tr}(a^* a)={\rm tr}(a a^*)\) for any \(a\in\mathscr{M}_{\psi}\). It is called finite if it values in \(\mathbb{R}\) and called semi-finite if it is finite only on a nontrivial subset of \(\mathscr{M}_{\psi}\).↩︎

13. This is a continuous version of direct sum.↩︎

14. The uniqueness is proved in Ref. [97].↩︎

15. More precisely the canonical extension if \(\psi\) is not a factor.↩︎

16. In Ref. [82], the notation \(P_{\mathscr{A}}(f_{1},f_{2})\) is used for fidelity↩︎

17. The topology on \(\mathrm{Aut}(\mathscr{A})\) is specified by Eq. 3 . This is called strong topology in mathematical literature.↩︎

18. Note that the lemma fails if one removes the condition of real rank 0, See [104] for a counterexample.↩︎

19. This means they fit into the following short exact sequence \(1\hookrightarrow H\to \tilde{H}\twoheadrightarrow G\to 1\) and we do not assume this is a central extension.↩︎

20. As noted before, \(\mathscr{A}\) is a UHF algebra and therefore nuclear, so the tensor product \(\mathscr{A}\otimes\mathscr{B}\) is well defined even when \(\mathscr{B}\) is a general \(C^*\) algebra [90].↩︎

21. This means \(f(r)\) is non-negative, decreasing and \(\lim_{r\to\infty}r^{n}f(r)=0\) for any \(n\in\mathbb{N}\).↩︎

22. In principle, one has to show that the phase \(\omega(g,h)\) is the same on each quantum state. This relies the coherence of these states and one can find the proof in Sec. 2.2 of Ref. [119].↩︎

23. Actually, this a subtler case because \(SO(3)\) is a Lie group so it requires more careful treatment, which will be left to later sections. We omit this subtlety for now.↩︎

24. Previously the notation \(\alpha\) is used to represent an operation acting on operators, but here we use it to represent a map from the symmetry group \(G\) to all possible QCA opetations \(\mathscr{G}^{\mathrm{QCA}}\).↩︎