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# Three dimensional Mirrors from IR N-ality

Anindya Dey<EMAIL_ADDRESS>Department of Physics and Astronomy,
Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA

###### Abstract

We discuss an explicit field theory construction of three dimensional mirrors
for a large sub-class of quiver gauge theories involving unitary and special
unitary gauge nodes with matter in fundamental and bifundamental
representations. For this sub-class of theories, one can deploy a sequence of
IR dualities and certain Abelian field theory operations to reduce the
aforementioned problem to a problem of finding the 3d mirror of a simpler
unitary quiver gauge theory. We illustrate the construction with a linear
quiver consisting of unitary and special unitary gauge nodes.

Introduction. 3d ${\cal N}=4$ theories have a rich landscape of IR dualities,
the most well-known of them being mirror symmetry Intriligator and Seiberg
(1996); de Boer  _et al._ (1997). This duality acts on a pair of quiver gauge
theories, which are distinct in the UV in terms of the gauge groups and the
matter content but flow to the same SCFT in the IR, by exchanging the Coulomb
branch (CB) of one theory with the Higgs branch (HB) of the other (and vice-
versa) in the deep IR. Mirror symmetry is the supersymmetric parent of a very
large class of non-supersymmetric boson/fermion-type dualities which arise
from the former by a soft supersymmetry-breaking mechanism Kachru  _et al._
(2016). Historically, the construction of mirror duals has relied heavily on
the realization of these theories in terms of brane systems in String Theory
and deploying String dualities. The Hanany-Witten Hanany and Witten (1997)
construction in the Type IIB String Theory is the earliest avatar of this
procedure. For a large class of quiver gauge theories, 3d mirrors have also
been constructed via the magnetic quiver approach which makes use of five-
dimensional brane webs Bourget  _et al._ (2020, 2022).

Recently, there have been renewed efforts to probe 3d mirror symmetry from an
exclusively field theory perspective Dey (2021, 2022a); Hwang  _et al._ (2020,
2022). In particular, Dey (2021) focussed on constructing mirror duals for
quiver gauge theories with unitary gauge nodes given by a generic graph,
provided the theories are good in the Gaiotto-Witten sense Gaiotto and Witten
(2009). While these theories generically do not have a known string theory
realization, one can construct the 3d mirrors of a large subclass in a
systematic fashion starting from the well-understood duality of linear
($A$-type) quivers, using a generalized version of the $S$-operation Witten
(2003) on a 3d CFT.

In this paper, we present a field theory construction of 3d mirrors for quiver
gauge theories with unitary as well as special unitary gauge nodes and
hypermultiplets in the fundamental/bifundamental representations, as shown in
Fig. 1. Our algorithm does not rely on any String Theory realization of these
theories, and may be applied to quiver gauge theories given by generic graphs,
as long as the theories are good or ugly in the Gaiotto-Witten sense. The
diagnosis of bad theories in this class of quivers and the construction of
associated 3d mirrors will be discussed in a separate paper.

$N_{1}$ $N_{2}$ $N_{\alpha}$ $N_{\alpha+1}$ $N_{\alpha+2}$ $M_{\alpha+2}$
$M_{1}$ $M_{2}$ $M_{\alpha}$ $M_{\alpha+1}$

Figure 1: A generic quiver with unitary/special unitary gauge nodes and
(bi)fundamental matter. Black and yellow circular nodes denote unitary and
special unitary gauge nodes respectively, black boxes denote fundamental
hypers, and black lines connecting two gauge nodes denote bifundamental
hypers.

For a generic quiver of the form given in Figure 1, a unitary gauge node
$U(N_{\alpha})$ is balanced if the balance parameter
$e_{\alpha}=N^{\alpha}_{f/bf}-2N_{\alpha}=0$, where $N^{\alpha}_{f/bf}$ is the
total number of fundamental and bifundamental hypers associated with the node.
In contrast, a special unitary node $SU(N_{\alpha})$ is balanced if the
balance parameter $e_{\alpha}=N^{\alpha}_{f/bf}-2N_{\alpha}=-1$. A quiver with
at least one balanced special unitary node has an emergent 0-form global
symmetry on the CB Dey (2022b, c, 2023) in the IR. For such a theory, the rank
of the IR symmetry and that of the UV-manifest symmetry (realized as
topological symmetries associated with the unitary gauge groups) are
different, and the IR SCFT has mass deformations which are not visible in the
UV Lagrangian.

In Dey (2022c, 2023), such theories were shown to admit a sequence of IR
dualities which are distinct from 3d mirror symmetry. Each of these IR
dualities maps the CB and the HB of one theory to the respective CB and the HB
of the dual theory in the deep IR. Therefore, the duality sequence leads to a
set of quiver gauge theories which flow to the same IR SCFT and have
isomorphic CBs and HBs in the deep IR. This phenomenon was referred to as IR
N-ality. For a quiver gauge theory ${\cal T}$ of the form given in Fig. 1 with
one or more balanced special unitary nodes, it was also shown that there
exists a preferred N-al theory ${\cal T}_{\rm maximal}$ (generically not
unique) for which the rank of the UV-manifest CB symmetry is maximized. If the
quiver ${\cal T}$ is given by a tree-graph (i.e. without any loops), the rank
of this maximal UV-manifest CB symmetry is equal to the rank of the emergent
CB symmetry. The theory ${\cal T}_{\rm maximal}$ will play a crucial role in
our construction of the 3d mirror of ${\cal T}$.

$N_{1}$ $N_{2}$ $N_{\alpha}$ $N_{\alpha+1}$ $M_{\alpha\,\alpha+1}$
$N_{\alpha+2}$ $M_{\alpha+2}$ $F$ $M_{1}$ $M_{2}$ $M_{\alpha}$
$(\widetilde{Q}^{1},\widetilde{Q}^{2})$ $P$ $Q^{1}$ $Q^{2}$ $Q^{\alpha}$

Figure 2: A generic quiver with unitary/special unitary gauge nodes, with
(bi)fundamental and Abelian hypermultiplets. The charges $\\{Q^{i}=\pm
N_{i}\\}$ encode whether the Abelian hypermultiplet transforms in the
determinant or the anti-determinant representation of a given unitary gauge
node with $N_{i}$ being the rank of the unitary gauge node $i$.

As discussed in Dey (2022c, 2023), the theory ${\cal T}_{\rm maximal}$ (as
well other theories N-al to ${\cal T}$) is generically not of the type shown
in Fig. 1. In addition to unitary/special unitary gauge nodes and
fundamental/bifundamental matter, the theory ${\cal T}_{\rm maximal}$ has
hypermultiplets that transform in powers of the determinant and/or the anti-
determinant representations of the unitary gauge nodes. We will collectively
refer to these matter multiplets as Abelian hypermultiplets. The quiver
notation for a generic theory in this class is shown in Fig. 2. A blue square
box with label $F$ represents $F$ hypermultiplets in the determinant
representation of the unitary gauge node. A thin blue line connecting two or
more unitary gauge nodes is an Abelian hypermultiplet associated with those
gauge nodes, while a thick blue line with a label $P$ denotes a collection of
$P$ Abelian hypermultiplets.

Quiver Mutations and IR Duality Sequence. Let us now briefly review the
construction of the theory ${\cal T}_{\rm maximal}$ given a quiver gauge
theory ${\cal T}$ with at least one balanced special unitary gauge node. For
more details on the construction, we refer the reader to Dey (2022c).

$N_{\alpha_{2}}$ $N_{\alpha}$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$
$N_{\alpha_{4}}$ $({\cal T})$
---
${\cal O}_{I}$
$N_{\alpha_{2}}$ $N_{\alpha}-1$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$
$N_{\alpha_{4}}$ $({\cal T}^{\vee})$
$N_{\alpha_{2}}$ $N_{\alpha}$ $N_{\alpha_{3}}$ $N_{\alpha_{1}}$
$N_{\alpha_{4}}$ $\bm{Q}$ $({\cal T})$
${\cal O}_{III}$
$N_{\alpha_{2}}$ $N_{\alpha}-1$ $N_{\alpha_{3}}$ $1$ $1$ $N_{\alpha_{1}}$
$N_{\alpha_{4}}$ $(1,\bm{Q}^{\prime})$ $({\cal T}^{\vee})$
Figure 3: The mutation ${\cal O}_{I}$ and the mutation ${\cal O}_{III}$ (for
$P=1$).

The theory ${\cal T}_{\rm maximal}$ can be obtained from the quiver ${\cal T}$
by sequentially implementing a pair of quiver mutations, which we will refer
to as mutation $I$ and mutation $III$ (and denote as ${\cal O}_{I}$ and ${\cal
O}_{III}$ respectively in quiver diagrams). The former, shown in the top block
of Figure 3, involves replacing a balanced $SU(N_{\alpha})$ node by a
$U(N_{\alpha}-1)$ node and a single Abelian hypermultiplet with the following
charge vector $\bm{Q}$:

$\bm{Q}=\Big{(}0,\ldots,N_{\alpha_{1}},N_{\alpha_{2}},-(N_{\alpha}-1),N_{\alpha_{3}},N_{\alpha_{4}},\ldots,0\Big{)},$
(1)

where $-(N_{\alpha}-1)$ is the charge under the $U(N_{\alpha}-1)$ gauge node,
and $\\{N_{\alpha_{i}}\\}$ are the charges under those gauge nodes
$\\{U(N_{\alpha_{i}})\\}$ which are connected to $U(N_{\alpha}-1)$ by
bifundamental hypers. The hypermultiplet is not charged under any other gauge
node in the quiver.

Mutation $III$, shown in the bottom block of Figure 3, involves replacing a
$U(N_{\alpha})$ gauge node, with balance parameter $e_{\alpha}=-1$ and $P\geq
1$ Abelian hypermultiplets, by a $U(N_{\alpha}-1)$ node and a $U(1)$ node
where the latter node has a single fundamental hyper. In addition, the $P$
Abelian hypermultiplets get mapped by the mutation to $P$ Abelian
hypermultiplets with different charges. Without loss of generality, we can
choose the Abelian hypers in the quiver ${\cal T}$ to have charge $N_{\alpha}$
under $U(N_{\alpha})$ such that the charge vectors have the generic form

$\bm{Q}^{l}=\Big{(}Q^{l}_{1},\ldots,Q^{l}_{\alpha_{1}},Q^{l}_{\alpha_{2}},N_{\alpha},Q^{l}_{\alpha_{3}},Q^{l}_{\alpha_{4}},\ldots,Q^{l}_{L}\Big{)},$
(2)

where $l=1,\ldots,P$, the total number of unitary nodes is $L$ and
$\\{Q^{l}_{\alpha_{i}}\\}$ are the charges under those gauge nodes
$\\{U(N_{\alpha_{i}})\\}$ which are connected to $U(N_{\alpha})$ by
bifundamental hypers. The charge vectors after the mutation are then given by:

$\displaystyle\bm{Q}^{\prime l}=\Big{(}$ $\displaystyle
1,Q^{l}_{1},\ldots,Q^{l}_{\alpha_{1}}+N_{\alpha_{1}},Q^{l}_{\alpha_{2}}+N_{\alpha_{2}},-(N_{\alpha}-1),$
$\displaystyle
Q^{l}_{\alpha_{3}}+N_{\alpha_{3}},Q^{l}_{\alpha_{4}}+N_{\alpha_{4}},\ldots,Q^{l}_{L}\Big{)},$
(3)

where the first entry denotes the charge under the new $U(1)$ node and
$-(N_{\alpha}-1)$ is the charge under the $U(N_{\alpha}-1)$ node. For the
other nodes, only the charges associated with those connected with
bifundamental hypers to $U(N_{\alpha}-1)$ get transformed under the mutation.

$\widetilde{{\cal T}}_{\rm good}$$\widetilde{{\cal T}}_{\rm
unitary}$$\widetilde{{\cal T}}_{\rm maximal}$${{\cal T}}_{\rm good}$${{\cal
T}}_{\rm unitary}$${{\cal T}}_{\rm maximal}$${\cal T}$Ab. $S$-type Op.Ab.
gaugingAb. $S$-type Op.Ab. gaugingIR dualities3d mirror3d mirror3d mirror3d
mirror

Figure 4: Construction of the 3d mirror.

Given the quiver mutations $I$ and $III$, the theory ${\cal T}_{\rm maximal}$
can be obtained in the following fashion.

  * •

We first implement mutation $I$ at each balanced $SU$ node in the quiver
${\cal T}$.

  * •

The mutation can render certain overbalanced $SU$ nodes balanced, and we
implement mutation $I$ sequentially until we arrive at a quiver which has only
overbalanced $SU$ nodes as well as unitary nodes with different balance
parameters ($\geq-1$) plus a number of Abelian hypers.

  * •

In the next step, we implement mutation $III$ at the unitary gauge nodes that
admit it. These operations will generically alter the balance parameters of
both unitary and special unitary nodes in the quiver, thereby creating new
nodes which admit mutation $III$ or mutation $I$.

  * •

We implement these mutations sequentially until we arrive at a quiver where
none of the gauge nodes admits either mutation $I$ or mutation $III$. This is
where the duality sequence terminates and the resultant quiver is a candidate
for ${\cal T}_{\rm maximal}$.

The theory ${\cal T}_{\rm maximal}$ obtained in this fashion is a quiver of
generic type given in Figure 2, with the property that every $SU$ node is
overbalanced, while every non-Abelian unitary node is either balanced or
overbalanced. The quiver may contain certain $U(1)$ nodes which are attached
to Abelian hypers and have balance -1, as these cannot be reduced any further.

$M_{1}$ $N_{1}$ $N$ $N_{2}$ $M_{2}$ $({\cal T})$ |  ${\cal O}_{I}$ |  $(-(N_{1}-1),N)$ $M_{1}$ $N_{1}-1$ $N$ $N_{2}$ $M_{2}$ $({\cal T}^{\vee}_{1})$
---|---|---
|  |  ${\cal O}_{III}$
$(1,-(N-1))$ $((N-1),-(N_{2}-1))$ $M_{1}$ $N_{1}-1$ $N-1$ $N_{2}-1$ $M_{2}$ 1 1 $({\cal T}^{\vee}_{3}={\cal T}_{\rm maximal})$ |  ${\cal O}_{I}$ |  $(1,-(N-1))$ $M_{1}$ $N_{1}-1$ $N-1$ $N_{2}$ $M_{2}$ 1 1 $({\cal T}^{\vee}_{2})$
Figure 5: Derivation of ${\cal T}_{\rm maximal}$ for a linear quiver ${\cal
T}$.

Prescription for the 3d mirror. We can now write down the general recipe for
the construction of the 3d mirror of a quiver gauge theory ${\cal T}$ in the
class of theories given in Fig. 1. The construction involves the following
steps:

  1. 1.

Given a quiver ${\cal T}$ containing at least one balanced special unitary
gauge node, we obtain the theory ${\cal T}_{\rm maximal}$ following the
procedure outlined above. If ${\cal T}$ has any ugly unitary node (i.e.
balance parameter $e=-1$), we first apply the Gaiotto-Witten duality Gaiotto
and Witten (2009) to write ${\cal T}$ in terms of a good quiver and a set of
free twisted hypermultiplets, and then implement the quiver mutations on the
good quiver. Also, if ${\cal T}$ has no balanced special unitary node, we
simply have ${\cal T}={\cal T}_{\rm maximal}$.

  2. 2.

Given ${\cal T}_{\rm maximal}$, we replace all the overbalanced special
unitary nodes by unitary nodes of the same label, i.e. we replace $SU(N)$ by
$U(N)$. We will refer to this theory as ${\cal T}_{\rm unitary}$. ${\cal
T}_{\rm unitary}$ can be mapped back to ${\cal T}_{\rm maximal}$ by gauging
the topological $\mathfrak{u}(1)$ symmetries of the replaced unitary gauge
nodes. Note that ${\cal T}_{\rm unitary}$ consists of unitary gauge nodes
only, while the matter sector includes fundamental/bifundamental hypers as
well as Abelian hypers. If ${\cal T}_{\rm maximal}$ has no overbalanced
special unitary node, we have ${\cal T}_{\rm maximal}={\cal T}_{\rm unitary}$.

  3. 3.

In the next step, we strip off from ${\cal T}_{\rm unitary}$ the Abelian
hypers and the Abelian quiver tails introduced by the quiver mutations $I$ and
$III$. The resultant theory is a standard quiver gauge theory with unitary
gauge nodes and matter in the fundamental/bifundamental representation, and
may consist of a number of possibly ugly $U(1)$ nodes (coming from $U(1)$
nodes with balance -1 attached to Abelian hypers in ${\cal T}_{\rm maximal}$).
Using the Gaiotto-Witten duality to remove these ugly nodes, we obtain a
theory where the gauge nodes are all either balanced or overbalanced. For tree
graphs this balance condition ensures that the theory is good, and we will
refer to it as ${\cal T}_{\rm good}$. For a generic graph, one has to check if
this theory is indeed good before proceeding further.

The theory ${\cal T}_{\rm unitary}$ can be obtained from ${\cal T}_{\rm good}$
by attaching to the gauge nodes of ${\cal T}_{\rm good}$ one or more Abelian
hypers which are in turn connected to Abelian quiver tails. This can be
performed by introducing a decoupled Abelian quiver tail ${\cal T}_{\rm
decoupled}$ with the theory ${\cal T}_{\rm good}$ and gauging a certain linear
combination of the topological symmetries of gauge nodes in ${\cal T}_{\rm
decoupled}$ and ${\cal T}_{\rm good}$. This procedure can be thought of as
mild generalization of the $S$-operation Witten (2003) on ${\cal T}_{\rm
good}$.

  4. 4.

We then find the 3d mirror of ${\cal T}_{\rm good}$, which we will refer to as
$\widetilde{{\cal T}}_{\rm good}$. This theory might be difficult to find, but
the problem greatly simplifies if ${\cal T}_{\rm good}$ is an (affine) $A$ or
$D$-type quiver or a quiver which is related to quivers in these classes by a
set of Abelian gauging operations. In a generic situation, one can try to
engineer $\widetilde{{\cal T}}_{\rm good}$ using the technology of non-Abelian
$S$-type operations Dey (2021).

  5. 5.

Given the mirror pair ${\cal T}_{\rm good}$ and $\widetilde{{\cal T}}_{\rm
good}$, we perform a set of QFT operations on ${\cal T}_{\rm good}$ as
described above which takes us back to ${\cal T}_{\rm unitary}$. On the dual
side, these operations act on $\widetilde{{\cal T}}_{\rm good}$ to produce a
theory which is 3d mirror of ${\cal T}_{\rm unitary}$ – we will refer to this
theory as $\widetilde{{\cal T}}_{\rm unitary}$.

  6. 6.

In the final step, we recall that ${\cal T}_{\rm maximal}$ can be obtained
from ${\cal T}_{\rm unitary}$ by gauging certain topological $\mathfrak{u}(1)$
symmetries in the latter quiver. On the dual side, this amounts to gauging
certain Abelian subgroups of the HB global symmetry of $\widetilde{{\cal
T}}_{\rm unitary}$. The resultant theory is the 3d mirror of ${\cal T}_{\rm
maximal}$ – we will refer to this as $\widetilde{{\cal T}}_{\rm maximal}$.
Since ${\cal T}_{\rm maximal}$ is related to ${\cal T}$ by a sequence of IR
dualities, the quiver $\widetilde{{\cal T}}_{\rm maximal}$ is the 3d mirror of
${\cal T}$ (up to the free hypermultiplets if any in Step 1).

$M_{1}$ $N_{1}-1$ $N-1$ $N_{2}-1$ $M_{2}$ 1 1 1 1 1 $({\cal T}_{\rm good})$ |  $S_{1}$ |  $((N-1),-(N_{2}-1))$ $M_{1}$ $N_{1}-1$ $N-1$ $N_{2}-1$ $M_{2}$ 1 1 1 $({\cal T}_{\rm intermediate})$ |  $S_{2}$ |  $(1,-(N-1))$ $((N-1),-(N_{2}-1))$ $M_{1}$ $N_{1}-1$ $N-1$ $N_{2}-1$ $M_{2}$ 1 1 $({\cal T}_{\rm maximal})$
---|---|---|---|---
$\Sigma_{1}$$\Sigma_{2}$ $M_{1}+3r$ $M_{1}+3r$ 1 1 2 1 1 1 2 $(\widetilde{{\cal T}}_{\rm good})$ |  $S_{1}$ |  $\Sigma_{1}$$\Sigma_{2}$ $M_{1}+3r$ $M_{1}+3r$ 1 1 2 1 1 2 $(\widetilde{{\cal T}}_{\rm intermediate})$ |  $S_{2}$ |  $\Sigma_{1}$$\Sigma_{2}$ $M_{1}+3r$ $M_{1}+3r$ 1 2 1 2 $(\widetilde{{\cal T}}_{\rm maximal})$
Figure 6: Abelian quiver operations for constructing the 3d mirror of ${{\cal
T}}_{\rm maximal}$.

The above construction is schematically shown by a flow-chart in Fig. 4. Note
that the sequence of IR dualities plays a crucial role in reducing the theory
${\cal T}$ to a theory ${\cal T}_{\rm good}$ which is given by a standard
quiver with unitary gauge nodes. Assuming the 3d mirror of the latter is
known, the 3d mirror of ${\cal T}$ can be found by implementing a sequence of
well-defined QFT operations involving Abelian gauging of global symmetries on
the mirror pair $({\cal T}_{\rm good},\widetilde{{\cal T}}_{\rm good})$. In
the rest of the paper, we illustrate the above recipe with a concrete example.

Illustrative Example : A Linear Quiver. In this section, we will construct the
3d mirror of the following quiver gauge theory ${\cal T}$ with two special
unitary gauge nodes and a single unitary gauge node on a linear graph:

$M_{1}$ $N_{1}$ $N$ $N_{2}$ $M_{2}$ $({\cal T}):$

We will consider the case where the left $SU$ node and the central $U$ node
are balanced while the right $SU$ node is overbalanced with balance parameter
$e=0$. The integers $\\{M_{1},N_{1},N,N_{2},M_{2}\\}$ therefore obey the
following constraints : $M_{1}+N=2N_{1}-1$, $N_{1}+N_{2}=2N$ and
$M_{2}+N=2N_{2}$, leading to a 2-parameter family of quiver gauge theories.
For the sake of concreteness, we will assume that $M_{1}<N_{1}-1<M_{2}$, with
$N_{1}-1=M_{1}+r$ for a positive integer $r$.

The first step is to determine the quiver ${\cal T}_{\rm maximal}$. Starting
from the quiver ${\cal T}$, the duality sequence leading to ${\cal T}_{\rm
maximal}$ is given in Figure 5 and involves the following steps:

  1. 1.

We implement mutation $I$ on the balanced $SU(N_{1})$ node of ${\cal T}$
following Figure 3 to obtain the quiver ${\cal T}^{\vee}_{1}$. The charge of
the Abelian hypermultiplet in ${\cal T}^{\vee}_{1}$ can be read off from the
equation (1).

  2. 2.

The $U(N)$ gauge node in ${\cal T}^{\vee}_{1}$ is attached to an Abelian
hypermultiplet and has balance parameter $e=-1$. Therefore, we can implement
mutation $III$ at this node following Figure 3 to obtain the quiver ${\cal
T}^{\vee}_{2}$. The charge of the Abelian hypermultiplet in ${\cal
T}^{\vee}_{2}$ can be read off from the equation (Three dimensional Mirrors
from IR N-ality).

  3. 3.

The $SU(N_{2})$ gauge node of ${\cal T}^{\vee}_{2}$ is balanced, which implies
that we can implement another mutation $I$. This leads to the quiver ${\cal
T}^{\vee}_{3}$.

  4. 4.

The quiver ${\cal T}^{\vee}_{3}$ has no special unitary nodes, while all the
unitary nodes (whether connected to Abelian hypers or not) are either balanced
or overbalanced. Therefore, the duality sequence terminates and we can
identify ${\cal T}^{\vee}_{3}={\cal T}_{\rm maximal}$.

The IR CB symmetry algebra of ${\cal T}$ can be read off from the quiver
${\cal T}_{\rm maximal}$. Noting that the $U(1)$ and the $U(N_{1}-1)$ gauge
nodes are balanced while the other two are overbalanced, we have

$\mathfrak{g}^{\rm IR}_{\rm C}({\cal T}_{\rm maximal})=\mathfrak{g}^{\rm
IR}_{\rm C}({\cal
T})=\mathfrak{su}(2)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1).$
(4)

The next step is to find the quiver ${\cal T}_{\rm good}$. Since ${\cal
T}_{\rm maximal}$ does not have any special unitary node, we have ${\cal
T}_{\rm unitary}={\cal T}_{\rm maximal}$. Stripping off the the Abelian hypers
and attached Abelian quiver tails introduced by the quiver mutations from
${\cal T}_{\rm maximal}$, we obtain the quiver ${\cal T}_{\rm good}$:

$M_{1}$ $N_{1}-1$ $N-1$ $N_{2}-1$ $M_{2}$ $({\cal T}_{\rm good})$

In the present case, the theory ${\cal T}_{\rm good}$ is an $A$-type (linear)
quiver which is manifestly good in the Gaiotto-Witten sense. This implies that
the 3d mirror of ${\cal T}_{\rm good}$ is also a good $A$-type quiver
$\widetilde{{\cal T}}_{\rm good}$, which can be found by the standard Hanany-
Witten realization, and has the following form:

$\Sigma_{1}$$\Sigma_{2}$ $M_{1}+3r$ $M_{1}+3r$ 3 1 $(\widetilde{{\cal T}}_{\rm
good})$

The grey nodes, labelled $\Sigma_{1}$ and $\Sigma_{2}$, denote $A$-type quiver
tails, and their explicit forms are given as follows:

$\Sigma_{1}$ : $1$ $2$ $M_{1}$ $M_{1}+3$ $M_{1}+6$ $M_{1}+3(r-1)$
---
$\Sigma_{2}$ : $M_{1}+3r-1$ $M_{1}+3r-2$ $1$

Note that the total number of gauge nodes in $(\widetilde{{\cal T}}_{\rm
good})$ is $M_{1}+M_{2}-1$. One can readily check that global symmetry
algebras across the duality agrees: $\mathfrak{g}^{\rm IR}_{\rm C}({\cal
T}_{\rm good})=\mathfrak{g}^{\rm UV}_{\rm H}(\widetilde{{\cal T}}_{\rm
good})=\mathfrak{su}(3)\oplus\mathfrak{u}(1)$, while $\mathfrak{g}^{\rm
UV}_{\rm H}({\cal T}_{\rm good})=\mathfrak{g}^{\rm IR}_{\rm
C}(\widetilde{{\cal T}}_{\rm
good})=\mathfrak{su}(M_{1})\oplus\mathfrak{su}(M_{2})\oplus\mathfrak{u}(1)$,
as expected.

The final step is to implement the Abelian $S$-type operations on ${\cal
T}_{\rm good}$ that leads to the quiver ${\cal T}_{\rm maximal}$ – the precise
operations are shown in the top row of Fig. 6. The corresponding operations on
the mirror side, which are shown in the bottom row, gives the 3d mirror of
${\cal T}_{\rm maximal}$. We add to the quiver ${\cal T}_{\rm good}$ a pair of
decoupled Abelian quiver gauge theories :

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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{147.37766pt}{56.90552pt}\pgfsys@lineto{166.98355pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{140.84116pt}{73.39766pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\minipage[t]{2.84528pt}\leavevmode\hbox{}\raggedright{1}\@add@raggedright\endminipage}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{{{
{}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{169.29391pt}{72.0168pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\minipage[t]{2.84528pt}\leavevmode\hbox{}\raggedright{1}\@add@raggedright\endminipage}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\qquad{\cal
T}^{(2)}_{\rm decoupled}=\scalebox{0.7}{ \leavevmode\hbox to66.78pt{\vbox
to31.94pt{\pgfpicture\makeatletter\hbox{\hskip-23.69711pt\lower
51.79166pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{90.07214pt}{56.90552pt}\pgfsys@curveto{90.07214pt}{59.50894pt}{87.9617pt}{61.61938pt}{85.35828pt}{61.61938pt}\pgfsys@curveto{82.75485pt}{61.61938pt}{80.64441pt}{59.50894pt}{80.64441pt}{56.90552pt}\pgfsys@curveto{80.64441pt}{54.3021pt}{82.75485pt}{52.19165pt}{85.35828pt}{52.19165pt}\pgfsys@curveto{87.9617pt}{52.19165pt}{90.07214pt}{54.3021pt}{90.07214pt}{56.90552pt}\pgfsys@closepath\pgfsys@moveto{85.35828pt}{56.90552pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{85.35828pt}{56.90552pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
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}{}\pgfsys@rect{25.11975pt}{53.57251pt}{6.66602pt}{6.66602pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{80.24442pt}{56.90552pt}\pgfsys@lineto{62.01938pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{51.79166pt}{56.90552pt}\pgfsys@lineto{32.18576pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{83.93564pt}{73.39766pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\minipage[t]{2.84528pt}\leavevmode\hbox{}\raggedright{1}\@add@raggedright\endminipage}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{\minipage[t]{2.84528pt}\leavevmode\hbox{}\raggedright{1}\@add@raggedright\endminipage}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{{{
{}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$

On the mirror side, we add the 3d mirrors of the aforementioned decoupled
Abelian theories (a set of free hypermultiplets) to the theory
$\widetilde{{\cal T}}_{\rm good}$:

$\widetilde{{\cal T}}^{(1)}_{\rm decoupled}=\scalebox{0.7}{ \leavevmode\hbox
to37.96pt{\vbox to29.18pt{\pgfpicture\makeatletter\hbox{\hskip
4.75565pt\lower-3.733pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{-3.33301pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{25.11975pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.45276pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.733pt}{0.0pt}\pgfsys@lineto{24.71976pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} { {}{}{}}{{{ {}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.42264pt}{15.11128pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\minipage[t]{2.84528pt}\leavevmode\hbox{}\raggedright{1}\@add@raggedright\endminipage}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{{{
{}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.03012pt}{15.11128pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\minipage[t]{2.84528pt}\leavevmode\hbox{}\raggedright{1}\@add@raggedright\endminipage}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\qquad\widetilde{{\cal
T}}^{(2)}_{\rm decoupled}=\scalebox{0.7}{ \leavevmode\hbox to37.96pt{\vbox
to29.18pt{\pgfpicture\makeatletter\hbox{\hskip 4.75565pt\lower-3.733pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{-3.33301pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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In the first step, we gauge a linear combination of the $\mathfrak{u}(1)$
topological symmetries associated with a subset of unitary gauge nodes in the
quivers ${{\cal T}}^{(1)}_{\rm decoupled}$ and ${{\cal T}}_{\rm good}$. The
gauge nodes in question are marked by the red dashed arrows in the top left
figure. This operation introduces an Abelian hyper charged under two nodes of
the quiver ${\cal T}_{\rm good}$, and gives the quiver ${\cal T}_{\rm
intermediate}$, as shown in the top middle figure. On the mirror side, this
operation corresponds to identifying the Abelian flavor nodes marked by red
dashed arrows in the bottom left figure and gauging the identified flavor
node. This gives the quiver $\widetilde{{\cal T}}_{\rm intermediate}$ as shown
in the bottom middle figure – the 3d mirror of ${\cal T}_{\rm intermediate}$.

In the next step, we perform an analogous operation on the quivers ${{\cal
T}}^{(1)}_{\rm decoupled}$ and ${\cal T}_{\rm intermediate}$, as shown in the
top middle figure, which gives ${\cal T}_{\rm maximal}$. The operation on the
mirror side is shown in the bottom middle figure, and gives the quiver
$\widetilde{{\cal T}}_{\rm maximal}$ – the 3d mirror of ${\cal T}_{\rm
maximal}$. One therefore ends up with the following mirror pair:

$M_{1}$ $N_{1}$ $N$ $N_{2}$ $M_{2}$ $({\cal T}):$
---
$\Sigma_{1}$$\Sigma_{2}$ $M_{1}+3r$ $M_{1}+3r$ 1 2 1 2 $(\widetilde{{\cal
T}}):$

Note that the emergent CB symmetry of the quiver ${\cal T}$ is realized as a
UV-manifest HB global symmetry in the theory $\widetilde{{\cal T}}$, i.e.

$\displaystyle\mathfrak{g}^{\rm IR}_{\rm C}({\cal T})=\mathfrak{g}^{\rm
UV}_{\rm H}(\widetilde{{\cal
T}})=\mathfrak{su}(2)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1),$
(5)

where the latter can be simply read off from the quiver $\widetilde{{\cal
T}}$.

Conclusion and Outlook. We have presented an explicit field theory algorithm
for constructing the 3d mirror of a unitary-special unitary quiver gauge
theory of generic shape with matter in the fundamental/bifundamental
representation, where the quiver has at least a single balanced special
unitary node and the theory is good or ugly in the Gaiotto-Witten sense. One
can use a sequence of IR dualities to reduce a quiver ${\cal T}$ in the
aforementioned class to a quiver ${\cal T}_{\rm maximal}$ which consists only
of unitary gauge nodes but involves hypermultiplets in determinant/anti-
determinant representation in addition to the fundamental/bifundamental
matter. The quiver ${\cal T}_{\rm maximal}$ can in turn be reduced to a
unitary quiver gauge theory ${\cal T}_{\rm good}$ with only
fundamental/bifundamental hypermultiplets, by a sequence Abelian field theory
operations. In this fashion, one reduces the problem of finding the 3d mirror
of a unitary-special unitary quiver to the problem of finding the 3d mirror of
a simpler unitary quiver. Given the 3d mirror of ${\cal T}_{\rm good}$, we can
retrace our steps and construct the 3d mirror of ${\cal T}$.

We illustrate the construction by finding the mirror of a linear unitary-
special unitary quiver gauge theory. The theory has an emergent CB symmetry in
the IR, which appears in the 3d mirror as a UV-manifest Higgs branch symmetry.
Generally speaking, if ${\cal T}$ is a linear graph, then the quiver ${\cal
T}_{\rm good}$ is a linear quiver with unitary gauge nodes and therefore has a
3d mirror which is also a linear quiver with unitary gauge nodes. This makes
the above construction particularly easy to implement. However, examples of 3d
mirrors for theories beyond linear quivers are reasonably straightforward to
construct using this algorithm and some of them will appear in future papers.

It is well known that the circle reduction for a large class of Argyres-
Douglas theories leads to unitary-special unitary quivers Closset  _et al._
(2021) of the type studied in this paper. The algorithm presented here
therefore gives an alternative way to construct the 3d mirrors of circle-
reduced Argyres-Douglas theories, without resorting to either the class ${\cal
S}$ technology or the magnetic quiver approach. This topic will be explored in
an upcoming paper.

Acknowledgments The author acknowledges the hospitality of the Simons Summer
Workshop 2023 during the completion of this work. The author is supported in
part at the Johns Hopkins University by the NSF grant PHY-2112699.

## References

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# The wave-function as a true ensemble

J.R. Hance<EMAIL_ADDRESS>Quantum Engineering Technology
Laboratories, Department of Electrical and Electronic Engineering, University
of Bristol, Woodland Road, Bristol, BS8 1US, UK  S. Hossenfelder  Frankfurt
Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am
Main, Germany

###### Abstract

In quantum mechanics, the wave-function only predicts probabilities of
measurement outcomes, not individual outcomes. This suggests that it describes
an ensemble of states with different values of a hidden variable. Here, we
analyse this idea with reference to currently known theorems and experiments.
We argue that the $\psi$-ontic/epistemic distinction fails to properly
identify ensemble interpretations and propose a more useful definition. We
then show that all $\psi$-ensemble interpretations which reproduce quantum
mechanics violate Statistical Independence. Finally, we explain how this
interpretation helps make sense of some otherwise puzzling phenomena in
quantum mechanics, such as the delayed choice experiment, the Elitzur-Vaidman
bomb detector, and the Extended Wigner’s Friends Scenario.

##  I Introduction

The wave-function is weird. Its most salient feature – that it merely predicts
probabilities for measurement outcomes, rather than the outcomes themselves –
suggests that quantum mechanics is an emergent, average description of an
underlying, more complicated dynamics. In this underlying theory, the time-
evolution of the system would be determined and measurement outcomes could be
predicted. It is because we lack information about the details of the initial
state that we can only make a probabilistic prediction.

That the wave-function might be an emergent description of yet-to-be-
discovered underlying physics is often called a hidden variable
interpretation. The hidden variables are the information that is missing in
quantum mechanics. This straightforward explanation for the strange properties
of the wave-function, however, has seemingly been disfavoured by Bell’s and
related theorems [3, 6], and, more recently, by the Pusey-Barrett-Rudolph
theorem [23, 19] (hereafter PBR-theorem). We here want to look closer at what
these theorems actually say about the wave-function as an ensemble of
different values of the hidden variables.

The PBR-theorem in particular builds on a classification of models put forward
by Harrigan and Spekkens [15]. Their framework attempts to mathematically
formalise the distinction between a wave-function that represents reality
($\psi$-ontic) and one that represents knowledge ($\psi$-epistemic). That this
is a false dichotomy was previously pointed out in [12]. We here want to spell
out another problem with the $\psi$-ontic/epistemic framework – it doesn’t
identify the hidden variables theories that Bell was trying to rule out.

We begin with defining in what sense we take the wave-function to be an
ensemble in Section II, and explain the relation to the $\psi$-ontic/epistemic
framework in Section III. In Section IV we will discuss the most important
consequence, that is the different definition of what the hidden variables
describe, and how to interpret it. In Section V, we will revisit the PBR-
theorem in this light. In Section VI we discuss Statistical Independence and
locality. Finally, in Section VII, we will explain how the $\psi$-ensemble
interpretation helps making sense of quantum mechanics.

##  II The $\psi$-ensemble interpretation

We will begin by explaining what exactly we mean by the wave-function being an
ensemble. We do not mean the Ensemble, or Statistical, Interpretation of
quantum mechanics [2], according to which the wave-function describes an
ensemble of identically prepared states in identical experiments. It arguably
does, but if the states are indeed identical, that’s not much of an ensemble.
Similarly, we do not mean Smolin’s ensemble interpretation [32], where the
ensemble considered is the ensemble of all the systems in the same quantum
state in the universe. We mean instead that the supposedly identically
prepared states in supposedly identical experiments are in fact different:
they have different hidden variables and this is the reason why the
measurement outcomes can be different for identical wave-functions. We will
refer to this interpretation of the wave-function as the $\psi$-ensemble
interpretation.

In our $\psi$-ensemble interpretation, we have an underlying theory with
variables that we will collectively name $\kappa$ and we will denote the
state-space of all $\kappa$ with $K$. The $\kappa$ are generically time-
dependent, $\kappa(t)$. These $\kappa(t)$s are not the same as Bell’s hidden
variables, as will become clear in a moment.

We use the term ‘hidden variables’ because it has become common terminology.
However, we want to stress that these variables are not a priori unmeasurable.
They are ‘hidden’ merely in the sense that they do not appear in quantum
mechanics. If the underlying theory was better understood, they could well
become measurable one day.

$|\psi\rangle$ is, as usual, an element of a projected Hilbert space, ${\cal
H}$. It fulfils the Schrödinger equation. We are considering a generic quantum
mechanics measurement, in which $|\psi\rangle$ is prepared at time $t_{\rm p}$
and measured at time $t_{\rm m}$. The measurement is described by an
orthonormal basis of pointer states that we will denote $|I\rangle$, where
$I\in\\{0...N-1\\}$ and $N$ is the dimension of the Hilbert-space. $|I\rangle$
could be describing multiple different detectors, and might be a product-state
or an entangled state when expressed in a basis of the individual detectors.
The basis $|I\rangle$ implicitly contains the measurement settings at the time
of measurement.

For simplicity we will in the following assume that the basis $|I\rangle$ is
time-independent. This does not mean that the detector setting cannot change.
It merely means that if the detector setting before the measurement was
different from the setting at the actual measurement, then it wasn’t described
by $|I\rangle$.

In quantum mechanics now, the Schrödinger evolution of $|\psi\rangle$ will
generically result in a state that is not one of the detector eigenstates at
the time of measurement. In this case, we assume that $|\psi(t_{\rm
m})\rangle$ is an ensemble of different $\kappa(t_{\rm m})$, that is, a
distribution $\rho(\kappa(t))$ over $K$. This means in the $\psi$-ensemble
theory we have a map

${\cal P}(K)\ni\rho(\kappa(t))\to|\psi(t)\rangle\in{\cal H}~{},$ (1)

where ${\cal P}(K)$ is the space of all normalisable probability distributions
on $K$. That is to say, we interpret $|\psi(t_{\rm m})\rangle$ as an ensemble
because we empirically know it corresponds to different measurement outcomes.
The measurement then merely reveals which of the hidden variables described
the actual state.

The reason we have for considering this interpretation is that the wave-
function update is non-local when interpreted as a physical process. This
makes it difficult to combine quantum theory with general relativity. If we
hence introduce a $\psi$-ensemble interpretation, our motivation is to develop
an underlying theory which restores locality, causality, and determinism.
However, of course one can conceive of $\psi$-ensembles that are not local,
for example Bohmian Mechanics [5].

##  III Difference between $\psi$-ensemble and $\psi$-epistemic

The introduction of the $\psi$-ensemble in the previous section sounds
superficially very similar to the definition of a $\psi$-epistemic model
introduced by Harrigan and Spekkens [15]. According to Harrigan and Spekkens
(hereafter HS), a model is $\psi$-epistemic if one particular value of a
hidden variables can correspond to more than one wave-function.

As a simple example, consider you have a two-state system with detector
eigenstates $\left|0\right\rangle$ and $\left|1\right\rangle$. You could
prepare the state so that, under the Schrödinger-evolution it goes to
$\left|0\right\rangle$ with certainty. Or you could prepare it so that it goes
to $(\left|0\right\rangle+\left|1\right\rangle)/\sqrt{2}$. If the wave-
function was $\psi$-epistemic, then the latter case must have some overlap
with the former in the underlying hidden-variables theory because it can
result in the same measurement outcome. At least that is the idea.

Figure 1: Harrigan and Spekkens’s $\psi$-epistemic and $\psi$-ontic models of
reality [15]. Wavefunctions $\psi$ and $\varphi$ each have probability
distributions over state-space. In a $\psi$-epistemic model, these can overlap
over a subspace. A state $\kappa$ within this overlap can then be represented
both by $\psi$ and $\varphi$. However, in $\psi$-ontic models, each state can
be represented by at most one wave-function. The amount of overlap is
quantified by the parameter $\Delta$. For $\psi$-ontic models, $\Delta=0$.

Formally, HS first define the probability $p(\kappa|\psi)$ that the quantum
wave-function $\psi$ is described by $\kappa$, where $\kappa\in K$ [14, 20].
Next, they define the probability of getting a particular measurement outcome
given $\kappa$. While their framework can be used for any measurement, here we
merely need the probability $\mathcal{A}(\psi|\kappa)$ that the state $\kappa$
is measured as $\psi$. From this one obtains the support $K_{\psi}\subset K$
for the wave-function $\psi$

$\forall~{}\kappa\in K_{\psi},\;\mathcal{A}(\psi|\kappa)=1~{}.$ (2)

They then take a second wave-function, $\varphi$, with its own probability
distribution $p(\kappa|\varphi)$, and ask what is the probability that the
$\kappa$s which contribute to $\varphi$ also contribute to $\psi$. They
quantify this overlap as

$\Delta=\int_{K_{\psi}}p(\kappa|\varphi)d\kappa~{}.$ (3)

If $\Delta>0$, HS interpret the wavefunctions in such a model as epistemic
states, because the same underlying $\kappa$ could contribute to both $\psi$
and $\varphi$. Models with non-zero overlap $\Delta$ are correspondingly
called $\psi$-epistemic, whereas models without overlap are $\psi$-ontic (for
illustration, see Fig.1).

There are several problems with the HS $\psi$-ontic/epistemic distinction. One
of them is purely linguistic. According to the HS definition, the wave-
function in quantum mechanics itself is ontic. It seems odd to declare
something ontic that can’t be observed, especially given that Bohr already
argued $\psi$ is far better interpreted as knowledge. Settling a hundred-year
old debate by definition is not insightful.

One might put this aside as historical nitpicking, but this linguistic issue
reflects a deeper problem. If there’s no observational requirement tied to
calling something ontic, then every model can be made ontic. This was pointed
out already in [30]. Suppose you have a theory that isn’t ontic because two of
the hidden variables map to the same wave-function. Well, then you can just
declare the wave-function to be part of the hidden variables, so that the new
hidden variables – now including the wave-function – will always map to only
one wave-function. Such an easily malleable definition of ‘ontic’ is not what
one wants to base theorems on. (See also the discussion in [19].)

It is also highly confusing that, according to the HS-definition, most hidden
variables models which have been proposed so far are actually $\psi$-ontic.
Think back of the example with the two state system that evolves into
$(|0\rangle+|1\rangle)/\sqrt{2}$ but is measured in either $|0\rangle$ or
$|1\rangle$. While the $(|0\rangle+|1\rangle)/\sqrt{2}$ state must have been
made up of different underlying states (because otherwise it couldn’t give
rise to different outcomes) there is no reason why any one of those underlying
states must have been _the same_ as those of a system prepared to evolve into
$|0\rangle$ or $|1\rangle$. Hence, according to the HS-definition, a
$\psi$-ontic model can solve the measurement problem as well as a
$\psi$-epistemic one.

We believe our definition of a $\psi$-ensemble to be more descriptive of
hidden variables models, and less ambiguous than the $\psi$-epistemology
proposed by HS. Whether or not the state prepared to evolve into
$(|0\rangle+|1\rangle)/\sqrt{2}$ had overlap with any other wave-function, it
will always be an ensemble if the underlying hidden variables theory solves
the measurement problem. That is to say a $\psi$-ensemble could be either
$\psi$-ontic or $\psi$-epistemic according to the HS-definition.

##  IV Hidden variables label trajectories

In our definition of a $\psi$-ensemble, we have been careful to point out that
naturally the underlying hidden variables theory will be dynamic, hence the
hidden variables will be functions of time. This is of utmost importance for
locality considerations, as those usually pertain arguments about the
distribution of the hidden variables. (We will discuss this in Section VI.)

If we evaluate the distribution of hidden variables at the time of
measurement, then we can infer the measurement outcome directly from that.
However, for Bell’s theorem, and also for the PBR-theorem, one normally
considers the distribution of variables at the time of preparation instead.
These two distributions will in general not be identical. The problem is, if
one merely takes the distribution at the time of preparation, this will _not_
constitute the hidden variables of Bell’s theorem.

To see this, note that since we assume the underlying theory is deterministic,
we can always take the distribution at the time of preparation, $t_{\rm p}$,
and evolve it forward with whatever is the transport function of the theory.
Assuming time-translation invariance, let us denote the transport function as
$T(t_{1}-t_{0},\cdot)$ where the free slot is for the hidden variables. It has
the property that

$\kappa(t_{1})=T(t_{1}-t_{0},\kappa(t_{0}))~{},~{}T(0,\cdot)\equiv{\rm
Id}~{}.$ (4)

Now arguably the entire information that is necessary to determine the outcome
at $t_{\rm m}$ is _both_ the initial distribution at preparation
$\kappa(t_{\rm p})$ and the transport function $T(t_{\rm m}-t_{\rm p},\cdot)$.

Bell, therefore, in the derivation of his inequality, correctly defines the
hidden variables as the ‘full specification’ of the information necessary to
predict the outcome [4]. If we call Bell’s hidden variables (as usual)
$\lambda$, then $\lambda(t_{0})=(\kappa(t_{0}),T(t_{\rm
m}-t_{0},\kappa(t_{0}))$ and one can now treat the hidden variables as time-
independent. The additional information in the transport function becomes
redundant in case one defines the distribution of the hidden variables at the
time of measurement already ($t_{0}=t_{\rm m}$) but normally one chooses
$t_{\rm p}<t_{0}<t_{\rm m}$.

Figure 2: Sketch to illustrate the relation between the hidden variables
$\kappa(t)$ which define the state, and the hidden variables $\lambda$ which
define the entire trajectory. $|I\rangle$ and $|J\rangle$ depict two different
detector eigenstates.

Pictorially this means that Bell’s hidden variables (the $\lambda$) are really
labels for histories (in state space), whereas the HS hidden variables (that
we called $\kappa$) define a state (see Fig. 2). For this reason, assuming
that Bell’s hidden variables belong to a particular moment in the history of
the (entire) state makes no sense. We will comment in Section VI on what this
means for locality requirements that one normally uses for the distribution of
the hidden variables.

A particularly illustrative example for hidden variables that define
trajectories is the model proposed in [21, 22]. In this case, the hidden
variables simply are the detector eigenstates towards which the prepared state
evolves. Given that we cannot measure the state before we measure the state,
this is the most minimal assumption that one can make.

##  V The PBR theorem revisited

Let us then look at what happens when one confuses these two sets of hidden
variables. The PBR-theorem [23], in a nutshell, shows that models which are
$\psi$-epistemic according to the HS-definition are incompatible with
observations. This superficially sounds similar to Bell’s theorem which takes
on hidden variables theories, but the PBR-theorem works entirely differently.
For this reason, one cannot draw conclusions from it about $\psi$-ensemble
theories.

This can be seen as follows. The hidden variables in the HS-definition merely
define the state. They do not also specify the time-evolution of the
underlying hidden variable system. If one hence does not define the state at
the time of measurement, then the hidden variables in the HS-definition do not
determine the measurement outcome. How could they if they do not contain
information about the evolution law? Now one may say that a definition is just
that, a definition. Alright, but the HS-definition has been used in the PBR-
theorem _as if_ the hidden variables actually did determine the measurement
outcome.

For the PBR-theorem one considers the distribution of hidden variables at
preparation in the HS-definition. However, those don’t tell us what the
outcome is, so no conclusions about whether the theory is viable or not can be
derived. If one instead replaces the variables in the PBR-theorem with the
variables which actually determine the outcome – i.e., those which contain
information about the transport function – then one needs an additional
assumption. This assumption is that the complete variables are not correlated
with the detector settings. That is, one needs the assumption that Bell called
Statistical Independence. This is because if Statistical Independence is not
fulfilled, then one of the assumptions for the PBR-theorem, the Preparation
Independence Postulate (PIP) is not fulfilled.

Bell defined Statistical Independence (SI) as the absence of correlations
between the hidden variables and the two detector settings which he considered
in a particular experiment. However, of course one can define Statistical
Independence in general as

$\rho(\lambda|S)=\rho(\lambda)~{},$ (5)

where $S$ are the detector settings (of whatever experiment). As laid out in
[13], violating this condition does not necessarily require correlations
between the settings and the distribution of the hidden variables over the
state space; the correlations can instead be induced by the structure of the
state-space.

Statistical Independence sneaks into the PBR-theorem because the HS-definition
for a $\psi$-epistemic model assumes that the hidden variables which define
the state at preparation have nothing to do with the measurement. And that
sounds reasonable, until one realises that if one wants to say anything about
measurement outcomes one needs to know the time-evolution of the hidden
variables too.

The PIP now says that if we prepare two independent (spatially separate)
experiments, then the distributions of the hidden variables should factorise.
In the PBR argument one first considers two experiments in isolation. Then one
combines the two so prepared states – using PIP – and measures them instead
with a smartly chosen basis of distinguishable states. From this, one derives
a contradiction: If the distribution of the hidden variables was actually the
product of the distributions for the separate measurements, then one cannot
reproduce the predictions of quantum mechanics.

Needless to say, the theorem is correct for what the mathematics is concerned,
but the relevant physical assumption was that the distributions of the hidden
variables for the individual experiments already didn’t depend on the
measurement settings. So of course if one multiplies them, they still do not
depend on the measurement settings and one cannot reproduce the predictions of
quantum mechanics. That is, in the $\psi$-epistemic framework, we have SI
$\Rightarrow$ PIP111Note that it’s not in general the case that $\neg$ PIP
$\Rightarrow\neg$ SI because it could be that other assumptions of the
$\psi$-epistemic framework are violated rather than SI..

However, the relevant assumption was the independence of the complete hidden
variables (the $\lambda$s which determine the outcome) from the measurement
settings, not the factorisation. In particular, the initial distribution of
the variables, $\kappa(t_{\rm p})$, might well factorise at preparation.

An example for a model in which this happens is [7]. This model uses one
(complex valued) hidden variable for each dimension of the Hilbert-space.
These hidden variables, which correspond to $\kappa$, are uniformly
distributed in the unit disk. Hence, the model fulfils the requirements for
the PBR-theorem and yet reproduces the predictions of quantum mechanics,
seemingly defying the theorem. The reason is that the dynamical law of the
model depends on the detector settings, and so do the full hidden variables
$\lambda$. When one uses these variables, the model violates PIP, hence no
contradiction with the theorem arises. However, since the variables then no
longer describe the initial state at preparation, it’s unclear why they should
fulfil PIP in the first place.

We may note that this issue does not appear in Bell’s theorem, simply because
Bell considers a situation where the state is prepared at one place but
measured in two, whereas PBR consider a situation where the state is prepared
in two places but measured in one. In Bell’s case one has no reason to assume
that the distribution of the hidden variables factorises.

##  VI Properties of $\psi$-ensemble theories

###  VI.1 Statistical Independence

We know from Bell’s theorem that any $\psi$-ensemble theory which is locally
causal (in Bell’s sense) and reproduces the predictions of quantum mechanics
must violate Statistical Independence. Indeed, that one can reproduce quantum
mechanical correlations while respecting local causality provided Statistical
Independence is violated is quite possibly the reason why Bell (and others
after him) sought other ways to rule out Statistical Independence, notably by
arguing that it requires finetuning (discussed below).

However, one does not need Bell’s theorem to see that local and causal
$\psi$-ensemble models will violate Statistical Independence. We just need to
note that if we want states in the hidden variables theory to evolve locally
and within the light-cone into detector eigenstates, then we need information
about the detector before the prepared state reaches the detector – otherwise
how do we know what state to evolve into?

To illustrate this point consider the generic setting of an experiment with
two detectors (Fig. 3). The state is prepared at P and measured at detectors
D1 and D2. One may have in mind for example a single photon that goes through
a beam splitter. The state does not have to be entangled. We want to know how
it evolves in the underlying hidden variables theory.

When the prepared state arrives at D1 it needs to know the setting at D2. Yet,
it is clear that if that information wasn’t available already at P, then it
can’t have gotten there within the light cone. Hence, both detector settings
D1 and D2 must have been either in the initial distribution of hidden
variables or the evolution law. In either case, Statistical Independence is
violated.

To be sure, we have drawn an extreme case where the prepared state actually
moves with the speed of light. If one took a massive particle, the last moment
in which information about D2 must be available for the underlying state going
to D1 could be after preparation. But that doesn’t change the fact that
Statistical Independence must have been violated because information about the
detector setting must have been in the full specification of the hidden
variables.

It is a corollary of this – and not the starting point – that any
$\psi$-ensemble theory that solves the measurement problem and reproduces
quantum mechanics violates the assumptions for Bell’s theorem and the PBR-
theorem. It follows that experiments which show that Bell’s inequality or the
CHSH inequality are violated [1], or that the measurement proposed in the PBR-
theorem agrees with the predictions of quantum mechanics [25] cannot rule out
$\psi$-ensemble theories. A different type of experiment is necessary to test
such theories.

Figure 3: A diagram showing the two backwards light cones from detectors D1
and D2, and their overlap at point of preparation P .

Understanding that Statistical Independence is violated _because_ that is
required to solve the measurement problem also explains why one need not
expect such violations to occur in non-quantum experiments, like the often
quoted tobacco trials to infer a correlation between smoking and cancer (see
[17] and references therein). No preparation of such an experiment will ever
result in a superposition of measurement outcomes. Even if one used a quantum
experiment to, say, sample mice into two groups (one exposed to tobacco smoke,
the other not), then the quantum superposition would be destroyed before one
even wrote down the outcome of the quantum experiment. But Statistical
Independence is only violated between the preparation of an experiment that,
under the Schrödinger-evolution, evolves into a superposition of detector
eigenstates, and the measurement of that state.

It must be understood here that we use the term ‘measurement’ in the same
sense as in quantum mechanics. That is, it does not necessarily require an
observer or even a detector, it just requires a sufficient amount of
interactions with the environment. We don’t need to be specific about this
because this aspect of the measurement problem was already solved by
decoherence and einselection [36]: A detector is any system that decoheres the
prepared state quickly enough, and the detector eigenstates are defined as
those which are stable in that process. We merely need violations of
Statistical Independence to explain why we always measure a detector
eigenstate and not, as decoherence predicts, a mixed state.

It is worth stressing that the statements made in this section are not
suggested interpretations of mathematical expressions, they are conclusions
drawn from empirical facts. We know, as a matter of fact, that measurements
yield definite results. If the results come about locally, causally, and
deterministically from the prepared state, then the underlying theory must
violate Statistical Independence. We also know empirically that we never
observe superpositions of detector eigenstates, hence violations of
Statistical Independence are negligible for macroscopic objects.

###  VI.2 Locality

It is quite common to get confused here about how such a requirement – that
the evolution law depends on the measurement settings at the time of
measurement – can possibly be local. However, keep in mind that the hidden
variables $\lambda$ label trajectories, not initial states, of the underlying
theory. The trajectories necessarily connect the prepared state with the
detectors locally and causally (if they didn’t, we couldn’t detect the
prepared state). Hence, the requirement that the hidden variables are
correlated with the detector settings at measurement can come about entirely
by local propagation and by local interactions. (Again, we do not claim that
such a theory necessarily has to be local. It’s just that this is the case we
are interested in.)

Such $\psi$-ensemble theories that violate Statistical Independence fulfil
Bell’s criterion of local causality because if the information about both
detector settings was available already at preparation, then it is unnecessary
to later add it.

###  VI.3 Fine-tuning

It is frequently argued that theories which violate Statistical Independence
would need a lot of detail to specify the initial state for the detector
setting at the time of measurement, and that they are thence fine-tuned
“conspiracies” [31]. This initial state is often believed to be required long
before the experiment even began, possibly as far back as the Big Bang.

However, the idea that we need to define an initial state for the detector
setting is patently absurd. In quantum mechanics, we also never define the
initial condition at the Big Bang that will give rise to a specific detector
setting. Indeed, we never do this in any theory. It is in practice unnecessary
to ever write down this initial state, regardless of whether it is feasible in
the first place. We simply make predictions for any possible measurement
setting. Hence, violating Statistical Independence does not require a lot of
detail; it merely requires the detector settings at the time of measurement –
an input we also use in ordinary quantum mechanics.

A second finetuning argument [34] has it that it would require specifying a
lot of details for the distribution of hidden variables in order to prevent
violations of Born’s rule that would allow for superluminal signalling.
However, leaving aside that the absolute impossibility of superluminal
signalling (rather than its observed rarity) is a postulate, not an empirical
fact, we can arguably always achieve that Born’s rule is fulfilled simply by
postulating that it is fulfilled, which one also does in quantum mechanics.

That is, these two arguments use a double-standard: Assumptions are not called
fine-tuned when used in normal quantum mechanics (detector settings at
measurement and Born’s rule are required to calculate probabilities) but are
called fine-tuned when used in other frameworks. As already argued in [16],
such a notion of fine-tuning is scientifically meaningless and can at best be
considered metaphysical.

##  VII Using the $\psi$-ensemble to make sense of quantum experiments

Of course using a $\psi$-ensemble, even if deterministic, local, and causal
doesn’t remove the weirdness of quantum mechanics. It remains in the property
that for a spatially distributed detector, a state in the underlying theory
that goes to one particular eigenstate must have had information about that
part of the detector to which it _didn’t_ go. Again, this requirement can be
formulated purely in local terms (if the state would have gone to the other
part of the detector, it would have interacted with it), but this is where the
famed quantum weirdness goes. If the underlying hidden variables theory is
local and causal and deterministic, then the evolution which the actual state
takes – and which we ultimately observe – takes into account what would have
happened if the state had gone elsewhere. We will now explain how this nicely
explains some otherwise very unintuitive quantum behaviour.

###  VII.1 The Double-Slit Experiment

To warm up, we consider the familiar double-slit experiment. The odd thing
about this experiment is that the behaviour of the quantum particle changes
depending on whether we know which slit the particle goes through. If we don’t
know, the particle seems to go through both slits and interferes with itself,
like we expect for a wave. But if we measure which slit the particle goes
through, the interference pattern vanishes.

This seemingly strange behaviour makes total sense from the $\psi$-ensemble
perspective because the trajectory of the state in the underlying hidden
variables theory depends on what is being measured.

###  VII.2 Wheeler’s Delayed-Choice Experiment

The delayed choice-experiment [33] comes in a large number of variations, but
in the simplest version, one changes the variable that is being measured after
the prepared state has begun its propagation. In normal quantum mechanics,
this is even more difficult to understand than the double slit experiment
because it seems that the prepared state must somehow ‘change its mind’ as the
setting is changed.

However, from the $\psi$-ensemble perspective this is clearly not what
happened. This is because the trajectories depend on the detector setting _at
the time of measurement_. What the settings were before that or how often they
were changed is irrelevant.

###  VII.3 The Elitzur-Vaidman Bomb Detector

The Elitzur-Vaidman bomb detector allows one to detect whether a bomb is live
(would blow up when it detects a photon) without triggering the bomb [8]. In
this experiment, the bomb acts as a detector. Whether the bomb is live or a
dud therefore constitutes two different detector settings. From the
$\psi$-ensemble perspective, it is hence rather unsurprising that the case in
which the bomb does not blow up contains information about whether the bomb is
live, because this is just a type of detector setting.

###  VII.4 Counterfactual Computation/Communication

In counterfactual computation [18] one infers the result of a computer query
without interacting with the computer. This experiment is a variant of the
Elitzur-Vaidman experiment and can be explained the same way. The computer
acts as a detector depending on what the outcome of the calculation is. In the
$\psi$-ensemble interpretation, the particle which one measures contains
information about the setting of the detectors which it wasn’t detected in,
which allows one to infer the result of the computation.

The same logic also applies to Salih et al’s counterfactual communication
device and related protocols [28, 29, 10], and so protocols for imaging [11]
and for quantum information transfer derived from it [26, 27].

###  VII.5 The Extended Wigner’s Friend Scenario

The previous examples all work more or less the same way. The Extended
Wigner’s Friend experiment [9] is a more recent proposal that beautifully
highlights the problems one runs into when rejecting hidden variables
theories, but works entirely differently.

The Extended Wigner’s Friend Scenario is a Bell-type test with two entangled
particles, one measured by Charlie and one by Debbie (the friends) in
spatially separated laboratories. Each of the two observers is observed by a
super-observer, Alice and Bob respectively. Charlie and Debbie measure an
entangled state and Alice and Bob then further measure correlations between
Alice’s and Bob’s measurement outcomes. It turns out that, given suitable
measurement settings, the observers cannot all agree on what the measurement
results are.

The issue with this experiment is a fuzzy definition of what constitutes a
measurement. As already pointed out in [24, 35], one can interpret the
friends’ measurements as either having resulted in a definite outcome, or as
still being in an entangled quantum state. The resulting contradiction does
not occur if one has a theory in which a measurement outcome is an unambiguous
result of the time-evolution.

In a $\psi$-ensemble theory, the friends either make a measurement and
violations of Statistical Independence are for all practical purposes
destroyed. Then the super-observers can no longer measure quantum correlations
between the friends. Or the friends really just make a transformation on a
quantum state, rather than a measurement, in which case the super-observers
can well observe further quantum effects.

##  VIII Conclusion

We have argued here that the $\psi$-ontic/epistemic distinction is not
descriptive of hidden variables models that can restore locality and causality
by giving a physical meaning to the wave-function update. Instead, they can
better be described as $\psi$-ensemble theories. If they are local, causal,
deterministic and reproduce the predictions of quantum mechanics, then such
theories violate the assumptions of both Bell’s theorem and the PBR-theorem.
We have shown that taking this possibility seriously can help making sense of
some otherwise strange quantum phenomena.

##  IX Acknowledgements

We thank James Ladyman, Sophie Inman, Tim Palmer and John Rarity for useful
discussions. SH acknowledges support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under grant number HO 2601/8-1. JRH is
supported by the University of York’s EPSRC DTP grant EP/R513386/1, and the
EPSRC Quantum Communications Hub (funded by the EPSRC grant EP/M013472/1).

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