Subsystem Phase Matrices

• Subsystem Phase Matrices are constructions that encode phase information on a reduced, subsystem level, capturing relative phase data rather than global scalars.
• They play a key role in distinguishing strong versus weak SSPT orders and reformatting phase transitions in monitored quantum circuits, with measurable effects on entanglement properties.
• The concept extends to singular value and Schmidt decompositions and network theory, where diagonal phase factors and reduced matrices preserve phase characteristics under truncation and composition.

Searching arXiv for the cited papers and closely related terminology.
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{"query":"id:2311.16523 OR \"Phase Preservation of N-Port Networks under General Connections\"","max_results":5,"sort_by":"relevance"}
{"query":"id:2205.07607 OR \"On the Phases of a Semi-Sectorial Matrix\"","max_results":5,"sort_by":"relevance"}
{"query":"id:2203.12579 OR \"Phase Factors in Singular Value Decomposition and Schmidt Decomposition\"","max_results":5,"sort_by":"relevance"}
{"query":"id:2511.18969 OR \"Web of Non-invertible Dualities for (2+1) Dimensional Models with Subsystem Symmetries\"","max_results":5,"sort_by":"relevance"}
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Subsystem phase matrices, as an Editor's term, denote matrix-valued, cocycle-valued, or sector-valued encodings of phase information attached to subsystems rather than to an undivided global system. In current literature the expression is not a single standardized formal object. Instead, it names a family of constructions that recur in several technical settings: the commutator data of left/right subsystem symmetries across a cut in strong SSPT classification; diagonal phase-factor matrices acting on one subsystem in Schmidt decomposition; phase lists, phase intervals, and phase cones associated with compressions, Schur complements, and interconnected subnetworks; and, more indirectly, the symmetry-resolved phase structure of subsystem-symmetric monitored circuits and subsystem dualities [1808.05300] [2203.12579] [2205.07607] [2311.16523] [2407.18340].

1. Conceptual scope

A common structural feature across these settings is that “phase” is not attached merely to a scalar order parameter. It is attached to a reduced object: a half-space symmetry action, a subsystem basis, a compression to a subspace, a Schur complement, or a subnetwork. This suggests that subsystem phase matrices are best understood as reduced or relative phase data that remain meaningful after factorization, truncation, or composition.

Setting	Subsystem notion	Phase data
Strong SSPT classification	Left/right symmetry factors near a cut	(\phi(g_1,g_2)), (\beta(g))
Schmidt decomposition	Local subsystem (A) or (B)	(P_A), (P_B)
Semi-sectorial and network phase theory	Compression, Schur complement, subnetwork	(\phi_i(C)), ([\underline{\phi},\overline{\phi}])

In the SSPT setting, the relevant object is not a single 1D projective representation but the projective noncommutativity between left and right pieces of subsystem symmetry. In Schmidt decomposition, the relevant object is a diagonal phase rotation on one subsystem alone. In semi-sectorial matrix theory and in (n)-port network theory, the relevant object is the phase sector occupied by a reduced or interconnected matrix. The monitored-circuit and duality literatures use “phase” differently, but they supply the physical contexts in which subsystem symmetry changes phase structure, criticality, and sector organization.

2. Cocycle-valued phase data in strong SSPT order

The most direct algebraic realization of a subsystem phase matrix appears in the classification of 2D SSPTs with linear subsystem symmetries. For a finite abelian onsite group (G_s), the subsystem symmetries act along rigid lines,
[
S^{v}{x}(g)=\prod{y=-\infty}^{\infty}u_{xy}(g),\qquad
S^{h}{y}(g)=\prod{x=-\infty}^{\infty}u_{xy}(g),
]
and the classification distinguishes weak SSPTs from strong SSPTs by introducing strong equivalence under a linearly symmetric local unitary (LSLU). Under this notion, weak SSPTs built from decoupled 1D SPT chains are trivial background structure, whereas strong SSPTs are intrinsically 2D [1808.05300].

The classification theorem is
[
\mathcal{C}[G_s] = \mathcal{H}^{2}[G_s^2,U(1)]/(\mathcal{H}^{2}[G_s,U(1)])^3.
]
Here the numerator classifies projective representations of the product group (G_s^2=G_s^{\mathrm{left}}\times G_s^{\mathrm{right}}), which is the symmetry seen when a 2D system is cut and the effective 1D degrees of freedom near the cut are examined. The quotient removes precisely the 1D-chain ambiguities compatible with subsystem geometry, so the result is “2D modulo 1D.”

The phase data are carried by a factor system (\omega) for a projective representation (V(g)),
[
V(g_1)V(g_2)=\omega(g_1,g_2)V(g_1g_2),
]
with cocycle condition
[
\omega(g_1,g_2)\omega(g_1g_2,g_3)=\omega(g_1,g_2g_3)\omega(g_2,g_3),
]
and rephasing equivalence
[
\omega'(g_1,g_2)=\frac{\alpha(g_1g_2)}{\alpha(g_1)\alpha(g_2)}\,\omega(g_1,g_2).
]
For finite abelian groups, the associated commutator phase
[
\phi(g_1,g_2)=\frac{\omega(g_1,g_2)}{\omega(g_2,g_1)}
]
is the basic matrix-like entry. The strong invariant is the cross-term
[
\beta(g)=\phi(g^L,g^R)=\frac{\omega(g^L,g^R)}{\omega(g^R,g^L)},
]
with (g^L) and (g^R) denoting the same group element acting on the left and right sides of the cut. In this sense, the “matrix” is the table of pairwise commutation phases between symmetry generators across the cut.

The paper also gives an alternative algebraic description,
[
\mathcal{C}[G_s]=\mathcal{A}[G_s]/\mathcal{A}^{\mathrm{anti}}[G_s],
]
where (\mathcal{A}[G]) is the group of bilinear functions (G\times G\to U(1)). This is explicitly matrix-like: the phase class is encoded by a bilinear commutation form, modulo antisymmetric pieces removable by 1D stacking. Examples emphasize how many independent entries survive the quotient:
[
\mathcal{C}[\mathbb{Z}2\times \mathbb{Z}_2]=(\mathbb{Z}_2)^3,\qquad
\mathcal{C}[\mathbb{Z}_N]=\mathbb{Z}_N,
]
and
[
\mathcal{C}[\mathbb{Z}_n\times\mathbb{Z}_m]=\mathbb{Z}_n\times\mathbb{Z}_m\times\mathbb{Z}{\gcd(n,m)}.
]

A recurrent misconception is that any nonzero spurious topological entanglement entropy identifies a strong SSPT. The classification result is narrower: strong SSPTs necessarily exhibit spurious topological entanglement entropy on a cylinder, but the converse is false. A spurious TEE can also arise from weak or disjoint SSPT structure.

3. Subsystem symmetry and phase transitions in monitored circuits

In subsystem-symmetric monitored quantum circuits, the phrase is not used as a literal matrix object, but subsystem symmetry reorganizes the phase structure in a way that makes the preceding SSPT data physically consequential. The studied setup is a monitored Clifford circuit on an (L\times L) torus, evolved one operation at a time. At each step one chooses among
[
U \quad\text{(5-qubit Clifford unitary)} \quad p_u,\qquad
M^z \quad\text{(single-site computational-basis measurement)} \quad p_M^z,\qquad
M^s \quad\text{(cluster-stabilizer measurement)} \quad p_M^s,
]
with probabilities summing to one. The unitary (U) acts on a “(+)”-shaped five-qubit stencil centered on a randomly chosen site, and (M^s) measures the corresponding 2D cluster-state stabilizer. Three ensembles are compared: unconstrained Cliffords with trivial symmetry (\mathds{I}), globally symmetric Cliffords respecting checkerboard (\mathbb{Z}_2\times\mathbb{Z}_2), and subsystem-symmetric Cliffords respecting (\mathbb{Z}_2^{2L-1}), with hierarchy
[
\mathds{I}\subset \mathbb{Z}_2\times\mathbb{Z}_2\subset \mathbb{Z}_2^{2L-1}.
]
Across all ensembles, the phase diagram contains one volume-law phase and two distinct area-law phases: a trivial area-law phase and a cluster area-law phase associated with 2D cluster-state stabilizers [2407.18340].

The relevant diagnostics are entanglement-based observables. The 7-term topological entanglement entropy is
[
S_{\text{top}} \equiv S_{AB}+S_{BC}+S_{AC}-S_A-S_B-S_C-S_{ABC},
]
evaluated on cylindrical subregions. The ancilla entropy is
[
S_{\text{anc}}=S_{\mathrm{vN}}(\rho_{\text{anc}}),
]
and the dumbbell entropy distinguishes the cluster area-law phase from the trivial area-law phase, especially in the pure-measurement limit. Finite-size scaling uses
[
\Delta(L,p)\approx L^{\gamma}F!\left((p-p_c)L^{1/\nu}\right),
]
with (\gamma) found to be approximately zero in practice and set to zero in the reported figures.

The critical results are numerically sharp:

Ensemble	Volume (\to) cluster area-law	Volume (\to) trivial area-law
Unconstrained Cliffords	(p_c\approx 0.068,\ \nu\approx 0.90)	(p_c\approx 0.193,\ \nu\approx 0.85)
Global (\mathbb{Z}_2\times\mathbb{Z}_2)	(p_c\approx 0.084,\ \nu\approx 0.83)	(p_c\approx 0.170,\ \nu\approx 0.83)
Subsystem (\mathbb{Z}_2^{2L-1})	(p_c\approx 0.176,\ \nu\approx 0.38)	(p_c\approx 0.612,\ \nu\approx 0.52)

The unconstrained and globally symmetric ensembles remain near the ((2+1))D percolation value (\nu\approx 0.87\text{–}0.88), and the pure-measurement critical point near (p_c\approx 0.5) with (\nu\approx 0.84) is also reported as consistent with 2D percolation. By contrast, the subsystem-symmetric SSPT ensemble is the outlier: for the volume-to-cluster transition the fitted (\nu) remains in the range (\nu\approx 0.387\text{–}0.406), depending slightly on geometry and fitting routine.

The authors interpret these distinct (\nu) values as distinct spacetime response scales. The broad phase structure is similar across symmetry classes, but subsystem symmetry changes the criticality qualitatively and enlarges the area-law regions because the allowed unitaries are less entangling. A common overgeneralization would be to infer that any symmetry protection modifies monitored-circuit universality. The numerical result is more specific: global (\mathbb{Z}_2\times\mathbb{Z}_2) symmetry leaves the percolation-like criticality largely intact, whereas extensive subsystem symmetry reshapes the transition itself. The paper further links this to MBQC, arguing that the 2D cluster state is relevant precisely because its extensive subsystem symmetry underlies universal MBQC.

4. Diagonal phase-factor matrices in SVD and Schmidt decomposition

A narrower but literal meaning of subsystem phase matrices arises in singular value decomposition and Schmidt decomposition. For a complex matrix (A),
[
A = UDV^\dagger,
]
but the singular vectors are unique only up to unit-modulus phase factors. The paper therefore introduces diagonal phase-factor matrices and first writes
[
A = U_0 D_0 V_0^\dagger,
]
where (U_0) and (V_0) are built from eigenvectors of (AA^\dagger) and (A^\dagger A), ignoring phases, while (D_0) is a complex diagonal matrix. If the diagonal entries of (D_0) are (d_j), then
[
d_j = \sigma_j e^{i(\alpha_j+\beta_j)}, \qquad \sigma_j = |d_j|.
]
The phase product
[
e^{i(\alpha_j+\beta_j)} = \frac{d_j}{|d_j|}
]
is fixed, although the split into (\alpha_j) and (\beta_j) is arbitrary [2203.12579].

The consistent reconstruction is
[
U = U_0 D_\alpha,\qquad V = V_0 D_\beta,
]
with
[
D_\alpha = \operatorname{diag}(e^{i\alpha_j}), \qquad
D_\beta = \operatorname{diag}(e^{-i\beta_j}),
]
so that (A=UDV^\dagger). The paper summarizes this as a three-step procedure: construct (U_0) and (V_0) from eigenvectors, solve (U_0D_0V_0^\dagger=A), then distribute the phases into (U) and (V). All the phase may be assigned to one side, or split arbitrarily, provided the product constraint is respected.

For a bipartite state
[
|\psi\rangle = \sum_{jk} a_{jk} |j\rangle |k\rangle,
]
the coefficient matrix (A=(a_{jk})) yields the Schmidt decomposition. The phase-factor matrices then become phase rotation operators acting on one subsystem alone. Writing
[
|i_A\rangle = e^{i\alpha_i}|u_i\rangle, \qquad
|i_B\rangle = e^{i\beta_i}|v_i\rangle,
]
one may introduce
[
P_A = \operatorname{diag}(e^{i\alpha_0}, e^{i\alpha_1}, \dots), \qquad
P_B = \operatorname{diag}(e^{i\beta_0}, e^{i\beta_1}, \dots),
]
so that the same physical state is represented with locally rephased subsystem bases. In this formulation, a subsystem phase matrix is literally a local diagonal unitary on subsystem (A) or (B). The entanglement content is unchanged; only the local basis choice varies.

5. Phase cones for reduced matrices and interconnected subnetworks

A different formalization arises from the phase theory of sectorial and semi-sectorial matrices. For a sectorial matrix (C\in\mathbb C^{n\times n}), one has a decomposition
[
C=T^*DT,
]
where (T) is nonsingular and (D) is diagonal unitary. The phases are the ordered arguments of the diagonal entries of (D),
[
\overline{\phi}(C)=\phi_1(C)\ge \cdots \ge \phi_n(C)=\underline{\phi}(C),
\qquad
\overline{\phi}(C)-\underline{\phi}(C)<\pi.
]
The theory extends these notions to semi-sectorial matrices, including singular cases and rotated Hermitian degeneracies. The subsystem-relevant operations are compression and generalized Schur complementation. For a full-column-rank (X), a compression
[
\widetilde C=X^*CX
]
is again semi-sectorial when nonzero, and its phases interlace those of (C):
[
\phi_j(C)\ge \phi_j(\widetilde C)\ge \phi_{r-\tilde r+j}(C),\qquad j=1,\dots,\tilde r.
]
For the generalized Schur complement
[
C\backslash_{11}=C_{22}-C_{21}C_{11}^\dagger C_{12},
]
analogous interlacing holds under the stated range conditions. The pseudoinverse reverses and negates the phase list,
[
\phi_i(C^\dagger)=-\phi_{r-i+1}(C),\qquad i=1,\dots,r.
]
These results make phase data stable under subsystem reduction and elimination [2205.07607].

The same language becomes explicitly subsystem-based in (n)-port LTI network theory. For an impedance matrix (Z(s)), the frequency response (Z(j\omega)) is called frequency-wise (semi-)sectorial when each (Z(j\omega)) is (semi-)sectorial. The frequency-wise phase sector set is
[
\mathcal{Z}[\alpha(\omega),\beta(\omega)] =
\left{Z(j\omega)\in\mathbb{C}^{n\times n}:
Z(j\omega)\ \text{is (semi-)sectorial and }
[\overline{\phi}(Z(j\omega)),\underline{\phi}(Z(j\omega))]
\subset[\alpha(\omega),\beta(\omega)]
\right},
]
and the frequency-independent class is
[
\mathscr{Z}[\alpha,\beta] =
\left{Z\in\mathcal{R}^{n\times n}:
Z\ \text{is frequency-wise (semi-)sectorial and }
[\overline{\phi}(Z),\underline{\phi}(Z)]\subset[\alpha,\beta]
\right}.
]
Passivity is the special case
[
Z\in \mathscr{Z}\big[-\pi/2,\pi/2\big].
]

The central preservation theorem states that if
[
Z^a,\ Z^b\in \mathcal{Z}[\alpha,\beta],
]
then standard interconnections preserve the same phase sector. This includes open, shorted, series, parallel, hybrid, cascade, and hybrid-cascade connections, with representative operations such as
[
Z^c=Z^a+Z^b,\qquad
Z^c=Z^a\vcentcolon Z^b = Z^a(Z^a+Z^b)^\dagger Z^b,\qquad
Z^c=Z^a/_{22}.
]
The theory also provides a unified subtraction result: when (Z^b) and (Z^c) are known and their phase intervals are disjoint modulo (2\pi), the unknown subnetwork (Z^x) lies in an explicitly determined maximal allowable phase range independent of the subtraction type. The final extension to confluence-induced connections shows that all physically realizable network connections preserve the phase sector [2311.16523].

A plausible implication is that, in this matrix-analysis and network-theory lineage, “subsystem phase matrices” are most naturally the phase-bounded reduced matrices associated with compressions, Schur complements, and subnetworks, rather than a new primitive object.

6. Symmetry-twist sector maps and non-invertible subsystem dualities

In two-dimensional lattice models with subsystem symmetries, phase data can also be organized by symmetry sectors and twist sectors. On an (L_x\times L_y) square lattice with a spin-(\tfrac12) on each vertex, the subsystem (\mathbb{Z}2\times\mathbb{Z}_2) symmetry is generated by
[
U^x_j = \prod{i=1}^{L_x}\sigma^x_{i,j}, \qquad
U^y_i = \prod_{j=1}^{L_y}\sigma^x_{i,j},
]
subject to
[
\prod_{i=1}^{L_x} U^y_i \prod_{j=1}^{L_y} U^x_j = 1.
]
There are therefore (L_x+L_y-1) independent generators and (2^{L_x+L_y-1}) symmetry sectors. The twist variables (t^x_{j-\frac12}), (t^y_{i-\frac12}), together with (t^{xy}), are constrained in the same way, again leaving (2^{L_x+L_y-1}) twist sectors. The full Hilbert space decomposes into
[
4^{L_x+L_y-1}
]
symmetry-twist sectors [2511.18969].

The paper studies three phase types: spontaneous subsystem symmetry breaking (SSSB), a trivial paramagnet, and SSPT phases, including weak SSPTs and the strong 2D cluster-state SSPT. Two duality maps organize them. The subsystem Kramers-Wannier duality maps SSSB and trivial phases into one another. The subsystem Kennedy-Tasaki transformation, constructed as
[
N = N^\dagger U_{\text{DW}} N,
]
maps SSSB to strong SSPT while leaving the trivial phase invariant.

The phase-matrix aspect appears in the explicit sector maps. For the subsystem KW duality, symmetry sectors and twist sectors are exchanged:
[
\widehat u^x_{j-1}\longrightarrow t^x_{j-\frac12}, \qquad
\widehat u^y_{i-1}\longrightarrow t^y_{i-\frac12}, \qquad
t^x_j \longrightarrow u^x_j, \qquad
t^y_i \longrightarrow u^y_i.
]
For the subsystem KT map, the symmetry labels are fixed while the twist labels are shifted by symmetry charges:
[
{u'\sigma,u'\tau}={u_\sigma,u_\tau},
]
[
t'^x_{\sigma,j-1}=t^x_{\sigma,j-1}+u^x_{\tau,j-1},\qquad
t'^y_{\sigma,i-1}=t^y_{\sigma,i-1}+u^y_{\tau,i-1},
]
[
t'^x_{\tau,j}=t^x_{\tau,j}+u^x_{\sigma,j},\qquad
t'^y_{\tau,i}=t^y_{\tau,i}+u^y_{\sigma,i}.
]
This is matrix-like in the sense that one sector vector is transformed by exchange or shear operations.

Boundary conditions are decisive. On closed manifolds, both KW and KT become intrinsically non-unitary and non-invertible on the original Hilbert space. For KW,
[
N^\dagger N =
\frac12 \prod_{i=1}^{L_x}\left(1+(-1)^{t^y_i}U^y_i\right)
\prod_{j=1}^{L_y}\left(1+(-1)^{t^x_j}U^x_j\right)
\equiv \widehat P(t^x,t^y),
]
so the map is a projector onto a twist sector rather than a unitary. On open lattices, by contrast,
[
N^{\text{open}\,\dagger}N^{\text{open}}=I,
]
and the same holds for the open-boundary KT map. The non-invertibility on closed geometry reflects the mismatch of ground-state degeneracies and sector structure, not a failure of the duality construction.

The duality paper also reconnects with the bulk-edge invariants of strong SSPT order. For the strong 2D cluster SSPT, the repair operators satisfy
[
V_{ij}(g_\sigma)=\tau^z_{i-1,j-1},\quad
V_{ij}(g_\tau)=\sigma^z_{i-1,j-1},\quad
V_{ij}(g_\sigma g_\tau)=\tau^z_{i-1,j-1}\sigma^z_{i-1,j-1},
]
and the bulk invariant takes the values
[
\beta(g_\sigma)=\beta(g_\tau)=1,\qquad
\beta(g_\sigma g_\tau)=-1.
]
For the weak SSPT, all (\beta(g)=1). Under KT, the local repair operators become highly nonlocal in the dual SSSB description, but the commutation phase is preserved. This clarifies an important point: subsystem phase data need not remain local under duality, yet their algebraic content can remain intact.

In aggregate, the literature supports two complementary uses of subsystem phase matrices. One is algebraic and local, where phases are encoded by cocycles, commutator forms, or diagonal local rephasings. The other is structural and compositional, where reduced matrices, symmetry-twist sectors, and network interconnections carry phase information that survives truncation, composition, or duality.