Plus State Transfer in Quantum Walks

• Plus state transfer is the symmetric quantum evolution of two-vertex superposition states on a graph, key for understanding coherent state routing.
• It relies on spectral properties and strong cospectrality, with various Hamiltonians enabling perfect and pretty good state transfer in well-classified graph families.
• The framework bridges theoretical spectral graph analysis and practical applications in multi-qubit and photonic implementations by examining phase coherence in superposition states.

Plus state transfer is the transfer, under continuous-time quantum evolution on a graph, of a symmetric two-vertex superposition of the form

$$\frac{1}{\sqrt{2}}\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)$$

to another state of the same form. In the standard graph-based model, a real symmetric or Hermitian matrix $M$ associated with a graph $G$ generates a unitary walk

$U_M(t)=e^{itM},$

and plus perfect state transfer (plus PST) from $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$ to $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$ occurs when

$$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$

for some time $\tau>0$ and phase $\gamma$ with $|\gamma|=1$. In the recent literature, plus states appear as the $M$0 case of $M$1-pair states, as symmetric states attached to graph involutions, and as the symmetric counterpart of pair states $M$2; their transfer properties have been analyzed for adjacency, Laplacian, signless Laplacian, $M$3-Laplacian, and generalized Laplacian dynamics, with complete classifications in several graph families and with explicit links to fractional revival, quotient reductions, line graphs, and multi-qubit transfer on paths (Kim et al., 2024, Godsil et al., 12 Feb 2025, Mohapatra et al., 18 Mar 2026).

1. Definition and placement within state-transfer theory

A plus state is the symmetric superposition

$M$4

which is the $M$5 specialization of the $M$6-pair state

$M$7

The same framework contains pair states at $M$8, and more general real pure states beyond two-vertex support. In the pure-state formulation, plus PST is simply a special case of perfect state transfer between real pure states (Kim et al., 2024, Godsil et al., 12 Feb 2025).

The ambient transfer model is the continuous-time quantum walk. For a graph $M$9 with adjacency matrix $G$0, the foundational formulation uses

$G$1

and defines perfect state transfer from a state $G$2 to a state $G$3 by $G$4 for some phase $G$5. The same phase-invariant notion is used for Laplacian and other graph Hamiltonians, while pretty good state transfer (PGST) replaces exact equality by arbitrarily good approximation along a sequence of times (Godsil, 2011, Godsil et al., 12 Feb 2025).

The literature now treats plus states as a distinct transfer object rather than as a minor variant of vertex PST. That distinction is substantive. In one direction, plus states can display transfer behavior not shared by their individual vertex components. In another, transferring a known fixed plus state is much weaker than transferring an arbitrary unknown state: for a known $G$6, one can trivially achieve fidelity $G$7 by local preparation at the receiver, which does not constitute genuine universal state transfer capability (Bommel, 2019, Liu et al., 2012).

2. Spectral structure and necessary conditions

The central spectral decomposition is

$G$8

with $G$9 the spectral projectors. For a real pure state $U_M(t)=e^{itM},$0, its eigenvalue support is

$U_M(t)=e^{itM},$1

For plus states, the support is computed from $U_M(t)=e^{itM},$2, and all PST constraints are imposed only on that support (Godsil et al., 12 Feb 2025).

A necessary condition for PST between real pure states is strong cospectrality. For plus PST between $U_M(t)=e^{itM},$3 and $U_M(t)=e^{itM},$4, this takes the form

$U_M(t)=e^{itM},$5

The general real-state criterion then adds arithmetic restrictions on the supported eigenvalues. If the support has at least three eigenvalues and is closed under algebraic conjugation, the supported eigenvalues must be either all integers or all of the form

$U_M(t)=e^{itM},$6

with $U_M(t)=e^{itM},$7 square-free, together with a parity condition distinguishing the $U_M(t)=e^{itM},$8 and $U_M(t)=e^{itM},$9 spectral sectors. In the two-eigenvalue case, strong cospectrality alone is sufficient (Godsil et al., 12 Feb 2025).

Periodicity remains the bridge between structure and transfer. A real state is periodic iff its support satisfies the ratio condition, and every periodic real pure state admits PST with another real pure state at half the period. For $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$0-pair states, the paper on generalized pair transfer shows that $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$1 is periodic iff its support satisfies the ratio condition; for cospectral vertices, the $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$2 cases are exceptional in that periodicity of the combined state can occur even when the individual vertex states do not share the same behavior (Kim et al., 2024). This suggests that plus states are not merely reducible to two independent vertex channels.

3. Classified families and Hamiltonian dependence

Several graph families now have complete classifications of plus PST or plus PGST. The results depend sharply on the Hamiltonian.

Family	Hamiltonian	plus-state result
Complete graphs $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$3	Adjacency, hence also $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$4 for regularity	No perfect plus state transfer between distinct plus states
Paths $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$5	Adjacency	plus PST iff $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$6
Paths $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$7	Laplacian	plus PST iff $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$8
Paths $\frac{1}{\sqrt{2}}(\mathbf{e}_a+\mathbf{e}_b)$9	Unsigned Laplacian	plus PST iff $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$0
Cycles $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$1	Adjacency	perfect plus PST only on $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$2 and $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$3
Cycles $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$4	Unsigned Laplacian	plus PST iff $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$5
Cycles $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$6, $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$7	Adjacency, Laplacian, signless Laplacian on regular graphs	plus PGST iff $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$8 with $$\frac{1}{\sqrt{2}}(\mathbf{e}_\alpha+\mathbf{e}_\beta)$$9
Complete bipartite $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$0	Adjacency	explicit plus PST families including $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$1, and cases with a part of size $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$2
Complete bipartite $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$3	Laplacian	plus PST in $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$4 and in $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$5 or $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$6 with odd $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$7

These classifications come from explicit spectral analysis in paths and complete bipartite graphs, from the $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$8-pair theory on cycles, and from the complete characterization of pretty good plus state transfer in cycles and their complements (Godsil et al., 12 Feb 2025, Kim et al., 2024, Chen et al., 2019, Mohapatra et al., 18 Mar 2026).

The cycle case is especially revealing. For perfect transfer, the adjacency-based $$U_M(\tau)\bigl(\mathbf{e}_a+\mathbf{e}_b\bigr)=\gamma\bigl(\mathbf{e}_\alpha+\mathbf{e}_\beta\bigr)$$9-pair analysis yields plus PST on $\tau>0$0 and $\tau>0$1, whereas the unsigned-Laplacian treatment yields plus PST only on $\tau>0$2. For pretty good transfer, the later cycle theory proves that $\tau>0$3 and $\tau>0$4 admit pretty good plus state transfer if and only if $\tau>0$5 with $\tau>0$6, and moreover every plus state in $\tau>0$7 has plus PGST; for complements, every plus state in $\tau>0$8 has plus PGST for $\tau>0$9 (Kim et al., 2024, Chen et al., 2019, Mohapatra et al., 18 Mar 2026).

The path case shows equally strong Hamiltonian dependence. Under adjacency, plus PST occurs only for $\gamma$0, $\gamma$1, and $\gamma$2, with the explicit transfers

$\gamma$3

at times $\gamma$4, $\gamma$5, and $\gamma$6, respectively. Under the Laplacian, only $\gamma$7 survives, with plus PST between $\gamma$8 and $\gamma$9 at time $|\gamma|=1$0. Under the unsigned Laplacian, plus PST occurs exactly on $|\gamma|=1$1 and $|\gamma|=1$2 (Godsil et al., 12 Feb 2025, Chen et al., 2019).

The complete-graph obstruction is absolute in the unweighted regular setting. For adjacency, there is no perfect $|\gamma|=1$3-pair state transfer between distinct $|\gamma|=1$4-pair states in $|\gamma|=1$5, hence no plus PST; because $|\gamma|=1$6 is regular, the same obstruction carries over to Laplacian and signless Laplacian evolution up to global phases (Kim et al., 2024).

4. Involutions, quotient reductions, and other transfer mechanisms

Graphs with non-trivial involutions provide the most systematic reduction theory for plus states. If $|\gamma|=1$7 is an involution, the symmetric combinations

$|\gamma|=1$8

span the $|\gamma|=1$9-eigenspace of the involution operator, and the antisymmetric combinations span the $M$00-eigenspace. The generalized-Laplacian and $M$01-Laplacian analyses show that the walk block-diagonalizes with respect to this decomposition, and plus PST on the full graph is equivalent to vertex PST on a smaller induced block $M$02 or its $M$03-Laplacian analogue (Ojha et al., 25 Sep 2025, Ojha et al., 22 Apr 2026).

This reduction is explicit rather than heuristic. In the involution basis, the propagator takes a block form in which the symmetric block governs plus states and the antisymmetric block governs pair states. Consequently, plus-state transfer in the full graph can be decided by solving an ordinary vertex-PST problem in the symmetric half-graph with potentials. The 2025 $M$04-Laplacian paper states this equivalence for pair or plus states in graphs with involutions, while the 2026 generalized-Laplacian paper formulates the same principle for $M$05 and uses it to construct large families with state transfer after adding only a few loops or edges (Ojha et al., 25 Sep 2025, Ojha et al., 22 Apr 2026).

A second mechanism comes from double covers and fractional revival. The 2026 cycle paper proves that fractional revival in a graph lifts to fractional revival between plus states in its double cover. In particular, vertex FR in a base graph $M$06 corresponds to FR, and hence to plus PGST in special cases, between states of the form

$M$07

in the double cover. This is the mechanism behind the complete cycle/complement characterization and its translation to weighted paths with potentials (Mohapatra et al., 18 Mar 2026).

A third equivalence relates plus states on edges to line-graph vertex states. If $M$08 is a graph and $M$09 its line graph, then PST between vertices in $M$10 relative to the adjacency matrix is equivalent, under explicit necessary and sufficient conditions, to PST between the plus states formed by the corresponding edges in $M$11 relative to the signless Laplacian. For trees and non-bipartite unicyclic graphs, this equivalence becomes exact because the incidence matrix has full column rank (Kim et al., 2024).

Finally, on bipartite graphs there is a direct signed/unsigned Laplacian correspondence. If $M$12 is the diagonal sign matrix for the bipartition, then

$M$13

As a result, perfect pair state transfer under the Laplacian is equivalent to perfect plus state transfer under the unsigned Laplacian between corresponding edge-supported states. This equivalence explains why several path and even-cycle classifications appear in parallel pair/plus forms (Chen et al., 2019).

5. Pretty good transfer, superposition advantage, and plus states on paths

The plus-state literature is not confined to exact transfer. In unmodulated XX chains modeled by path graphs, the multi-qubit PGST framework treats arbitrary single-excitation states

$M$14

and their mirror states $M$15. In this setting, a plus state is a two-site superposition such as

$M$16

and PGST depends only on the eigenvalue support of the state, not on whether the state is a single vertex or a superposition (Bommel, 2019).

The structural result is that mirror states on paths are strongly cospectral, with

$M$17

so the PGST problem becomes number-theoretic. Theorem 4.3 in that paper gives a necessary and sufficient condition for PGST between $M$18 and $M$19 in terms of integer relations among the supported eigenvalues. This lifts the usual single-vertex criterion to arbitrary states, including plus states (Bommel, 2019).

A central example is $M$20, where neither $M$21 nor $M$22 has PGST individually, but

$M$23

does admit pretty good mirror transfer to

$M$24

This directly refutes the misconception that superposition transfer is merely inherited componentwise from vertex transfer. In these path models, plus-type states can behave better than the basis states from which they are built (Bommel, 2019).

The same paper gives an infinite family of path lengths $M$25, with $M$26 an odd prime and $M$27, for which two-site plus states

$M$28

have PGST to their mirrors whenever $M$29 is odd and $M$30. This is one of the clearest demonstrations that passive, Hamiltonian-only transfer of superposition states can persist well beyond the range of exact single-site PST (Bommel, 2019).

6. Physical interpretation and relation to broader quantum-state transfer

In operational terms, plus state transfer is a transfer problem for a single known symmetric superposition, not for an arbitrary unknown input. That distinction is explicit in the bipartite-operation formulation of quantum state transfer power. The latter averages over all input pure states and optimizes over receiver initialization and basis identification, whereas a fixed plus state $M$31 can be reproduced locally at the receiver with fidelity $M$32 even by a useless channel. For that reason, “plus-state transfer” is a strictly weaker task than unknown-state transfer, and good performance on $M$33 alone says little about universal state-transfer capability (Liu et al., 2012).

At the same time, plus states are physically meaningful test states because they probe coherent preservation of relative phase. In the photonic implementation of perfect state transfer, the polarization qubit state

$M$34

plays the role of a plus-type state. The experiment included $M$35 and $M$36 in single-qubit process tomography and demonstrated that the engineered PST device preserved polarization superpositions while relocating the spatial mode. After compensation, the average process fidelity was

$M$37

and for entangled-state routing across the three implemented transfers the average polarization-state fidelity was

$M$38

Because the Hamiltonian acted identically on $M$39 and $M$40, any superposition, including $M$41, was transferred correctly up to a calibrated local unitary (Chapman et al., 2016).

A plausible implication is that plus-state transfer occupies an intermediate position between vertex transfer and full unknown-state transfer. It is more sensitive than basis-state transport because it tests coherent interference, yet less demanding than universal transfer because the target manifold is only a one-parameter slice of the Bloch sphere or, more generally, a special family of real pure states. The current graph-theoretic literature reflects exactly that position: plus states are rich enough to expose new phenomena—such as involution reductions, double-cover correspondences, and superposition-only PGST—while still admitting explicit classifications that remain out of reach for general pure-state transport (Godsil et al., 12 Feb 2025, Mohapatra et al., 18 Mar 2026).

The modern theory therefore treats plus state transfer as a distinct and technically structured subject: a special case of real pure-state transfer, a symmetric counterpart of pair-state transfer, and a bridge between spectral graph theory, fractional revival, and experimentally relevant coherent routing of superposition states.