I.2 Quantum seed rank and cyclicity
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I.2.1 Finite observable algebras
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Let a finite-dimensional unital \(*\)-subalgebra of operators act in its standard multiplicity form
on
Here every \(d_i\) is a positive integer and every multiplicity \(m_i\) is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector \(\psi_i\in{\mathbb C}^{d_i}\otimes{\mathbb C}^{m_i}\) is identified with a \(d_i\times m_i\) coefficient matrix \(C_i\).
Theorem I.2.1: Orbit dimension of one quantum seed
For integers \(d\ge1\) and \(m\ge0\), consider the action of \(M_d({\mathbb C})\otimes I_m\) on \({\mathbb C}^d\otimes{\mathbb C}^m\). Let \(C\) be the coefficient matrix of a vector \(\psi\). Then
The seed is cyclic if and only if \({\operatorname{rank}} C=m\), which is possible only when \(m\le d\).
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Proof of Theorem I.2.1
Under the identification by coefficient matrices, \((A\otimes I_m)\psi\) corresponds to \(AC\). Every row of \(AC\) lies in the row space of \(C\). Conversely, any \(d\times m\) matrix whose rows lie in the row space of \(C\) equals \(AC\) for a suitable \(A\): choose each row of \(A\) to produce the desired row combination. The orbit therefore consists of all matrices with \(d\) independently chosen rows in the \({\operatorname{rank}} C\)-dimensional row space of \(C\), and has dimension \(d{\operatorname{rank}} C\). It equals the full \(dm\)-dimensional space exactly when \({\operatorname{rank}} C=m\).
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Theorem I.2.2: Quantum seed-rank formula
For
the minimum number of seeds required to generate \({\mathcal H}\) is
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Proof of Theorem I.2.2
Each seed contributes a coefficient matrix of rank at most \(d_i\) in the \(i\)th block. Thus \(k\) seeds can cover at most a \(kd_i\)-dimensional subspace of the multiplicity factor \({\mathbb C}^{m_i}\), so \(k\ge\lceil m_i/d_i\rceil\) for every \(i\).
For the converse, take
In each block, partition a basis of \({\mathbb C}^{m_i}\) into at most \(k\) groups of size at most \(d_i\). Choose the coefficient matrix of seed \(a\) so that its nonzero rows span the \(a\)th group. The combined row spaces of the \(k\) seeds then span \({\mathbb C}^{m_i}\) in every block. The preceding theorem shows that their \({\mathcal A}\)-orbits generate the whole direct sum.
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Corollary I.2.3: Local orbit and Schmidt rank
Let \(\psi\in{\mathcal H}_A\otimes{\mathcal H}_B\), with \(\dim{\mathcal H}_A=d_A\ge1\), and let \(r\) be its Schmidt rank. The reduced positive operator on \(B\) is \(\rho_B={\operatorname{Tr}}_A({\left|\psi\right\rangle}{\left\langle\psi\right|})\). Normalization of \(\psi\) does not affect its support. Then
In particular, local operators on \(A\) generate the complete bipartite Hilbert space exactly when the reduced state on \(B\) has full support.
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Proof of Corollary I.2.3
A Schmidt decomposition of length \(r\) gives a coefficient matrix of rank \(r\). Applying Theorem Theorem I.2.1 gives the orbit dimension \(d_A r\). The range of \(\rho_B\) is the span of the \(r\) Schmidt vectors on \(B\), so full support is equivalent to \(r=\dim{\mathcal H}_B\). Because \(d_A>0\), this is equivalent to the orbit having the full bipartite dimension.
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The conclusion concerns the algebraic span obtained from arbitrary local operators. A normalized unitary gate orbit need not contain every vector, and the span statement does not supply a deterministic protocol for preparing arbitrary remote states. Entanglement controls the multiplicity support appearing in that algebraic span.
I.2.2 A computational-basis seed
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Theorem I.2.4: One seed for the computational basis
For an integer \(n\ge0\), let
and let \(X_j\) denote the Pauli \(X\) operator on qubit \(j\). Then
is the complete computational basis. Hence the seed is cyclic under the algebra generated by the \(X_j\).
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Proof of Theorem I.2.4
For a binary address \(a=(a_1,\ldots,a_n)\),
Distinct binary addresses produce distinct orthonormal basis vectors. There are \(2^n\) of them, equal to the dimension of the Hilbert space.
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Numerical certificate I.2.5: Three-qubit seed orbit
The original numerical implementation generated all eight computational basis states from \({\left|000\right\rangle}\) using the three local \(X\) address operators. It reported an orbit rank of \(8/8\).
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