--- title: Choi Rank in Quantum Channels url: https://www.emergentmind.com/topics/choi-rank type: topic --- # Choi Rank in Quantum Channels Choi rank is the rank of the Choi matrix of a quantum channel and, equivalently, the minimum number of nonzero Kraus operators required in an operator-sum representation. For a channel \(\Phi:M_m\to M_n\), this invariant is also the rank of the associated Choi state/operator and, in the operational formulation used for channel-state duality, it equals the minimal environment dimension in a Stinespring dilation. Recent work uses Choi rank as a structural invariant for channel decompositions, a parameter in constrained channel classification, and an operational limit in entanglement-assisted communication tasks [2003.14405, 1710.11416, 2406.08360]. ## 1. Definition and basic equivalences For a quantum channel \(\Phi:M_n\to M_n\) with Kraus representation \[ \Phi(X)=\sum_{k=1}^r A_k X A_k^*, \] the Choi rank is the minimum possible number of nonzero Kraus operators in such a representation. In the standard matrix formulation, the Choi matrix may be written as \[ J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k}, \] or equivalently, with the tensor factors ordered oppositely, \[ C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}). \] The Choi rank is then \[ \operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)), \] and this equals the minimum number of nonzero terms in any Kraus form [2003.14405, 1710.11416]. The same invariant appears in the channel-state picture. For a channel \(\Phi:M_n(\mathbb C)\to M_m(\mathbb C)\), the Choi-Jamiolkowski isomorphism identifies \(\Phi\) with \(J(\Phi)\), and for a channel \(n^{-1}J(\Phi)\) is a density matrix. In the normalized state language, the Choi rank is simply the matrix rank of that bipartite state. In the operational formulation of the Choi state used for dense-coding-type tasks, \[ J_{\mathcal N_{\rm AB}}=(\mathcal N_{\rm A}\otimes \mathcal I_{\rm B})\!\left(\ketbra{\Phi^+}_{\rm AB}\right), \] and the paper on \(k\)-state exclusion defines \[ r_c^{\mathcal N} \coloneqq \operatorname{rank}(J_{\mathcal N}). \] That paper also states that \(r_c^{\mathcal N}=1\) iff \(\mathcal N\) is unitary [2211.11909, 2406.08360]. These equivalences make Choi rank the intrinsic “Kraus length” of a channel. In finite dimensions, it is the channel invariant that simultaneously packages Kraus minimality, Choi-matrix rank, and the state-space support size of the associated Choi state. ## 2. Choi-Jamiolkowski conventions and what rank does not depend on The numerical value of Choi rank is stable under the standard variants of the Choi-Jamiolkowski correspondence, even though the positivity interpretation depends on which convention is used. One formulation reviewed in "Variations on the Choi-Jamiolkowski isomorphism" writes the basis-independent Jamiołkowski operator as \[ \rho_\phi^{J}=J(\phi):=\sum_k e_k^*\otimes \phi(e_k), \] while the standard Choi form is \[ \rho_\phi^{C}=C(\phi):=\sum_{ij}|i\rangle\langle j|\otimes \phi(|i\rangle\langle j|). \] The inverse of the latter uses basis-dependent transposition, \[ C^{-1}(\rho)(a)=[\rho(a^T\otimes \mathbbm 1_B)]. \] The paper emphasizes that these versions differ by transposition, adjoint-twisted bookkeeping, and identification of \(H_A\) with \(H_A^*\), but not in the numerical rank of the resulting operator [2211.16533]. This convention-stability should be distinguished from a different rank statement that appears in "Choi matrices revisited". That paper studies when a generalized tensor \[ E\in M_n\otimes M_n \] can replace the standard maximally entangled tensor in the Choi construction while preserving the correspondence \[ \Phi \text{ is CP } \Longleftrightarrow C_\Phi\ge 0. \] Its main characterization is that the generalized tensor must be a positive rank-one matrix whose range vector has full Schmidt rank, equivalently \(E=|v\rangle\langle v|\) with \(v\) of full Schmidt rank, or equivalently the associated map must be \(\operatorname{Ad}_s\) for some nonsingular \(s\in M_n\). In the basis formulation, the Choi correspondence is preserved exactly when the basis has the form \[ b_{ij}=|\xi_i\rangle\langle \xi_j|. \] Here the relevant rank-one condition concerns the auxiliary tensor \(E\), not the channel invariant usually called Choi rank [2206.09324]. ## 3. Relations to other decomposition lengths Choi rank is often the smallest decomposition length among several natural channel representations, but it does not generally determine the minimal length of more restricted decompositions. For mixed-unitary channels, the mixed-unitary rank \(N\) is the minimum number of unitary conjugations in a convex decomposition \[ \Phi(X)=\sum_{k=1}^N p_k\,U_k X U_k^*. \] If \(r\) denotes the Choi rank, then \[ N\ge r, \] because every mixed-unitary decomposition is also a Kraus representation. The main structural bound proved in "On the mixed-unitary rank of quantum channels" is \[ N \le r^2-r+1, \] and more generally \[ N\le r^2-s+1 \] when \(s=\dim(\mathcal S_\Phi)\) and \[ \mathcal{S}_{\Phi} = \operatorname{span}\{A_j^*A_k : 1\le j,k\le r\}. \] The same paper proves that \(N=2\) when \(r=2\), and that if \(r=3\) then \(N\le 6\). It also gives the first explicit examples with \(N>r\): there exist mixed-unitary channels having Choi rank \(d+1\) and mixed-unitary rank \(2d\) for infinitely many positive integers \(d\), including every prime power \(d\) [2003.14405]. For entanglement-breaking channels, the corresponding invariant is the entanglement breaking rank, the least number of rank-one Kraus operators in a decomposition \[ \Phi(X)=\sum_{i=1}^r (u_iv_i^*)X(u_iv_i^*)^*. \] The general inequality is \[ \operatorname{Choi\text{-}rank}(\Phi)\le \operatorname{EB\text{-}rank}(\Phi), \] because a rank-one Kraus decomposition is still a Kraus decomposition. "Entanglement Breaking Rank via Complementary Channels and Multiplicative Domains" identifies a rigid equality case: if \(J(\Phi)\) is a rank-\(d\) projection and \(\Phi\) is entanglement breaking, then \[ \operatorname{EB\text{-}rank}(\Phi)=\operatorname{Choi\text{-}rank}(\Phi)=d. \] The same paper shows that such channels are exactly, up to unitary equivalence, the complements of Schur product channels. It also stresses that this equality fails when the Choi matrix is only a scalar multiple of a projection; the Werner-Holevo channel provides a counterexample with \[ \operatorname{Choi\text{-}rank}(\Phi)=\frac{d(d+1)}{2}, \qquad \operatorname{EB\text{-}rank}(\Phi)=d^2 \] [2211.11909]. These comparisons place Choi rank at the base of a hierarchy: it is the minimal Kraus length, a universal lower bound on more structured decomposition lengths, and sometimes—under exact projection structure—the structured length itself. ## 4. Attainable Choi ranks under output constraints Choi rank can be analyzed through bipartite-state marginal problems. Let \[ S(\sigma_1,\sigma_2)=\{\rho\in D_{mn}:\operatorname{tr}_1(\rho)=\sigma_2,\ \operatorname{tr}_2(\rho)=\sigma_1\}. \] If \[ \operatorname{rank}(\sigma_1)=r_1 \ge r_2=\operatorname{rank}(\sigma_2), \] then "Ranks of quantum states with prescribed reduced states" proves that there exists a minimum rank \(r\) with \(r