Foundations of quantum mechanics · Theory and numerical analysis
Qubit-environment entanglement in pure dephasing dynamics for mixed initial states
arXiv:2610.36936v1 ↗PDF ↗Submitted 27 October 2026
- Question
- When a qubit only dephases, the joint qubit–environment state for a pure initial qubit is separable exactly when the environment ends up in the same state whichever pointer state the qubit is in, \(\hat R_{00}(t)=\hat R_{11}(t)\). This test made it possible to tell decoherence by entanglement from decoherence by classical correlations, and even to detect the difference on the qubit alone. Real qubits are already partly mixed. Does the test survive, and does entanglement then evolve as it does for pure qubits?
- Result
- If \(\hat R_{00}(t)=\hat R_{11}(t)\), the joint state stays separable, and zero-discord, for any initial qubit coherence. Otherwise, the family of minors that gave the pure-state criterion becomes only a witness: a negative minor proves entanglement, but nonnegative minors prove nothing. A Schur complement reduces the exact partial-transpose test to a matrix \(\hat D\) of the environment’s dimension, whose diagonal entries are those minors. Mixed qubits show behaviour impossible for pure ones: a separable initial period followed by sudden birth of entanglement, death and rebirth, and no entanglement at all during an entangling evolution when the qubit is strongly mixed. In a model of an NV-centre spin with up to five nuclear spins, the minors miss entanglement, sometimes for the whole evolution.
The paper in detail
A pure-dephasing Hamiltonian has the form \(\hat H=\sum_i\varepsilon_i\ket i\bra i+\hat H_E+\sum_i\ket i\bra i\otimes\hat V_i\), so the evolution is \(\hat U=\sum_i\ket i\bra i\otimes\hat w_i(t)\). With the qubit initially \(\bigl(\begin{smallmatrix}|a|^2&ab^*\varepsilon\\a^*b\varepsilon&|b|^2\end{smallmatrix}\bigr)\), where \(\varepsilon\in[0,1]\) measures its initial coherence, and an uncorrelated environment \(\hat R(0)\), the joint state at time \(t\) depends on one environment state \(\hat R_{00}=\hat w_0\hat R(0)\hat w_0^\dagger\) and one unitary \(\hat w=\hat w_1\hat w_0^\dagger\):
\[\hat\sigma(t)=\begin{pmatrix}|a|^2\hat R_{00}&ab^*\varepsilon\,\hat R_{00}\hat w^\dagger\\ a^*b\varepsilon\,\hat w\hat R_{00}&|b|^2\,\hat w\hat R_{00}\hat w^\dagger\end{pmatrix}.\] If \(\hat R_{11}=\hat R_{00}\), then \([\hat R_{00},\hat w]=0\), the two share an eigenbasis \(\ket{n}\) with \(\hat w\ket n=e^{i\phi_n}\ket n\), and \(\hat\sigma\) is a mixture of products of qubit states with \(\ket n\bra n\). It is therefore separable for every \(\varepsilon\), even though the qubit’s coherence is multiplied by \(\sum_np_ne^{i\phi_n}\) and decays. Separability for pure qubits thus implies separability for mixed ones.
When \(\hat w\) does not commute with \(\hat R_{00}\), the partial transpose with respect to the qubit swaps the off-diagonal blocks. In the eigenbasis of \(\hat R_{00}\), with \(y_{kn}=\bra k\hat w\ket n\), the principal minors used for pure qubits become \(M_i\propto\sum_n\bigl(p_n|y_{in}|^2-\varepsilon^2p_i^2|y_{ni}|^2/p_n\bigr)\). For \(\varepsilon=1\) these decide separability; for \(\varepsilon<1\) they can all be positive while the state is still entangled, because other minors may be negative. The exact test is that the Schur complement of the upper-left block,
\[\hat D=|b|^2\bigl(\hat w\hat R_{00}\hat w^\dagger-\varepsilon^2\hat R_{00}\hat w^\dagger\hat R_{00}^{-1}\hat w\hat R_{00}\bigr),\] has a negative eigenvalue. Its diagonal entries are the minors \(M_i\) up to a constant, but its eigenvalues cross zero at other times. For a one-qubit environment with \(\hat w=e^{i\omega t}\ket+\bra++e^{-i\omega t}\ket-\bra-\) and \(p_0=3/4\), the evolution stays separable throughout for \(\varepsilon=0.3\), and for larger \(\varepsilon\) entanglement is born suddenly, dies and is reborn symmetrically about \(\omega t=\pi/2\). For an NV-centre spin coupled by hyperfine interaction to one, two, three or five \(^{13}\)C spins at \(B_z=0.1\) T, the minors report separability while \(\hat D\) shows repeated birth and death (two spins, \(\varepsilon=0.5\)), or flag entanglement only in a short window while it is present continuously (five spins, \(\varepsilon=0.3\)). Whether measures cheaper than diagonalising \(\hat D\) can track weak entanglement in large environments remains open.
Transferable methods
Schur-complement reduction of a partial-transpose test
A block matrix \(\bigl(\begin{smallmatrix}\hat A&\hat C\\\hat C^\dagger&\hat B\end{smallmatrix}\bigr)\) with \(\hat A>0\) is congruent to \(\mathbb 1\oplus(\hat B-\hat C^\dagger\hat A^{-1}\hat C)\), so it has a negative eigenvalue exactly when the Schur complement does. Applied to the partial transpose of the qubit–environment state, this halves the size of the eigenvalue problem and exposes how the old minors sit inside the exact test.
With \(\hat U_1=\hat A^{-1/2}\oplus\mathbb 1\) and \(\hat U_2=\bigl(\begin{smallmatrix}\mathbb 1&0\\-\hat C^\dagger\hat A^{-1/2}&\mathbb 1\end{smallmatrix}\bigr)\), the matrix \(\hat U_2\hat U_1\) is invertible, and Sylvester’s law of inertia says congruence preserves the number of negative eigenvalues. For the partially transposed state, \(\hat A=|a|^2\hat R_{00}\), \(\hat C=a^*b\varepsilon\,\hat w\hat R_{00}\) (transposed blocks) and \(\hat B=|b|^2\hat w\hat R_{00}\hat w^\dagger\), which gives the \(\hat D\) quoted above. In the eigenbasis of \(\hat R_{00}\), the diagonal entry \(D_{ii}=|b|^2\sum_n(p_n|y_{in}|^2-\varepsilon^2p_i^2|y_{ni}|^2/p_n)\) is the minor \(M_i\) divided by \(|a|^{2N}\prod_kp_k\).
Editorial derivation of the separable start: at \(t=0\), \(\hat w=\mathbb 1\) and \(\hat D(0)=(1-\varepsilon^2)|b|^2\hat R_{00}\), whose smallest eigenvalue is \((1-\varepsilon^2)|b|^2p_{\min}>0\) when \(\varepsilon<1\) and \(\hat R_{00}\) has full rank. By continuity, \(\hat D(t)\) stays positive for a finite time, so no entanglement detectable by the partial transpose can appear immediately. For a pure qubit, \(\varepsilon=1\) gives \(\hat D(0)=0\), and entanglement can start at once, as known.
Editorial check: for the one-qubit environment with \(p_0=3/4\), \(|a|=|b|\), we computed the smallest eigenvalue of the full \(4\times4\) partial transpose and of \(\hat D\) on 2001 times in \(0\le\omega t\le\pi\); they agree in sign at every point. Entanglement first appears at \(\omega t\approx0.85\), 0.55 and 0.29 for \(\varepsilon=0.5\), 0.7 and 0.9, never for \(\varepsilon=0.3\), and occupies 46%, 65% and 82% of the interval, against 41%, 62% and 81% flagged by the minors.
How to reuse it
Any test of “is this block operator positive” for a large system with a known positive block can be reduced by a Schur complement, and the diagonal of the complement gives cheap necessary conditions that are exactly the \((N+1)\)-dimensional principal minors. The \(t=0\) argument generalises: if the initial complement is strictly positive with margin \(m\), entanglement cannot appear before the perturbation of \(\hat D\) reaches \(m\), which gives a computable lower bound on the onset time.
Common eigenbasis as a certificate of classical dephasing
If the environment state and the relative evolution commute, the environment acts on the qubit like a classical random phase. A shared eigenbasis gives an explicit separable decomposition and shows the correlations are classical on the environment side.
From \(\hat R_{00}=\hat w\hat R_{00}\hat w^\dagger\) and unitarity, \([\hat R_{00},\hat w]=0\). With \(\hat R_{00}=\sum_np_n\ket n\bra n\) and \(\hat w=\sum_ne^{i\phi_n}\ket n\bra n\), the joint state becomes
\[\hat\sigma(t)=\sum_np_n\begin{pmatrix}|a|^2&ab^*\varepsilon e^{-i\phi_n}\\a^*b\varepsilon e^{i\phi_n}&|b|^2\end{pmatrix}\otimes\ket n\bra n,\] a convex mixture of product states in which the environment states are orthogonal, so the discord with respect to the environment is zero. Tracing out the environment multiplies the qubit coherence by \(\sum_np_ne^{i\phi_n}\), which has modulus at most one: the qubit decoheres exactly as if a phase \(\phi_n\) had been drawn with probability \(p_n\).
How to reuse it
To decide whether a dephasing environment can be replaced by classical phase noise, check whether the conditional evolution \(\hat w_1\hat w_0^\dagger\) commutes with the conditional environment state at the times of interest. If it does, the noise can be simulated as a random unitary without loss, whatever the qubit’s initial mixedness; if not, the Schur complement above tells when a quantum treatment is needed.
Context & limitations
Why it matters. Dephasing dominates decoherence of many solid-state qubits, and whether it entangles the qubit with its environment affects how noise propagates through algorithms. The paper shows that the realistic, partly mixed case behaves qualitatively differently, and that a cheap witness can be badly wrong precisely when the entanglement is weak.
Limitations. Only entanglement detected by the partial transpose is studied; bound entanglement is possible for mixed qubits but not analysed. Apart from a one-qubit environment, the eigenvalues of \(\hat D\) are computed numerically. The NV examples use at most five nuclear spins, with no interaction between them and an uncorrelated, partially polarised initial state, and there is no experiment.