The entanglement-breaking index of a quantum channel is the number of self-compositions after which the channel destroys all entanglement with any reference system. For qubit channels it is decided by a partial transpose, and to our knowledge it has never been measured. We give a certified design for measuring and controlling it on programmable hardware, using one round of a collision model: a message qubit meets two fresh thermal ancillas. The ancilla states are diagonal, so the bath polarisation p is programmed exactly by classical sampling; the round is affine on the Bloch vector, so composite channels are computed exactly, and every quantity called certified is a two-sided enclosure or a proven integer in 256-bit ball arithmetic. We certify a target list for the integer staircase n_EB(p), with decision margins between 0.025 and 0.109 at twelve points, show that it survives per-round depolarising noise of strength 5e-3 as certified integers, and give exact and tight gate counts (seven CNOTs all-to-all, five native arbitrary-angle gates, ten CNOTs on heavy-hex). A gated search of 33 million circuits shows that the contrast of thermal valleys, circuits whose index dips below its infinite-temperature value on an interior window of p, has an interior maximum near index 90 at a certified 2.85e-3, below what standard tomography resolves. Exact composition over rounds of different polarisation then turns a valley into a switch: a centred pulse of exactly eight rounds restores entanglement, a 53-round pulse beats the static protocol by a certified margin, a certified depth-three valley is walked down its staircase by pulse duration, and a 1.2 per cent coherent tilt of the ancilla opens the switch and triples the static signal at fixed population. Valley signatures die at per-round noise near 1e-4. Certificates, code, search output and an independent re-derivation script are deposited.
Comments
19 pages, 6 tables. v2: self-contained revision prepared for journal submission; complete search output and an independent re-derivation script added to the ancillary files; one shot-count estimate corrected
The entanglement-breaking index of a quantum channel is the number of self-compositions after which the channel destroys all entanglement with any reference system. For qubit channels it is decided by a partial transpose, and to our knowledge it has never been measured. We give a certified design for measuring and controlling it on programmable hardware, using one round of a collision model: a message qubit meets two fresh thermal ancillas. The ancilla states are diagonal, so the bath polarisation p is programmed exactly by classical sampling; the round is affine on the Bloch vector, so composite channels are computed exactly, and every quantity called certified is a two-sided enclosure or a proven integer in 256-bit ball arithmetic. We certify a target list for the integer staircase n_EB(p), with decision margins between 0.025 and 0.109 at twelve points, show that it survives per-round depolarising noise of strength 5e-3 as certified integers, and give exact and tight gate counts (seven CNOTs all-to-all, five native arbitrary-angle gates, ten CNOTs on heavy-hex). A gated search of 33 million circuits shows that the contrast of thermal valleys, circuits whose index dips below its infinite-temperature value on an interior window of p, has an interior maximum near index 90 at a certified 2.85e-3, below what standard tomography resolves. Exact composition over rounds of different polarisation then turns a valley into a switch: a centred pulse of exactly eight rounds restores entanglement, a 53-round pulse beats the static protocol by a certified margin, a certified depth-three valley is walked down its staircase by pulse duration, and a 1.2 per cent coherent tilt of the ancilla opens the switch and triples the static signal at fixed population. Valley signatures die at per-round noise near 1e-4. Certificates, code, search output and an independent re-derivation script are deposited.
The entanglement-breaking index of a channel is the number of self-compositions after which all entanglement with a reference system is destroyed. In a thermal qubit collision loop whose round is affine in the bath polarisation p, some circuits have thermal valleys: on an interior window of p the index falls below its infinite-temperature value. Rounds at different p compose exactly, so every schedule of polarisations defines a composite channel whose entanglement-breaking status can be certified in ball arithmetic. A gated search of 33 million random circuits finds no valley with contrast above about 3e-3, with an interior maximum near index 90, and yields six certified targets. At the reference target, held inside its window for 88 rounds, a centred excursion to p = 0.05 restores entanglement at exactly eight rounds: every shorter centred pulse certifiably fails, and so do the same eight rounds at the start. A 53-round pulse certifiably beats the static protocol, a depth-three valley is walked down its staircase by pulse duration, and a 1.2 per cent coherent tilt of the ancilla opens the switch at fixed population. Closing the switch costs 21 to 35 rounds. The asymmetry is not one of local rates, which agree to within about two per cent at matched circuit positions (certified). It is one of distance and response: closing must remove an entanglement margin 2.2 times larger, and while hot rounds added to a window hold act almost independently, window rounds added to a hot hold act collectively. Near the crossing, the lever of the log-margin is that of a linear approach to the entanglement-breaking boundary, with no free parameter. At the reference target, depolarising noise of about 4ν/n per round kills a signature of size ν read after n rounds: 1.3e-4 for the static signal, 6.5e-6 for the eight-round switch.
A feedback loop that returns a message into the process it describes defines a quantum channel on the message. We ask when such a loop preserves entanglement with a reference, that is, when its channel is not entanglement breaking. For any qubit loop, the negativity of the Choi state is at most half the fidelity between the states that the two values of the message leave in the discarded registers, in any basis. This channel analogue of the duality between fringe visibility and which-way information is attained by pure dephasing, and a perfect record of the message in the discarded registers makes the loop entanglement breaking. We then solve a three-qubit loop in which the message steers a future event, a probe reads the event, and a partial swap feeds the probe back. Every partial feedback is a strict contraction, and the loop is entanglement breaking exactly when an explicit trigonometric polynomial in its coupling angles is not positive. A partial record of the message in the future event can restore entanglement that the feedback alone destroys. Stability, a coherent fixed point, information about the future event and entanglement preservation coexist on about three quarters of the parameter space.
A quantum feedback loop that returns information from a forward simulation to an earlier internal time induces a completely positive trace-preserving map on a message register; iterating that map eventually destroys its ability to carry entanglement. The entanglement-breaking index N = n_EB(Phi), the round at which this happens, is studied along two axes, interaction and bath. On the interaction axis, strict contraction onto a full-rank fixed point forces N finite in every dimension, via an explicit separable ball around the limiting Choi state, and across 3997 sampled channels index and stability gap are one clock, N [1 - rho(A)] in [0.60, 1.50]. On the bath axis, a thermal ancilla at polarisation p = tanh(hbar omega / 2 k_B T) makes every channel quantity an exact quadratic in p, and one-sided certificates (no eigensolver) establish N = 3 at the reference circuit at every temperature. The infinite-temperature index is an architecture floor with closed form and exact measure (46.427% of circuits break entanglement in one round at infinite temperature). For exchange collisions (generalised amplitude damping) the index has a closed form and is monotone in p; for the loop the floor is not a bound: certified circuits dip below it in interior temperature windows, at a rate with Gaussian cutoff in epsilon = 1 - ||A_0||_2 (marginal exponent 2.02 [1.90, 2.16], 4.1 million circuits), and shallow-floor valleys survive to epsilon = 0.125. On a certified coupling-temperature surface, temperature moves the index by at most 3/2, the interaction by 1213/3. Applied to quantum-cognition proposals, a per-bath polarisation table at 310 K shows nuclear spins maximally hot and intramolecular vibrations frozen, and a four-input checklist replaces decoherence times by round budgets; Fisher's nuclear-spin proposal specifies its bath and storage coherence, not its processing interaction or rate.
Discarding an output subsystem is a local post-processing operation, so an entanglement-breaking channel remains entanglement breaking after output reduction. The converse need not hold: a reduced channel can be entanglement breaking even when a richer output still carries reference entanglement. This preprint gives an exactly solvable two-parameter benchmark that quantifies this asymmetry. On two qubits, let (\Pi_\pm=(I\pm Z\otimes I)/2), let (\mathcal L(X)=\sum_s\Pi_sX\Pi_s) be sector dephasing, let (\mathcal M(X)=\sum_s\operatorname{Tr}(\Pi_sX)\Pi_s/2) be measure-and-prepare within the two sectors, and let (\mathcal D_4(X)=\operatorname{Tr}(X)I_4/4). The channel family is [\Phi_{t,\eta}=(1-\eta)\big[(1-t)\mathcal L+t\mathcal M\big]+\eta\mathcal D_4,\qquad 0\le t,\eta\le1.] For the reduced task, a variable second-qubit input is embedded with an arbitrary fixed first-qubit state and the first output qubit is discarded. The induced qubit channel is depolarizing with parameter (q=(1-\eta)(1-t)), and is entanglement breaking exactly when (q\le1/3). For the full four-dimensional output, the partially transposed normalized Choi state has one potentially negative eigenvalue, [\lambda_- = \frac{-4(1-\eta)+6(1-\eta)t+\eta}{16},] with multiplicity two. The block structure makes PPT sufficient for separability in this family. Hence, for (\eta<1), [t_{\rm red}(\eta)=\max!\left{0,1-\frac{1}{3(1-\eta)}\right},\qquadt_{\rm full}(\eta)=\max!\left{0,\frac{4-5\eta}{6(1-\eta)}\right}.] The exact region in which the reduced channel is already entanglement breaking while the full channel is not is (t_{\rm red}(\eta)\le t<t_{\rm full}(\eta)). Its width is [w(\eta)=\begin{cases}\eta/[6(1-\eta)],&0\le\eta\le2/3,\(4-5\eta)/[6(1-\eta)],&2/3\le\eta<4/5,\0,&4/5\le\eta\le1,\end{cases}] and reaches the exact maximum (w_{\max}=1/3) at (\eta=2/3). The Choi negativities are [\mathcal N_{\rm red}=\frac{[3(1-\eta)(1-t)-1]+}{4},\qquad\mathcal N{\rm full}=\frac{[4-5\eta-6(1-\eta)t]_+}{8}.] Therefore the maximal full-output Choi negativity compatible with an already entanglement-breaking reduced output is (1/12), attained at ((\eta,t)=(2/3,0)). The result is a controlled benchmark for output-access dependence: the same physical channel family can cross the entanglement-breaking boundary at different noise values solely because an output subsystem is retained or discarded.
We study a set of new functionals (called entanglement–breaking indices) which characterize how many local iterations of a given (local) quantum channel are needed in order to completely destroy the entanglement between the system of interest over which the transformation is defined and an external ancilla. The possibility of contrasting the noisy effects introduced by the channel iterations via the action of intermediate (filtering) transformations is analyzed. We provide some examples in which our functionals can be exactly calculated. The differences between unitary and non-unitary filtering operations are analyzed showing that, at least for systems of dimension d larger than or equal to 3, the non-unitary choice is preferable (the gap between the performances of the two cases being divergent in some cases). For d = 2 (qubit case), on the contrary, no evidences of the presence of such gap is revealed: we conjecture that for this special case unitary filtering transformations are optimal. The scenario in which more general filtering protocols are allowed is also discussed in some detail. The case of a depolarizing noise acting on a two–qubit system is exactly solved in a general case.
An operational way of measuring entanglement in a balanced two-path interferometers is presented, where path information is carried by some internal degree of freedom which, in turn, gets entangled with an ancilla system. The analysis is based on a tripartite description involving paths, an internal qubit degree of freedom, and some ancillary states entangled with the internal degree of freedom. It is then applied to two physically distinct experimental situations: a modified Stern-Gerlach interferometer with spin-1/2 particles and a Mach-Zehnder interferometer with photons carrying polarization. The ancilla degree of freedom may not be experimentally accessible. Tracing out the ancillary system, and employing a quantum erasing procedure based on the internal degree of freedom, it is demonstrated that a concurrence-based measure of the entanglement, between the internal degree of freedom and the ancilla, can be extracted directly from measurable asymmetry of the two output channels. These results show that loss of coherence, quantum erasure, and entanglement estimation in interferometric experiments arise from the same underlying correlation structure and provide a compact experimentally accessible framework for quantifying entanglement in qubit systems.