The halo of a decision: a two-way choice, its partner, and the event read as setting them

1 · Why a decision has a halo

Most of the decisions an engine is asked to make come down to two answers. A shipment leaves on Tuesday or it does not; a shift is covered or it stays open; a part is in the plan or out of it. Physics has a standard picture for a variable with two values. It was drawn for the simplest quantum system, the qubit, and it is called the Bloch sphere.

Every state of a two-valued variable is a point of a ball of radius one. The north pole is one answer, written |0⟩, and the south pole is the other, written |1⟩. A point at a pole is a decided variable, and the ground state a solver reads off a model of two-valued variables puts each variable at one pole or the other. A point inside the ball is a variable that has not decided yet, and how far up or down the axis it sits is how sure it is. We write that height as m. It is 1 at the north pole, −1 at the south pole and 0 at the centre, and it is the confidence of the decision.

That is the whole picture for one variable on its own. This paper is about what happens when a variable has a partner, which in a real decision it nearly always does: the shipment is tied to the truck that carries it, the shift to the person who covers it. A pair has a picture too, and in it the space around the axis is not empty. Two partly decided variables that came out of one event share a ring of correlations around their axis, at the height of their confidence. We call that ring the halo of the decision. When the decision is made, the halo shrinks to a point at the pole. While the decision is open, the halo holds what is known about the pair that neither half holds alone.

The mathematics of sections 2 to 4 is textbook; the rules of section 5 are the model's own; sections 7 and 8 are readings; and every number is copied from a recorded measurement. Nothing here describes how our engine finds an answer, and nothing claims that the engine contains a qubit. Section 9 says what the paper does not claim.

What this paper contributes, losses first, each with its basis and the sections that carry it:

1. Measured, one run each. Two registered tests of the lens, on an exact simulator, were falsified: its 95 % region covered 815 of 1,000 planted pairs, and its radius witness fires on a pair that is not entangled (section 6).
2. Measured in the model. The registered test that the phase clock's rate follows the balance of what flows in and out, on 200 planted beats run through the engine's code, and two registered tests of the margin measured on our rosters, were falsified (section 5).
3. Prediction. Three predictions about a clocked pair are registered, not run (sections 6 and 9).
4. Theorem in the model. The pair's three readings in closed form, checked against a direct computation from the amplitudes at 25 points, exact (sections 2 to 4).
5. Measured, one run. On a simulator the lens separates a mixture along the axis from an equatorial pair, and recovers a planted clock rate of 0.8 as 0.80 (section 6).
6. Reading. Three recorded margins read onto the sphere (section 8).

2 · The pair

No variable in a real decision is alone, so write two of them, a and b, as one state. The states of the pair live in the Hilbert space of two choices. The Schmidt decomposition says that any pure state of a pair can be written, in a suitable basis for each half, with two terms only. In the basis where the decision is read, the pair is

|ψ⟩  =  cos α |00⟩  +  e^{iθ} sin α |11⟩

m  =  cos 2α          α from 0 to π/2,   θ from 0 to 2π

The angle α sets how far the pair leans toward |00⟩ or toward |11⟩. The angle θ is a phase: the relative turn between the two terms. The two halves share one set of coefficients, cos α and sin α, so they lean together, and that sharing is what the Schmidt form says.

Read either half on its own and you get the same thing: a point on the axis at height m, with nothing off the axis. Both halves sit at one confidence. Each half alone looks like a partly decided variable, and nothing in either half shows that it has a partner, or what the phase θ is.

This is the first of three readings of one pure state, and it is the reading a solver can give: the answer as a pole, and the margin of the answer as the confidence. The other two readings are where the rest of the information is.

3 · The halo

The second reading is the correlations between the two halves. Measure a along one axis and b along another, many times over, and average the product of the two results. For the pair above, the two halves always agree along the axis where the decision is read, and every correlation that mixes that axis with the equator is zero. What remains lies flat, in the plane of the equator:

⟨X X⟩  =   sin 2α · cos θ          ⟨X Y⟩  =  sin 2α · sin θ
⟨Y Y⟩  =  −sin 2α · cos θ          ⟨Y X⟩  =  sin 2α · sin θ
⟨Z Z⟩  =  1                        every correlation of Z with X or Y  =  0

Those four equatorial numbers form a small table that is sin 2α times a reflection. Its size is sin 2α, which equals √(1 − m²), and its angle is θ. On the sphere, that is a ring at height m, of radius √(1 − m²), turned to the angle θ. That ring is the halo.

Its ends are the two ends of a decision. A decided variable, m = 1 or m = −1, has a halo of radius 0: a point at the pole, with nothing to turn. An even split, m = 0, has the equator itself, of radius 1. In between, the halo is a ring above or below the equator, and the more decided the pair, the smaller the ring.

This is why the equator matters. For a pair, the equator is where the halo is generated, because the only correlations that carry θ are the equatorial ones. For one variable on its own, the equator is where the state has no answer at all. A reading that looks only along the axis, which is what an answer from a solver is, never sees the phase.

Same observation, three vocabularies. Three fields (information theory, systems and graph theory) describe the same situation in different words, and an insight in one usually has a twin in the other two. This series names a twin only where its own model has the object, and says so where a twin is missing or runs the other way. Each twin is our reading, not an identity, unless a result the paper states makes the two one number. The halo's radius is one such observation, and here one twin runs the other way. In information theory, the wider the ring, the more each half's own reading leaves open: its entropy, which section 4 gives, is a full bit at the equator. Read as a system, the wider the ring, the less settled each half. Read as a graph, the two halves are two nodes joined by one tie, and the tie's weight, the concurrence, equals the ring's radius (section 4). So in a pure pair, the less settled its halves, the more tightly they are tied, where the usual twin of an unsettled part is a weakly linked one.

4 · The third reading

The third reading is the event that induced the pair. The pair as a whole is pure: it is one definite state, and nothing about it is unknown. Each half on its own is not. It carries uncertainty, and the textbook measure of that uncertainty is its entanglement entropy, S = H₂((1 + m)/2) bits, where H₂ is the entropy of a coin with that bias. At a pole S is 0; at the equator S is a full bit.

The textbook name for this relation is purification: a half that is uncertain on its own is exactly what is left of a whole that is certain, and the whole holds the certainty the half is missing. Read the other way, it is read as saying where a partly decided variable's uncertainty comes from: the half is read as unsure because it is tied to a partner by an event the half cannot see, and its uncertainty as its share of that event. The textbook measure of how tightly two halves are tied, the concurrence, comes out as sin 2α: the same number as the radius of the halo.

So one pure state carries three readings: the variable's own reading, a point on the axis at height m; the halo reading, a ring of radius √(1 − m²) at the angle θ; and the event that induced the pair, which makes the whole pure while each half carries S bits. The third reading is read as the independent one, because the other two are measured against it and only it can set the phase; the halves are read as uncertain because they are out of balance with the event that made them, while the whole is not.

We checked the formulas of sections 2 to 4 against a direct computation from the four amplitudes of the pair, exact and one run, on 3 October 2026. At 25 points of α and θ, every reading agreed with its formula to within 1e-12, and the largest difference measured was 8.9e-16, quoted as recorded. The one looser number is the concurrence, which the textbook computes through square roots of a matrix product: it agreed to within 1.3e-8.

5 · Waking up

The losses of this section come first. The registered test that the clock's rate follows the balance of what flows in and out of a pair was run on 200 planted beats through the engine's code and falsified: the correlation between the rate and that balance was −0.063, against a registered +0.5 or more, and none of the 20 beats with as much flowing in as out stopped turning, the slowest at a rate of 0.0726. It is a test on the engine's records, one run, of the model's rule, not of nature. Two registered tests of the margin measured on our own rosters were falsified too: the margin rose strictly with its lever at 23 of 24 roster temperatures, the one exception a tie in floating point, and on the frozen roster three readings sat at −0.76, −0.75 and −0.74 where the registration needed every reading above 0.9.

Corrected 10 October 2026. An earlier text of this paper, in its abstract, in section 1 and in the paragraph above, called the beats of the clock test the engine's own. They were 200 planted beats, run through the engine's code.

A picture of a decision should also say how a decision moves. Two numbers move, the confidence m and the phase θ, and in the model this paper describes they move by two rules:

θ'  =  θ + ΔE · dt                        the phase advances at the gap ΔE
m'  =  (1 − c) · m  +  c · sign(m)         the confidence moves toward its own pole

The first rule is the standard one for a system with two levels: when the two terms of the pair differ in energy by ΔE, their relative phase turns at that rate, so the gap is the pair's clock. The second rule is the model's own. The number c, between 0 and 1, is how firmly the latest evidence confirms the answer the variable already leans toward. At c = 0 the variable does not move up or down and only its halo turns; at c = 1 it goes straight to its pole.

In the reading our engine paper gives of its two halves, this turning is the current: a pair whose phase does not turn carries none, and only a reading taken on the clock can show one. Two clocks appear in our work and are kept apart by name. The temperature T sets how far a settle searches, and the phase θ turns the halo. Only θ is the clock of this section.

Three cases follow from the rules. A decided variable with full confidence is a fixed point: it sits at its pole, its halo is a point, and the turning phase moves nothing. An even split with no confidence is as awake as a pair can be: its halo is the equator, and the equator turns. In between, belief error is what keeps a pair awake. When the evidence stops agreeing with what the variable believed, c falls, the variable stops moving toward its pole, and its halo keeps turning. An example step from the record: from m = 0.3 with c = 0.5, the next m is 0.65, and the radius of the halo falls to 0.760.

The rules never move a variable away from its own pole. What opens a halo again is a new partner: a new event that ties the variable to something it was not tied to before. In this model a decision closes as its evidence agrees, and opens again only when something new arrives.

6 · The experiment the picture proposes, and the lens's two refuted tests

The phase θ has a special place in this picture. A solver that reads its answer along the axis supplies the poles and the margins, and from the margins the radius of each halo. It does not supply θ, and settling to a lowest state does not supply it either: for models of the kind such solvers settle, the lowest state carries a phase of 0 or a half turn, never one in between. The lowest state a solver returns for a model built only from terms along the axis is a set of poles, and its equatorial correlations are exactly zero. When two or more sets of poles tie for lowest, some of the other lowest states do carry a halo, but the set a solver returns does not. So in this paper θ is the model's clock, the first rule of section 5. Until it is measured on a real pair, it is a prediction. The converse is a prediction too: a pair driven on a clock must show a nonzero equatorial correlation somewhere on the clocked record, and if none ever does, the turning side of the reading is empty.

That is what makes the pair worth putting on quantum hardware. The experiment is defined by a difference in correlations: the model predicts zero equatorial correlation for a classical pair, a hardware pair would show a nonzero one, and that difference, not any speed, is what the experiment would measure; the witness for it exists in the lens. A pair this small is computed exactly on ordinary hardware, as every number in this paper was.

Reading a qubit is not like reading a variable in memory. One measurement of one copy gives one bit, along one axis, and it disturbs the copy it reads. So a state is read from many copies prepared the same way, each measured along one of several axes, and the averages are the correlations of section 3. The procedure is called state tomography. A pair has nine settings, the products of X, Y and Z on each half, and the four that carry the halo are XX, XY, YX and YY.

Two things make a reading honest. The first is a reference preparation. Before the pair is read, states whose answers are known in advance are prepared and read the same way, and the reader's own errors are fitted from them. A reading without that calibration is biased. On a simulator with readout errors of 0.03 and 0.05 planted, the north pole read 0.917 without the calibration and 0.986 with it, within one posterior standard deviation, from one run, of a reader with no error at all. The second is an interval. Every reading comes with a 95 % credible region, a Bayesian summary under a Hilbert-Schmidt prior, and the region is where the losses of this section sit.

The lens's two refuted tests. Both are registered tests run on planted pairs of an exact simulator, each one run of 7 October 2026. First, the region's coverage: the 95 % region held the true halo radius for 815 of 1,000 planted pairs, against a registered band of 0.932 to 0.968, and 183 of the 185 misses lie above the region, where the prior likely pulls the estimates low; near zero it held 248 of 250. So the test is falsified, and a halo decided by this region is decided at about 82 %, not 95 %. Second, the witness: on a pair that is not entangled, an even mixture of two product states, the lens read a halo radius of 0.934, from 0.890 to 0.942 over 20 source seeds, while the control along the axis read 0.089. The radius tests only against mixtures along the axis, as the next paragraph says, and the registered test that it stays below 0.9 on a pair that is not entangled is falsified.

The lens we built for this reads one variable or one pair in that way. It holds a belief over the state, chooses each next setting by the information it expects to gain from it, updates on every batch of outcomes, and stops when its belief stops moving. Its witness is the radius of the halo, set against what a classical pair of this kind predicts, which is 0: a mixture of answers along the axis has equatorial correlations of zero. That tests only against mixtures along the axis. Some pairs that are not entangled also show a halo, so the test for entanglement itself is a different one: the partial transpose, or a witness that also reads the correlation along the axis. These are the lens's readings of two planted pairs on an exact simulator, not on hardware, each one seeded run of 3,000 shots; each region is a 95 % credible region, whose measured coverage on planted pairs is about 82 %, against the textbook value 0 for any mixture along the axis:

The calibration was run on the simulator too, and it is in-sample. A pair prepared at m = 0.3, with its phase turning at a planted rate of 0.8, was read at six times. The model without its clock missed the measured phase by 1.71 radians, as a root mean square. Fitted to the same six readings, the clock came out at 0.80, with a 95 % interval from 0.78 to 0.82, and the miss fell to 0.066 radians, in-sample; no held-out time was read. At every one of the six times, the halo read clear of zero.

No hardware pair has been read. The lens refuses to talk to a live device until one is chosen, and nothing in this paper is a hardware measurement. When the first hardware pair is read, the comparison is the one the definition makes: the halo the model predicts from the margins, the zero a classical pair predicts, and the halo the hardware shows. Reading such a pair is itself routine: pairs of exactly this form have been read in laboratories since 1999, and two-qubit pairs on superconducting hardware since 2006.

7 · Needs the data has not decided

The same picture fits a kind of decision that is not physics at all. A need the data has not yet decided, a capability somebody will want or a gap nobody has filled, is a variable near the centre of the ball: m close to zero, its halo close to the equator. Nothing about it is settled, and what is known about it is held in its correlations with other variables, not in its own reading.

Reading such a need is what the lens does: it reports where the halo is and how wide, with an interval. Resolving the need is the move to a pole, north or south: the need is met, or it is ruled out. As an analogy, the halo also suggests where to look for whoever can meet the need: in the picture, it is shared with the partner the need is correlated with. A table of correlations names no partner outside the pair, so this is a way of reading, not an inference.

Read as a mechanism, this asks for no central planner. Each party keeps its own goals, and a need the data has not yet decided is filed as work for whoever can meet it. The paper describes how a decision can be read. It says nothing about how people should organise themselves.

8 · Three measured panels

The figure reads three recorded margins onto the sphere. They come from our periodic-table note, which fits a filling model to the elements and checks every answer NIST lists against it, in one run; the NIST data are used for research under the source's published terms, with written confirmation pending. For each element the margin is the gap between its lowest filling and the next one, divided by the widest gap on the ladder, so every m lies between 0 and 1. Three margins were picked by rule: the widest gap, the gap nearest half the widest, and the narrowest gap.

Krypton is one of four elements tied for the widest gap on the ladder, so its m is 1: decided, its halo a point at the pole, and no entropy in either half. Mercury sits close to halfway, and its halo is a ring above the equator. Chromium, one of the elements that break the filling order, is one of four tied for the narrowest gap, and its halo is almost the equator, with almost a full bit in each half. Each tie was broken to the lower atomic number.

A margin is a confidence reading, not a measured state. Nothing here prepared a pair of anything. The figure shows where a recorded margin puts a variable on the sphere, and the largest halo a partner could share with it: a radius of √(1 − m²) belongs to a pure pair, and a mixed pair at the same m shares a smaller one. All 118 margins on the ladder land inside the ball.

9 · Limits, and what this paper does not say

The open limits come first, each with what would close it.

• The lens's 95 % region covers about 82 % on planted pairs (section 6), so a halo it decides is decided at about 82 %. It would close with a calibrated region re-tested on planted pairs.
• The radius witness fires on a pair that is not entangled (section 6): it decides a halo only against mixtures along the axis. It would close with a witness that reads zero on every separable pair, re-run on the same pairs.
• No hardware pair has been read, and no phase has been measured. The phase is the model's clock until a hardware pair is measured. It would close with one pair prepared and read on quantum hardware, with the readout calibrated first, and its phase compared with the clock.
• The clock is not kept on a running record, and the registered test of its rate on the engine's beats was falsified (section 5). It would close with a clocked record that keeps the temperature and the phase as two named clocks and shows a turning halo on a driven pair.
• The rules of section 5 are the model's own, not measurements. They would close with a recorded sequence of decisions, read at each step and compared with the rules.
• Pairs only. Several variables tied by one event, and the halo they share, are untested. It would close with a run on three or more variables compared with the stated law.
• The body that induces a pair is a reading. The series' founding picture draws the event that induces a pair as a sphere of its own, with a pole, a halo and a clock whose turning sets the pair's clock. That is a model identification, not a result, and nothing in our engine carries such a body's state into a pair. If it holds, a step in that body's depth moves the clocked lock in proportion to the step, and a pair inside its reach turns at the gap the body sets, faster as the body deepens; both predictions are registered, not run. It would close with the wire built and measured on planted bodies, with its predictions registered before the run.
• The margins in the figure are confidence readings, gaps from a fitted filling model that misses some elements, not prepared states. They would close with a margin read from a prepared state in place of a model gap.

The standing statements, which no measurement closes:

• Computed, not hosted. Every number here was computed exactly on ordinary classical hardware. There is no qubit in our product, as our home page says: "the moving parts are two-way switches on classical hardware". This paper describes the textbook picture that such a switch can be read onto.
• The periodic-table note does not solve atoms, and neither does this paper.
• No method. How our engine reaches an answer is not described here, and will not be.

This paper reads a two-way decision onto the textbook picture of a qubit and says what an experiment on quantum hardware could measure. It claims no hardware result and no new physics. Every number is as of 3 October 2026, except the registered tests run on 7 October 2026.

10 · How this was made: the method and the chain of records

The question was set on 3 October 2026 as a written brief, under the series' scope rules: the picture is published in textbook terms, the engine's method is not described, and nothing claims more than was measured. The model is stated as the series' first postulate, the two-state pair: a two-valued variable is one half of a pair, and the pair carries three readings, on the variable, on the halo and on the event that set it; it is assumed and never derived, and the theorems of sections 2 to 4 rest on it.

Day drew the figure from three margins of the periodic-table ladder, a record it computed for the paper on the periodic table. Claude (Anthropic), the consultant named in the byline, designed and built the rest from the engineering side: the exact translation of the model into the standard form of section 2, the module that computes the three readings in closed form and the checks that hold it to a direct computation, the lens and its simulator runs, the figure, the tests that pin every number on this page to its record, and the draft of this paper.

The chain of records, in order:

• the closed forms, written and pinned against a direct computation from the amplitudes;
• the lens's simulator runs, written to their record and re-derived by its check;
• the predictions about a clocked pair, each frozen in a registration committed alone before any data, and not yet run;
• the lens's two registered tests, frozen in a registry committed alone at 65ff0d0b1 before their runs, and their falsified results;
• the tests that hold every number on this page to its record, and this page, generated from its source.

11 · Companion papers

Seven papers published beside this one take the same approach: each traces its figures to a recorded measurement and says what it did not measure.

• The Operational Unit: the model behind the engine, and how it was built.
• The periodic table as a ground-state ladder: each element read as the lowest-energy filling of its electrons, with every answer NIST lists checked against it.
• Closing the gaps: four sustainability problems an engine can schedule, and what it cannot do about them.
• Sense, remember, act: what the engine senses, remembers and learns.
• The Earth as a two-state recorder: the Earth's dipole read as a two-way choice from public records, with a prediction written before the data, and the planet's clocks.
• HOPE: recall in a library of Hilbert spaces: recall as settling to the nearest stored state, in a library of small spaces kept one per model, with the measured answer that they do not make one large space.
• The unit of work: a receipt for one settled problem, booked on a hash-chained ledger, and what it is not, with the economy's measured losses first.

Acknowledgements

• NIST Atomic Spectra Database (Kramida, Ralchenko, Reader and NIST ASD Team), DOI 10.18434/T4W30F. Reached through the periodic-table paper: the three recorded margins drawn on the sphere come from that paper's model, which is checked against it. Used for research under the source's published terms; written confirmation pending.

Replicate this paper, or tell us what is wrong

We invite replication and correction. The measured results come from records committed in our repository. The records are not yet published; until they are, write to us through the contact page, naming the paper and the number you want to rebuild or the error you found. We correct an error we can reproduce on the paper's web page, with its date.

The tests this paper registered before reading their data are frozen, each with the rule that would fail it, in docs/PAPERS_DUALFIELD_1_2026_10_04.md and docs/PAPER_EXPERIMENTS_REGISTRY_2026_10_07.md.

How to cite this paper

Please cite this paper as:

Agile Integrations LLC. 2026. "The halo of a decision: a two-way choice, its partner, and the event read as setting them." Version 1, 3 October 2026. https://agileintegrations.net/blog/the-halo-of-a-decision.html

Citation file: CITATION.cff.

12 · References

• Bennett, C. H., Bernstein, H. J., Popescu, S. and Schumacher, B. 1996. "Concentrating partial entanglement by local operations." Physical Review A 53, 2046–2052. The entropy of either half as the measure of a pure pair's entanglement. 10.1103/PhysRevA.53.2046
• Bloch, F. 1946. "Nuclear Induction." Physical Review 70, 460–474. 10.1103/PhysRev.70.460
• Bravyi, S., DiVincenzo, D. P., Oliveira, R. and Terhal, B. M. 2008. "The complexity of stoquastic local Hamiltonian problems." Quantum Information and Computation 8, 361–385. 10.26421/QIC8.5-1
• de Vicente, J. 2007. "Separability criteria based on the Bloch representation of density matrices." Quantum Information and Computation 7, 624–638. 10.26421/QIC7.7-5
• Fano, U. 1983. "Pairs of two-level systems." Reviews of Modern Physics 55, 855–874. The table of correlations of a pair. 10.1103/RevModPhys.55.855
• Feynman, R. P., Vernon, F. L. and Hellwarth, R. W. 1957. "Geometrical Representation of the Schrödinger Equation for Solving Maser Problems." Journal of Applied Physics 28, 49–52. The picture of any two-level system on a sphere. 10.1063/1.1722572
• Granade, C., Combes, J. and Cory, D. G. 2016. "Practical Bayesian tomography." New Journal of Physics 18, 033024. 10.1088/1367-2630/18/3/033024
• Gühne, O. and Tóth, G. 2009. "Entanglement detection." Physics Reports 474, 1–75. Entanglement witnesses. 10.1016/j.physrep.2009.02.004
• Horodecki, M., Horodecki, P. and Horodecki, R. 1996. "Separability of mixed states: necessary and sufficient conditions." Physics Letters A 223, 1–8. 10.1016/S0375-9601(96)00706-2
• Horodecki, R., Horodecki, P., Horodecki, M. and Horodecki, K. 2009. "Quantum entanglement." Reviews of Modern Physics 81, 865–942. 10.1103/RevModPhys.81.865
• Huszár, F. and Houlsby, N. M. T. 2012. "Adaptive Bayesian quantum tomography." Physical Review A 85, 052120. 10.1103/PhysRevA.85.052120
• Klassen, J. and Terhal, B. M. 2019. "Two-local qubit Hamiltonians: when are they stoquastic?" Quantum 3, 139. 10.22331/q-2019-05-06-139
• Kramida, A., Ralchenko, Yu., Reader, J. and NIST ASD Team. 2024. NIST Atomic Spectra Database. National Institute of Standards and Technology, Gaithersburg, MD. 10.18434/T4W30F, accessed 30 September 2026
• Nielsen, M. A. and Chuang, I. L. 2010. Quantum Computation and Quantum Information, 10th anniversary edition. Cambridge University Press. The standard textbook for the Bloch sphere, the Schmidt decomposition, purification, two-level systems and state tomography. 10.1017/CBO9780511976667
• Peres, A. 1996. "Separability Criterion for Density Matrices." Physical Review Letters 77, 1413–1415. The partial transpose test. 10.1103/PhysRevLett.77.1413
• Schmidt, E. 1907. "Zur Theorie der linearen und nichtlinearen Integralgleichungen." Mathematische Annalen 63, 433–476. 10.1007/BF01449770
• Steffen, M., Ansmann, M., Bialczak, R. C. and others. 2006. "Measurement of the Entanglement of Two Superconducting Qubits via State Tomography." Science 313, 1423–1425. 10.1126/science.1130886
• Werner, R. F. 1989. "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model." Physical Review A 40, 4277–4281. 10.1103/PhysRevA.40.4277
• White, A. G., James, D. F. V., Eberhard, P. H. and Kwiat, P. G. 1999. "Nonmaximally Entangled States: Production, Characterization, and Utilization." Physical Review Letters 83, 3103–3107. 10.1103/PhysRevLett.83.3103
• Wootters, W. K. 1998. "Entanglement of Formation of an Arbitrary State of Two Qubits." Physical Review Letters 80, 2245–2248. 10.1103/PhysRevLett.80.2245