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5  The local Bloch radius as a measure of purity and entanglement

Part I treated the qubit as a point on the surface of the sphere: a pure state, a probability \(p_0\) and its four readings. A qubit that belongs to a register is almost never pure: it is mixed by noise or by its entanglement with the others, and it sits inside the sphere. This chapter adds to the base of Chapter 1 the missing quantity, the local radius \(r\), and shows that the probability equation still holds with a single change: the polar bias is no longer \(\cos\theta\) but \(r\cos\theta\). From the radius come an entanglement measure, a way to separate the direction of the state from its purity, and wave–particle duality, all read in qg units.

5.1 The probability equation with a radius

For a qubit in a mixed state, its Bloch vector \((qg_X,qg_Y,qg_Z)\) has length \(r\le1\), and Born’s rule in the \(Z\) basis reads

\[ p_0=\frac{1+qg_Z}{2}=\frac{1+r\cos\theta}{2}. \]

It is the probability equation of Chapter 1 with the radius inside: the readings \(qg_Z=2p_0-1\) and \(m=2000\,p_0-1000\) m-qg do not change, but they no longer fix the angle, because the same \(p_0\) can come from a tilted pure state or from a mixed one closer to the pole. Each axis has its own probability, \(p_0^{(a)}\) in the basis \(a\in\{X,Y,Z\}\), and the squared radius is the sum of the three readings squared (§97):

\[ r^2=qg_X^2+qg_Y^2+qg_Z^2=\frac{m_X^2+m_Y^2+m_Z^2}{10^6}=2\operatorname{Tr}\rho^2-1 . \]

The radius is linear in the purity \(\operatorname{Tr}\rho^2\): it is 1 on the surface (pure state) and 0 at the centre (maximally mixed state). The sphere of radius \(r\) has area \(4\pi r^2\), and the surface deficit \(4\pi(1-r^2)\) is the area the qubit has lost; the library works with \(1-r^2\), which carries the same information.

5.2 The deficit as an entanglement measure

If the whole register is in a pure state, all the radius a qubit lacks is due to its entanglement with the rest. Then \(1-r_q^2=4\det\rho_q\) is the one-tangle of qubit \(q\), which for two qubits is the squared concurrence (Coffman et al. 2000), and its mean over the qubits is the global entanglement of Meyer and Wallach (2002):

\[ Q=\frac1n\sum_q\left(1-r_q^2\right). \]

Two examples set the scale. In the Bell state each qubit sits at the centre of its sphere, \(r=0\), and \(Q=1\). In the 3-qubit W state each has \(r^2=1/9\) and \(Q=0.889\). If the global state is mixed, the deficit mixes entanglement and noise and cannot separate them: the radius measures entanglement only when the register is known, or guaranteed, to be pure. Reading \(r^2\) takes three measurement settings for the whole register (every qubit in \(X\), in \(Y\) and in \(Z\)), not three per qubit.

5.3 The radius under gates and algorithms

A single-qubit gate rotates the sphere and preserves the radius, also for mixed qubits. SWAP exchanges the radii of two qubits. CNOT, CZ, iSWAP, Toffoli and Fredkin can take a qubit of a product input to the centre (\(r=0\), from inputs of the type \(|{+}0\rangle\) or \(|{+}{+}\rangle\)) and can also bring it back to the surface; on computational-basis states they create no deficit, because they act as classical permutations (§97). In algorithms, the radius of the read-out qubits says where the answer is:

Table 5.1: The radius of the read-out qubits in standard algorithms (§97).
algorithm \(r^2\) of the read-out qubits reading
Bernstein–Vazirani, constant Deutsch–Jozsa 1 on every qubit product output: the local \(qg_Z\) is the whole answer
Simon (\(s=110\)) 0 on every qubit the answer lies entirely in the correlations
Grover (\(n=5\)), \(k=0\ldots4\) \(1\to0.82\to0.66\to0.84\to0.998\) entangles and disentangles; back to 1 at the optimum
counting (\(N=16\), \(M=3\)) \((1-2M/N)^2=0.39\) deficit \(4(M/N)(1-M/N)=0.61\)
QAOA, ring of 4, \(p=1\) 0.25 the cost lives in \(ZZ\)

5.4 Definite weight: the radius from the Z basis

In a state of definite Hamming weight every local \(qg_X\) and \(qg_Y\) is zero, so \(r_q=|qg_Z^{(q)}|\) and the radius is read from \(Z\) shots alone. With the probability equation, the deficit becomes

\[ 1-r^2=1-\left(\frac{m}{1000}\right)^2=4\,p_0\,(1-p_0): \]

four times the binomial variance of the outcome, the same \(p_0(1-p_0)\) that governs the shot noise of Chapter 1. Figure 5.1 shows it on the milli-qang scale. The deficit is zero at the poles, largest on the equator and very sensitive near a pole: a qubit at 990 m-qg has a deficit of 0.02, and one at 800 m-qg, which is what 10 % of misplaced shots on a basis state gives, already reads 0.36.

Figure 5.1: The deficit of a definite-weight qubit against its milli-qang reading (bottom) and its probability (top). Near a pole, a few misplaced shots produce a large deficit.

This is where the symmetry filter (Chapters 20–21) and the radius meet. Under equal T1 on every qubit the filtered state is the noiseless pure state, so the filtered deficit is an exact entanglement measure. The predictions were committed before running (§98); on RBS circuits of 4 and 6 qubits:

Table 5.2: The deficit as an entanglement measure under noise, with and without the qg filter (§98).
noise exact error of the deficit, with qang without qang filter MSE lower
equal T1 \(<3\cdot10^{-15}\) 0.061–0.370 30 of 30
unequal T1 0.013–0.045 0.070–0.342 30 of 30
dephasing 0.01 0.031–0.307 0.080–0.352 24 of 30

On product states the filtered deficit is exactly 0 under all three noise models, while the raw read-out shows 0.60–0.95 on every excited qubit from decay alone: false entanglement. One prediction failed: the raw error was expected to be smaller at half filling, and it was larger (0.177 against 0.074), because the first-order cancellation holds only at exactly \(p=1/2\).

On noise models of real devices (three from IBM and two from IonQ, §100), the filter cut the error of the deficit 3.2–4.0× where the deficit is large, but it did not work as a test of “no entanglement”: on a product state it removed only 15–33 % of the false deficit. The cause is the one in Figure 5.1: errors inside the sector leave 10–21 % of misplaced shots, and near the pole that already reads as 0.36. A small deficit needs an echo calibration before it is interpreted.

5.5 Direction and radius apart

The polar bias \(qg_Z=r\cos\theta\) mixes the direction of the vector with its length. The usual reading \(\theta=\arccos(qg_Z)\) drifts towards the equator whenever noise shortens the vector. With the radius the two separate (§102):

\[ \theta=\operatorname{atan2}\!\left(\sqrt{qg_X^2+qg_Y^2},\,qg_Z\right)=\arccos\frac{qg_Z}{r}. \]

A state at \(30^\circ\) shortened to \(r=0.8\) by depolarizing noise has \(qg_Z=0.693\): \(\arccos(qg_Z)\) reads \(46.1^\circ\), and the separated angle reads the true \(30^\circ\). Figure 5.2 shows the effect for several radii: the naive angle flattens towards \(90^\circ\), and the separated one does not depend on \(r\).

Figure 5.2: The angle read against the true direction. The dashed curves are \(\arccos(qg_Z)\) for shortened vectors; the straight line is the angle separated from the radius, exact for any \(r\) under depolarizing noise.

The separation has a cost and does not always win. With the same total number of shots split over three bases:

  • Depolarizing: the separated angle is exact and \(\arccos(qg_Z)\) errs by up to 0.70 rad near the poles; with shots, the root-mean-square error drops 5.8×.
  • Dephasing: this is where it loses. Dephasing does not shorten \(qg_Z\), so \(\arccos(qg_Z)\) is exact, and the separated angle is pulled 0.03–0.12 rad towards the poles.
  • No noise: measuring three bases costs 1.6× in root-mean-square error.
  • Devices: on the IBM models the vector shrinks almost uniformly and the separated angle is 5–9× more accurate. On IonQ, \(\arccos(qg_Z)\) won at first, because the compiler had removed pairs of identical CX gates; in the native gate set the separated angle wins there too, 8.7–9.0× (§103).

The practical rule: if the measured radius is close to 1, use \(\arccos(qg_Z)\) with all shots in \(Z\); if it is clearly below 1 and the noise is not pure dephasing, use the separated angle. On the angular scale of Chapter 1 (Section 1.4), the example above is 83.3 qga true against 128.1 qga read.

The azimuthal phase does not depend on the radius. Since \(qg_X=r\sin\theta\cos\varphi\) and \(qg_Y=r\sin\theta\sin\varphi\), the ratio cancels \(r\) and \(\sin\theta\), and \(\varphi=\operatorname{atan2}(qg_Y,qg_X)\) is the same for the shortened vector. That holds whenever the noise shortens both transverse components equally: depolarizing noise, dephasing and T1 relaxation preserve the phase. A channel that treats \(X\) and \(Y\) differently shifts it: with a bit flip of probability 0.15, which keeps \(qg_X\) and shortens \(qg_Y\), a phase of 0.70 rad reads 0.53 rad.

Open proposals. The author’s working version adds a purity penalty \(\lambda_r(1-r^2)\) to the regularized energy of Chapter 1, damps the azimuthal part of the radiation function with \(r\) and, against noise-induced barren plateaus (Wang et al. 2021), proposes watching the radius layer by layer, discarding layers with \(r<0.3\) and compiling with minimum curvature \(\kappa_{qg}\), with a 35 to 60% reduction of exposure time. That noise produces plateaus is a known result; the three proposals and those figures have not been verified.

5.6 Wave–particle duality and entropic uncertainty

The radius also orders a classical result of the foundations. The predictability \(P=|qg_Z|=|m|/1000\) measures how much is known about the path (particle) and the visibility \(V=\sqrt{qg_X^2+qg_Y^2}\) measures the contrast of the interference (wave). Their relation is exactly the radius (Greenberger and Yasin 1988; Englert 1996):

\[ P^2+V^2=r^2\le1 . \]

The duality inequality is, in qg units, the statement that the state lies inside the sphere, and it is an equality for pure states (Figure 5.3). A state at \(60^\circ\) with \(r=0.9\) has \(P=0.45\), \(V=0.779\) and \(P^2+V^2=0.81\). It was checked on 2000 random states with a largest deviation of \(2.2\cdot10^{-16}\) (§127).

Figure 5.3: Wave–particle duality in qg units: each state is a point of the quarter disc, at distance \(r\) from the origin. Pure states lie on the arc \(r=1\).

The entropic version uses Chapter 1’s spread \(qg_S\) in two bases. The Maassen–Uffink relation, with the correction of Berta and co-workers for mixed states, says

\[ qg_S(Z)+qg_S(X)\ \ge\ 1+S(\rho),\qquad S(\rho)=H\!\left(\frac{1+r}{2}\right). \]

For the state above the sum is 1.349 and the bound 1.286. The bound is tight: the state \(|0\rangle\) meets it with equality. That gives a practical warning. With plain (uncorrected) entropy estimates, a pure state seems to violate the relation in 93 % of runs at 100 shots, because the estimated entropy is biased downwards; with the Miller–Madow correction, in 37 %. A test of the bound needs corrected estimators or intervals, like those of Chapter 4.

5.7 Estimating the radius from shots

Since \(r^2\) is a sum of squares, it inherits the bias of the quadratic estimators of Chapter 4 (Section 4.4): each axis adds \((1-qg^2)/N\), and an entangled state looks purer than it is. The exact correction \((N\widehat{qg}^{\,2}-1)/(N-1)\) is applied axis by axis (radius2_estimate), and in a definite-weight state the \(Z\) axis is enough.

5.8 Without qang and with qang

Table 5.3: The chapter’s actions without the qang scale and with it.
action without qang with qang difference
measure local purity tomography in three bases and \(\operatorname{Tr}\rho^2\) \(r^2=qg_X^2+qg_Y^2+qg_Z^2\); at definite weight, \(1-r^2=4p_0(1-p_0)\) from a single basis in favour: one basis instead of three, thanks to the probability equation together with the definite weight
measure the entanglement of a pure register one-tangle and Meyer–Wallach by definition the deficit \(1-r^2\), averaged none in value; with a mixed global state the deficit mixes entanglement and noise, like the one-tangle
direction of a noisy state \(\theta=\arccos qg_Z\) \(\theta=\arccos(qg_Z/r)\) in favour with depolarizing noise; against with pure dephasing
wave–particle duality visibility and predictability separately \(P^2+V^2=r^2\) reformulation: the same relation

Balance. The practical difference is reading purity at definite weight with a single measurement basis. The rest are known identities written on the qang scale, and the direction–radius split wins or loses depending on the noise.

5.9 In code

import math
import numpy as np
from qang import complementarity, formulation, statistics

bell = np.array([1, 0, 0, 1]) / math.sqrt(2)
formulation.radius_profile(bell, 2)          # [0.0, 0.0]: each qubit at the centre
w = np.zeros(8); w[[1, 2, 4]] = 1 / math.sqrt(3)
formulation.meyer_wallach(w, 3)              # 0.889: global entanglement of the W state
formulation.weight_sector_radius([0.99, 0.8])    # [0.0199, 0.36]: deficit from qg_Z (990 and 800 m-qg)

formulation.direction_from_qg(0.4, 0.0, 0.6928)  # (0.524, 0.0, 0.8): θ = 30° separated from r = 0.8
complementarity.duality_sum(0.779, 0.0, 0.45)    # 0.81 = P² + V² = r²
complementarity.entropic_sum(0.45, 0.779)        # 1.35 ≥ bound 1.29

The figures are generated by tools/make_cap05_figures.py.

5.10 Summary

  • For a mixed state, Chapter 1’s probability equation still holds with \(qg_Z=r\cos\theta\); the local radius \(r^2=qg_X^2+qg_Y^2+qg_Z^2=2\operatorname{Tr}\rho^2-1\) is the missing reading.
  • If the register is pure, the deficit \(1-r^2\) is the one-tangle and its mean is the Meyer–Wallach measure; with a mixed global state, it mixes entanglement and noise.
  • At definite weight, \(1-r^2=4p_0(1-p_0)\) is read in \(Z\) alone, and with the filter under equal T1 it is exact; near a pole a few misplaced shots inflate it, and on devices a small deficit needs calibration.
  • \(\theta=\arccos(qg_Z/r)\) separates direction from purity: it wins under depolarizing noise and loses under pure dephasing.
  • Wave–particle duality is \(P^2+V^2=r^2\), and entropic uncertainty bounds \(qg_S(Z)+qg_S(X)\) by \(1+S(\rho)\).

Bibliographic notes

The one-tangle and the concurrence are from Coffman et al. (2000), and global entanglement, from Meyer and Wallach (2002). Wave–particle duality is Greenberger and Yasin (1988) and Englert (1996); entropic uncertainty, Maassen and Uffink (1988) and, with quantum memory, Berta et al. (2010). The Deutsch–Jozsa, Bernstein–Vazirani, Simon and Grover algorithms are Deutsch and Jozsa (1992), Bernstein and Vazirani (1997), Simon (1997) and Grover (1996). The numbered research notes (§) cited in this chapter and the scripts that check it are in the repository (Monteverde 2026a); the definition of the qang unit is in Monteverde (2026b).