Quantum-Inspired Market Analysis

What this is, honestly. Stocks are not quantum systems and do not exhibit physical entanglement — nothing here involves a quantum computer or quantum physics. What is real is that the mathematics of quantum information theory applies to any positive semi-definite matrix with unit trace, and a correlation matrix divided by its dimension is exactly that: a valid density matrix ρ = C/N. Von Neumann entropy, reduced density matrices and quantum mutual information are then all well defined, and turn out to be genuinely useful measures of statistical dependence. "Entanglement" below means statistical inseparability, an analogy, not physics. The QUBO optimiser uses the problem format quantum annealers accept, solved classically.

Computed 4 Aug 2026 10:48 · Quantum-inspired analysis of 60 liquid ASX names

Market state — von Neumann entropy of ρ = C/N

Von Neumann entropy

3.5638

max 4.0775

Normalised entropy

87.4%

Effective dimension

35.3

of 59 stocks

Purity

0.05206

min 0.01695

Top eigenvalue share

17.22%

the "market mode" — how much of everything moves as one

Rolling normalised entropy and effective dimension over 120-day windows. When entropy falls, stocks are moving together and diversification is quietly disappearing — this is the measure that drops hardest in a crisis, when "everything correlates to one".

Sector "entanglement" — quantum mutual information

Sector pair	I(A:B)	Stocks
Communication Services ↔ Technology	0.1423	4/5
Industrials ↔ Real Estate	0.0951	8/6
Consumer Cyclical ↔ Real Estate	0.0877	6/6
Consumer Cyclical ↔ Financial Services	0.0781	6/13
Healthcare ↔ Technology	0.0754	8/5
Communication Services ↔ Consumer Cyclical	0.0744	4/6
Consumer Cyclical ↔ Industrials	0.0682	6/8
Financial Services ↔ Real Estate	0.0672	13/6
Consumer Cyclical ↔ Healthcare	0.066	6/8
Financial Services ↔ Industrials	0.0659	13/8
Consumer Cyclical ↔ Technology	0.0651	6/5
Real Estate ↔ Technology	0.0636	6/5
Communication Services ↔ Real Estate	0.0587	4/6
Basic Materials ↔ Industrials	0.0582	16/8
Communication Services ↔ Healthcare	0.0581	4/8
Industrials ↔ Technology	0.0543	8/5
Financial Services ↔ Healthcare	0.053	13/8
Financial Services ↔ Technology	0.0515	13/5
Healthcare ↔ Real Estate	0.0478	8/6
Healthcare ↔ Industrials	0.0462	8/8

Internal cohesion by sector

Sector	Norm. entropy	Eff. dimension	Stocks
Real Estate	72.2%	3.64	6
Basic Materials	77.5%	8.57	16
Technology	80.9%	3.68	5
Energy	81.6%	3.1	4
Financial Services	85.4%	8.94	13
Communication Services	88.0%	3.38	4
Consumer Cyclical	91.4%	5.14	6
Industrials	93.8%	7.03	8
Healthcare	94.3%	7.11	8
Utilities	98.8%	4.9	5

I(A:B) = S(ρ_A) + S(ρ_B) − S(ρ_AB): total correlation shared between two sectors. Low internal entropy means a sector's stocks move as one block — you get little diversification from holding several of them.

Hidden non-linear links — mutual information beyond correlation

Pair	MI	Corr	Gaussian MI	Non-linear excess
WOW ↔ COL	0.3029	0.403	0.0884	0.2145
IAG ↔ SUN	0.3829	0.569	0.1959	0.187
QBE ↔ SUN	0.2511	0.459	0.1184	0.1327
REA ↔ RMD	0.1489	0.196	0.0196	0.1293
REA ↔ CSL	0.1345	0.127	0.0082	0.1263
MQG ↔ WBC	0.2338	0.46	0.119	0.1148
CBA ↔ WBC	0.3712	0.634	0.2566	0.1146
NEM ↔ NST	0.3563	0.621	0.2433	0.113
ANZ ↔ NAB	0.343	0.607	0.2301	0.113
ANZ ↔ WES	0.1659	0.328	0.0568	0.1091
ANZ ↔ CBA	0.2739	0.531	0.1654	0.1085
IAG ↔ NAB	0.1278	0.2	0.0205	0.1073
ANZ ↔ MQG	0.192	0.401	0.0875	0.1045
WOW ↔ WES	0.1227	0.202	0.0209	0.1018
MQG ↔ CBA	0.1711	0.36	0.0695	0.1016

Binned mutual information minus the MI a Gaussian with the same correlation implies. A positive excess means these two move together in ways a correlation matrix — and therefore every mean-variance optimiser — cannot see. Binned estimators are biased upward in small samples, so compare pairs against each other rather than reading the level as absolute.

Lead–lag structure — transfer entropy

Information flows	Net flow	TE (nats)
GMG → MQG	0.09117	0.21926
RMD → BHP	0.08568	0.22244
AMC → WOW	0.07662	0.20939
CBA → QBE	0.06228	0.20773
QBE → NEM	0.0618	0.21639
SIG → GMG	0.0616	0.20224
CBA → CSL	0.06148	0.218
WOW → ANZ	0.06121	0.2092
WOW → NAB	0.06103	0.21995
WDS → TCL	0.05959	0.20468
SIG → QBE	0.05875	0.22258
TCL → CBA	0.05687	0.21624
ALL → TLS	0.05417	0.21161
FMG → BXB	0.052	0.18425
CSL → GMG	0.05085	0.19366

Transfer entropy TE(X→Y) measures how much X's past improves prediction of Y's next move beyond Y's own past. Unlike correlation it is directional, so it identifies which stock leads. Net flow is TE(X→Y) − TE(Y→X). Treat small values sceptically: binned estimators produce spurious flow on finite samples.

QUBO portfolio selection — simulated annealing

Optimal size

6 stocks

searched 2–8

Expected return

33.5%

Volatility

14.3%

Sharpe

2.067

Selected 6 of 40:

CPU QBE EVN TLS COL SIG

Size	Exp. return	Volatility	Sharpe	Holdings
2	58.3%	31.4%	1.729	EVN SIG
3	48.4%	23.4%	1.896	CPU EVN SIG
4	42.2%	19.4%	1.97	CPU QBE EVN SIG
5	37.2%	16.4%	2.029	CPU QBE EVN COL SIG
6 ★	33.5%	14.3%	2.067	CPU QBE EVN TLS COL SIG
7	31.8%	13.6%	2.043	CPU QBE EVN TLS COL SIG RIO
8	30.0%	12.8%	2.027	CPU QBE EVN TLS COL SIG WDS RIO
naive 6	40.5%	23.3%	1.565	highest expected returns, same size

Size is optimised, not assumed. The QUBO is annealed separately for every portfolio size from 2 to 8 and the results compared by Sharpe — raw QUBO energies are not comparable across sizes because each carries a different cardinality penalty. The curve above is the diversification trade-off: volatility usually falls steeply for the first few names and then flattens, so the best size is where extra diversification stops paying for the concentration it costs.
Choosing exactly K of N stocks is a genuinely binary problem, so it maps to a QUBO — the native format of quantum annealers and QAOA — rather than the continuous quadratic program mean-variance solves. The cardinality constraint is folded in as a penalty term exactly as it would be on quantum hardware; here it is annealed classically, which at this size is entirely sufficient. Expected returns are trailing estimates and carry all their usual noise.