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A Unique Qubit at an Extremal Discrimination Boundary

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Physics Letters A

Extremal states of qubit-qutrit system with maximally mixed marginals

S. Kanmani, S. V. M. Satyanarayana

Published Oct 26, 2020• 4 citations

Publisher

DOI: 10.1016/j.physleta.2020.126978

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A

, the marginal maximal-mixing defect is defined by

D_A=2^{|A|}\operatorname{Tr}(ρ_A^2)-1 =2^{|A|}\left\|ρ_A-\frac{I_A}{2^{|A|}}\right\|_2^2,

and

Q_{\mathrm{ex},\varepsilon}^{D}(n,k)

is defined as the maximum number of

k

-body marginals satisfying

D_A\leq\varepsilon

in an

n

-qubit pure state. This counting problem differs from approximate

k

-uniformity, which requires all

k

-body marginals to obey a common error bound. For pure states on

4m

qubits, the following local stability inequality is established:

\sum_{i\in T}D_{T\setminus\{i\}}\geq1 \qquad (|T|=2m+1).

It follows that, whenever

\varepsilon<1/(2m+1)

, the hypergraph of

\varepsilon

-good

2m

-subsets is

K_{2m+1}^{(2m)}

-free. Combined with the known exact eight-qubit construction, this yields the stability plateau

Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56, \qquad 0\leq\varepsilon<\frac15.

For odd systems of

2k+1

qubits, the exact forbidden hypergraph

H_k

is used to derive explicit finite-error stability radii. In particular,

Q_{\mathrm{ex},\varepsilon}^{D}(9,4)\leq120

for

0\leq\varepsilon<1/17

. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.