Quantum Tsallis Relative Entropy

• Quantum Tsallis relative entropy is a one-parameter divergence defined via trace functionals that generalizes Umegaki relative entropy as α approaches 1.
• It features nonadditive deformations and versatile formulations—including sandwiched, conditional, and operator-valued forms—that are applied in entanglement detection, coherence quantification, and resource theories.
• Its analytical bounds, variational representations, and sample-efficient estimation insights provide practical tools for quantum state discrimination and complexity analysis.

Searching arXiv for recent and foundational papers on quantum Tsallis relative entropy and closely related variants. Quantum Tsallis relative entropy is a one-parameter family of quantum divergences built from the trace functional $\Tr(\rho^{\alpha}\sigma^{1-\alpha})$ and used to quantify distinguishability between density operators. In the limit $\alpha\to 1$ it reduces to the Umegaki relative entropy, while for $\alpha\neq 1$ it furnishes a nonadditive deformation that has been developed in ordinary, sandwiched, conditional, and operator-valued forms. Across the literature it appears in entanglement detection, coherence and correlation theory, quantum information geometry, and, more recently, in sample-efficient estimation and complexity-theoretic classification (Rastegin, 2015, Nayak et al., 2014, Bao et al., 1 Oct 2025).

1. Standard definition and parameterizations

For density operators $\rho$ and $\sigma$ on a finite-dimensional Hilbert space, a standard definition is

$D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$

with the convention that for $\alpha>1$ the quantity is finite only when $\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$, and equals $+\infty$ otherwise (Rastegin, 2015). For $\alpha\in(0,1)$, the same quantity is often written in the equivalent form

$\alpha\to 1$0

which emphasizes the nonnegativity of the overlap term $\alpha\to 1$1 in that regime (Bao et al., 1 Oct 2025, Rastegin, 2011).

The literature uses both $\alpha\to 1$2 and $\alpha\to 1$3 as deformation parameters. In the notation based on the $\alpha\to 1$4-logarithm,

$\alpha\to 1$5

the divergence may also be written as

$\alpha\to 1$6

whenever $\alpha\to 1$7 is strictly positive on $\alpha\to 1$8 (Rastegin, 2015). The undeformed limit is

$\alpha\to 1$9

namely the usual quantum relative entropy of Umegaki–Lieb–Ruskai (Rastegin, 2015).

This standard trace form is the version most directly connected to the $\alpha\neq 1$0-affinity

$\alpha\neq 1$1

through

$\alpha\neq 1$2

and it is the form used in the recent estimation and complexity results for $\alpha\neq 1$3 (Bao et al., 1 Oct 2025).

2. Structural properties, inequalities, and variational representations

Several basic properties recur throughout the literature. Nonnegativity is standard:

$\alpha\neq 1$4

with equality if and only if $\alpha\neq 1$5 (Rastegin, 2015, 1811.11453). Monotonicity under trace-preserving completely positive or completely positive trace-preserving maps is also central. In one treatment, if $\alpha\neq 1$6 is any completely positive trace-preserving map, then for $\alpha\neq 1$7,

$\alpha\neq 1$8

and joint convexity holds in the same parameter range (Rastegin, 2015). Closely related summaries state the same monotonicity and joint convexity for $\alpha\neq 1$9 (1811.11453). A coherence-oriented treatment isolates $\rho$0 as the range in which the data-processing inequality is used directly and notes that for $\rho$1 the data-processing inequality generally fails in that setting (Vershynina, 2022). This suggests that the operational range depends on the exact formulation under discussion.

Quantitative comparison with norm distances is provided by Pinsker-type bounds. For $\rho$2, one recent estimate gives

$\rho$3

where $\rho$4 is the Schatten $\rho$5-distance (Bao et al., 1 Oct 2025). In the special case $\rho$6,

$\rho$7

so the Tsallis divergence becomes twice the quantum Hellinger distance (Bao et al., 1 Oct 2025). Earlier work also derives a family of lower bounds of Pinsker type for $\rho$8, upper continuity bounds for $\rho$9 in the commutative case, Tsallis-type Fano inequalities, and Fannes-type continuity estimates for Tsallis entropies (Rastegin, 2011).

A further analytic layer comes from variational and deformed-exponential methods. With the $\sigma$0-logarithm and $\sigma$1-exponential,

$\sigma$2

Shi and Hansen derive Legendre-type variational representations for the quantum Tsallis relative entropy and a $\sigma$3-deformed Golden–Thompson inequality for $\sigma$4 (Shi et al., 2019). In parallel, operator-level analyses produce upper and lower bounds for the Tsallis relative operator entropy and then, by taking traces, corresponding scalar bounds for the quantum Tsallis relative entropy (Furuichi et al., 2020, Furuichi, 2010).

3. Sandwiched and conditional forms

To treat noncommutativity explicitly, Rajagopal and coauthors introduced the sandwiched Tsallis relative entropy

$\sigma$5

If $\sigma$6, this collapses to the ordinary $\sigma$7, and by the Lieb–Thirring inequality one has

$\sigma$8

in the sense used in the separability analyses (Nayak et al., 2014, Rajagopal et al., 2013). The same line of work also records positivity and data-processing properties for the sandwiched quantity in the ranges stated there, together with the limit to Umegaki relative entropy as $\sigma$9 (Rajagopal et al., 2013, Nayak et al., 2016).

For a bipartite state $D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$0 with marginal $D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$1, the conditional sandwiched Tsallis relative entropy (CSTRE) is defined by choosing $D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$2:

$D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$3

where

$D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$4

In the special commuting case $D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$5, this reduces to the Abe–Rajagopal $D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$6-conditional Tsallis entropy

$D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$7

(Nayak et al., 2014).

The entanglement criterion is direct. For $D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$8, negativity of CSTRE implies bipartite entanglement; equivalently, all separable $D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\Bigl(\Tr[\rho^{\alpha}\sigma^{1-\alpha}]-1\Bigr),$9 satisfy

$\alpha>1$0

Hence $\alpha>1$1 is an entanglement witness (Nayak et al., 2014). The same framework is explicitly non-spectral: a well-known isospectral pair of two-qubit states with the same global and local spectra, one entangled and one separable, is distinguished by CSTRE, whereas the Abe–Rajagopal conditional entropy vanishes on both (Rajagopal et al., 2013).

4. Separability thresholds and entanglement applications

A major application of quantum Tsallis relative entropy is the determination of bipartite separability ranges in symmetric one-parameter families. For the noisy $\alpha>1$2-qubit W and GHZ families

$\alpha>1$3

with $\alpha>1$4 the projector onto the $\alpha>1$5-dimensional symmetric subspace, the $\alpha>1$6 separability thresholds obtained from the $\alpha>1$7 limit of CSTRE are

$\alpha>1$8

Thus the separability ranges are

$\alpha>1$9

for the noisy W family and

$\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$0

for the noisy GHZ family (Nayak et al., 2014). In these $\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$1 partitions the CSTRE criterion yields exactly the same critical values as the positive-partial-transpose test. The same analysis also displays the advantage over the Abe–Rajagopal criterion in the noisy W family, where the single-qubit marginal is not maximally mixed: Abe–Rajagopal gives $\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$2, whereas CSTRE gives the strictly smaller bound above and matches PPT (Nayak et al., 2014).

Related one-parameter families admit equally explicit thresholds. For pseudo-pure W and GHZ states in the $\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$3 cut, Nayak et al. obtain

$\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$4

while for Werner-like W and GHZ states they obtain

$\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$5

In each case, nonnegativity of the conditional sandwiched Tsallis relative entropy is necessary and sufficient for separability in the $\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$6 partition, because the $\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$7 limit reproduces the known PPT or Schmidt-coefficient thresholds (Nayak et al., 2016).

For the $\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$8-qudit Werner–Popescu states

$\mathrm{supp}(\rho)\subseteq \mathrm{supp}(\sigma)$9

the same method gives the exact $+\infty$0 separability range

$+\infty$1

again in the $+\infty$2 limit (Nayak et al., 2017). A plausible implication is that the conditional sandwiched Tsallis construction is especially effective on highly symmetric families, where it frequently reaches algebraic separability thresholds exactly.

5. Coherence, discord, and other resource measures

The Tsallis relative $+\infty$3-entropy induces a family of coherence quantifiers by minimizing over incoherent states. Given a reference basis $+\infty$4 and the set $+\infty$5 of diagonal states, one defines

$+\infty$6

For all $+\infty$7, $+\infty$8, the minimization has the closed form

$+\infty$9

For $\alpha\in(0,1)$0, these coherence measures satisfy nonnegativity, faithfulness, convexity in mixtures, monotonicity under incoherent operations, and a generalized strong monotonicity under selective measurements,

$\alpha\in(0,1)$1

which reduces to the usual form in the limit $\alpha\in(0,1)$2 (Rastegin, 2015). The same paper derives trade-off relations with mixedness, including

$\alpha\in(0,1)$3

where $\alpha\in(0,1)$4 (Rastegin, 2015).

Not every Tsallis-based coherence expression is a bona fide coherence monotone. The simple quadratic $\alpha\in(0,1)$5-measure $\alpha\in(0,1)$6 fails ordinary strong monotonicity (Rastegin, 2015). More generally, the distance-based definition

$\alpha\in(0,1)$7

fails strong monotonicity, whereas modified and convex-roof Tsallis coherence measures satisfy the full set of coherence axioms (Vershynina, 2019). A separate construction based on the entropy increment

$\alpha\in(0,1)$8

is nonnegative and asymptotically continuous, but it fails to be monotone under general incoherent operations or even genuine incoherent operations; monotonicity is recovered only for the restrictive class of $\alpha\in(0,1)$9-GIO maps obeying $\alpha\to 1$00 (Vershynina, 2022). This is one of the main misconceptions corrected in the recent literature: Tsallis-based coherence expressions are not interchangeable from the resource-theoretic viewpoint.

An operator-valued variant uses the Tsallis relative operator entropy

$\alpha\to 1$01

whose trace recovers a Tsallis relative entropy. The induced coherence

$\alpha\to 1$02

satisfies faithfulness, monotonicity and strong monotonicity under incoherent CPTP maps, convexity, and block additivity. In the limit $\alpha\to 1$03 it reproduces the relative-entropy coherence, while at $\alpha\to 1$04 it is closely related to geometric coherence (Guo et al., 2020).

Beyond coherence, Tsallis relative entropy supports discord-like and correlation measures. For classical–quantum states $\alpha\to 1$05, one defines

$\alpha\to 1$06

and an analytic formula is available:

$\alpha\to 1$07

with

$\alpha\to 1$08

The limit $\alpha\to 1$09 gives the relative-entropy discord, while $\alpha\to 1$10 reproduces the Hellinger-distance discord (1811.11453). Closely related work develops three discord measures and two correlation measures based on Tsallis relative entropy, with explicit pure-state formulas and tight upper and lower bounds (Vershynina, 2019). A further resource-theoretic extension quantifies imaginarity through

$\alpha\to 1$11

which is monotonic under real operations and admits closed-form evaluation for bosonic Gaussian states (Xu, 2023). In Grover’s search algorithm, the Tsallis relative $\alpha\to 1$12 entropy of coherence decreases with the increase of the success probability and obeys explicit complementarity relations with that probability (Ye et al., 15 Apr 2026).

6. Operator inequalities, quantum geometry, and estimation complexity

At the operator level, the Tsallis relative operator entropy

$\alpha\to 1$13

interpolates to the Fujii–Kamei relative operator entropy as $\alpha\to 1$14 (Furuichi et al., 2020). Furuichi and Moradi derive two-sided bounds for $\alpha\to 1$15, including a nontrivial upper bound, and improve monotonicity under unital positive maps (Furuichi et al., 2020). In the coherence-oriented operator formulation, Furuta-type inequalities yield the reverse Shannon inequality

$\alpha\to 1$16

and the extended Shannon inequality

$\alpha\to 1$17

from which $\alpha\to 1$18 if and only if $\alpha\to 1$19 (Guo et al., 2020).

Quantum Tsallis relative entropy also functions as a potential for information geometry. On the manifold of full-rank density matrices, one may define

$\alpha\to 1$20

and from it derive a family of quantum metrics through a coordinate-free differential calculus. The resulting metric can be written as

$\alpha\to 1$21

For qubits and qutrits this yields explicit decompositions into orbit and transversal parts; for qubits, the special value $\alpha\to 1$22 gives the Wigner–Yanase metric, and radial limits recover metrics on lower-rank strata, including pure states (Man'ko et al., 2016). In the tomographic picture, the Fisher–Rao metric on quantum tomograms can be used to reconstruct the quantum metric of density states, and the qubit Bloch-ball condition becomes the experimentally testable inequality

$\alpha\to 1$23

for spin-$\alpha\to 1$24 projections along three perpendicular directions (Man'ko et al., 2016).

Recent work addresses the estimation of the quantum Tsallis relative entropy itself. For any constant $\alpha\to 1$25 and unknown rank-$\alpha\to 1$26 states $\alpha\to 1$27, $\alpha\to 1$28 can be estimated to additive error $\alpha\to 1$29 with probability at least $\alpha\to 1$30 using

$\alpha\to 1$31

$\alpha\to 1$32

and

$\alpha\to 1$33

where $\alpha\to 1$34 hides $\alpha\to 1$35 factors (Bao et al., 1 Oct 2025). With purified query access to state-preparation unitaries, the same task requires only

$\alpha\to 1$36

in the same three parameter regimes (Bao et al., 1 Oct 2025). Since $\alpha\to 1$37 is twice the quantum Hellinger distance, this yields tolerant state certification with sample complexity $\alpha\to 1$38 or query complexity $\alpha\to 1$39, and the associated distinguishability problems are classified as $\alpha\to 1$40-complete in one regime and $\alpha\to 1$41-complete in the low-rank case (Bao et al., 1 Oct 2025).

Taken together, these developments place quantum Tsallis relative entropy at the intersection of operator inequalities, noncommutative entropic geometry, entanglement detection, resource theories, and quantum algorithms. The standard divergence, its sandwiched and conditional extensions, and its operator-valued analogues do not simply repackage the Umegaki case; they isolate parameter-dependent regimes in which noncommutativity, symmetry, and rank structure become analytically and operationally visible.