Mutual Trace Distance in Quantum Systems

• Mutual Trace Distance is defined as half the trace norm difference between two quantum states, capturing their distinguishability and total correlations.
• It is employed to quantify bipartite correlations, system–environment interactions, and block correlations in DMRG studies, providing insights into phase transitions and non-Markovian effects.
• Research reveals its non-monotonic behavior under tensor products and supports quantum algorithms for estimation, highlighting both its operational significance and practical challenges.

“Mutual trace distance” is not a standard standalone metric in quantum information theory. In the literature, the expression is used or interpreted in several closely related senses: as the standard trace distance between two quantum states; as a trace-distance quantifier of total correlations in bipartite systems; and as the trace distance between a composite state and the product of its marginals, especially for system–environment partitions in DMRG and open-system settings (Dajka et al., 2011, Aaronson et al., 2013, Luo et al., 2016, Maziero, 2015, Coles et al., 2019, Ghosh et al., 7 Apr 2026). Across these usages, the common core is the trace distance

$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1,$

an operationally meaningful distinguishability measure whose behavior under CPTP maps, tensor products, and subsystem reduction gives rise to distinct “mutual” constructions.

1. Foundational meaning of trace distance

For density operators $\rho$ and $\sigma$, trace distance is defined by

$$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$

with $0\le D(\rho,\sigma)\le 1$. In the unnormalized convention also used in the literature, $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$, so that $d_{\mathrm{tr}}=2D$ (Maziero, 2015). For commuting states or classical probability vectors $p=(p_i)$ and $q=(q_i)$, the same quantity reduces to total variation distance,

$D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$

Its operational role is fixed by binary minimum-error discrimination. For two equiprobable hypotheses, the optimal Helstrom measurement yields success probability

$\rho$0

equivalently minimum error probability

$\rho$1

The trace distance therefore quantifies the maximal bias achievable in distinguishing two states by measurement (Maziero, 2015, Ghosh et al., 7 Apr 2026).

As a geometric object, trace distance is a metric, is unitarily invariant, obeys the triangle inequality, and is contractive under CPTP maps:

$\rho$2

This contractivity is central to every later use of “mutual trace distance,” but several of the constructions discussed below probe situations in which the reference objects being compared are themselves composite, optimized, or varied under tensor powers, so the resulting behavior can be subtler than simple pairwise contractivity suggests (Ghosh et al., 7 Apr 2026).

2. Mutual trace distance as a measure of bipartite correlations

A precise use of the term appears in the hierarchy of trace-distance correlations for bipartite states. For a bipartite state $\rho$3, the relevant sets are the product states

$\rho$4

and the classical-quantum states on $\rho$5,

$\rho$6

The trace-distance-based measures are then defined as

$\rho$7

$\rho$8

$\rho$9

where $\sigma$0 is a closest classical-quantum state (Aaronson et al., 2013).

In this setting, “mutual trace distance” corresponds to the total correlations $\sigma$1, namely the minimal trace distance from $\sigma$2 to the set of uncorrelated product states. For Bell-diagonal two-qubit states,

$\sigma$3

the closest classical state is

$\sigma$4

where $\sigma$5. The trace-distance discord has the closed form

$\sigma$6

where $\sigma$7 is the median of $\sigma$8, while the classical correlations are

$\sigma$9

For total correlations, the optimization reduces to a one-parameter minimization over product states whose local Bloch vectors are aligned along the axis associated with $$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$0 (Aaronson et al., 2013).

The trace-distance hierarchy differs sharply from the relative-entropy hierarchy. The closest product state $$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$1 is not generally $$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$2; for Bell-diagonal states, this means that the closest uncorrelated state is generally not the product of the marginals. Moreover, the total correlations are generally strictly smaller than the sum of the quantum and classical parts,

$$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$3

This strict subadditivity is a defining structural feature of the trace-distance formulation (Aaronson et al., 2013).

For special Bell-diagonal families, explicit formulas are available. Werner states with $$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$4 satisfy

$$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$5

and

$$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$6

Rank-2 Bell-diagonal states with $$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$7 and $$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$8 satisfy

$$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_1=\frac{1}{2}\operatorname{Tr}|\rho-\sigma|,$$9

with piecewise expressions for $0\le D(\rho,\sigma)\le 1$0 across three intervals of $0\le D(\rho,\sigma)\le 1$1 (Aaronson et al., 2013).

The same framework also yields dynamical statements. Under local phase-flip channels and random external fields preserving Bell-diagonal form, $0\le D(\rho,\sigma)\le 1$2 exhibits freezing over a larger set of initial states than the corresponding relative-entropy or Hilbert–Schmidt measures. By contrast, $0\le D(\rho,\sigma)\le 1$3 generally does not freeze and varies smoothly because it depends on all three Bell-diagonal coefficients rather than only their median or maximum (Aaronson et al., 2013).

3. Reduced-state distinguishability and initial system–environment correlations

A second usage of the term arises when the relevant quantity is the trace distance between two reduced system states,

$0\le D(\rho,\sigma)\le 1$4

especially in the presence of initial system–environment correlations. In this setting the central issue is the failure of contractivity for the reduced dynamics when the initial total state is not factorized. For a positive, trace-preserving map $0\le D(\rho,\sigma)\le 1$5,

$0\le D(\rho,\sigma)\le 1$6

and for completely positive dynamical semigroups,

$0\le D(\rho,\sigma)\le 1$7

If the system and environment are initially uncorrelated, reduced dynamics is completely positive and contractive. Initial correlations can invalidate this conclusion, allowing the reduced trace distance to exceed its initial value (Dajka et al., 2011).

The open-system analysis was carried out for two pure-dephasing-type qubit models. The first is a qubit bilinearly coupled to an infinite bosonic environment with Hamiltonian

$0\le D(\rho,\sigma)\le 1$8

with $0\le D(\rho,\sigma)\le 1$9, a bosonic bath Hamiltonian $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$0, and bilinear interaction $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$1. Initial correlations are controlled by a parameter $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$2 through the bath state $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$3; $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$4 gives an uncorrelated total state, while $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$5 gives maximal entanglement within the class considered. The reduced qubit state has off-diagonal factor $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$6 determined by the spectral density

$$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$7

with ohmicity parameter $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$8 (Dajka et al., 2011).

In this infinite-environment model, only the trace distance and, for qubits, the equivalent Hilbert–Schmidt distance can increase above their initial values. Such growth occurs only in the super-ohmic regime $$d_{\mathrm{tr}}(\rho,\sigma)=\|\rho-\sigma\|_1\in[0,2]$$9. A necessary condition is that the compared initial total states differ in the environment component, specifically in the correlation parameter $d_{\mathrm{tr}}=2D$0; varying only the system amplitudes while keeping the same $d_{\mathrm{tr}}=2D$1 does not permit increasing growth. By contrast, the Bures distance, Hellinger distance, and quantum Jensen–Shannon divergence remain below their initial values, although they may display non-monotonic decay with partial recovery toward saturation (Dajka et al., 2011).

The second model is a qubit bilinearly coupled to a finite-size environment consisting of a single harmonic oscillator mode,

$d_{\mathrm{tr}}=2D$2

For coherent-state mixtures in the environment, the reduced dynamics is periodic, and all four distances studied—trace, Bures, Hellinger, and Jensen–Shannon—can increase above their initial values, reach a maximum, and oscillate with period $d_{\mathrm{tr}}=2D$3. The amplitude increases as $d_{\mathrm{tr}}=2D$4 and depends sensitively on the coherent-state amplitude $d_{\mathrm{tr}}=2D$5 and phase $d_{\mathrm{tr}}=2D$6. For number-state mixtures, only the trace distance and the Jensen–Shannon divergence can exceed their initial values, while Bures and Hellinger remain below their initial values for all times; the strongest increase occurs for the first excited state $d_{\mathrm{tr}}=2D$7 (Dajka et al., 2011).

These results give trace distance a special status as a witness of initial correlations. They also show that conclusions about “information backflow” or non-Markovianity depend strongly on the metric employed: some distances remain contractive in scenarios where trace distance does not (Dajka et al., 2011).

4. System–environment mutual trace distance in DMRG studies of quantum criticality

A third precise use appears in finite-system DMRG, where the chain is partitioned into a system block $d_{\mathrm{tr}}=2D$8 and an environment block $d_{\mathrm{tr}}=2D$9 at each step of a sweep. The relevant quantity is

$p=(p_i)$0

which quantifies total correlations, classical plus quantum, between the two DMRG blocks (Luo et al., 2016).

At zero temperature, $p=(p_i)$1 is the pure ground-state density matrix obtained from standard finite-size DMRG. At finite temperature, the thermal state

$p=(p_i)$2

is approximated at low temperature by targeting a set of 14 lowest-lying eigenstates in finite-size DMRG; the study focuses on $p=(p_i)$3. In both cases, the marginals $p=(p_i)$4 and $p=(p_i)$5 are obtained by partial trace in the DMRG basis. The “average correlation measured by the trace distance” is the sweep-average of $p=(p_i)$6 over all cuts of a full finite-system sweep (Luo et al., 2016).

The numerical implementation uses open boundary conditions, chain length $p=(p_i)$7, up to 5 sweeps at finite temperature, and maximum discarded weight approximately $p=(p_i)$8. Two Hamiltonians were studied. The first is the spin-1 XXZ chain with single-ion anisotropy,

$p=(p_i)$9

The second is the anisotropic spin-$q=(q_i)$0 Heisenberg chain with staggered coupling and staggered magnetic field,

$q=(q_i)$1

with

$q=(q_i)$2

For $q=(q_i)$3, the latter reduces to an exactly solvable staggered XY chain (Luo et al., 2016).

The central observation is that the sweep-averaged block correlation exhibits discontinuities at critical points of quantum phase transitions. In the spin-1 model, the discontinuities track the known phase boundaries separating Haldane, XY, ferromagnetic, and Néel phases. In the staggered spin-$q=(q_i)$4 chain at finite temperature, the same quantity displays sharp changes at the critical boundaries in the $q=(q_i)$5 and $q=(q_i)$6 planes. For $q=(q_i)$7, the resulting boundaries agree very well with exact analytic phase lines; as $q=(q_i)$8 increases, the boundaries shift toward smaller static fields $q=(q_i)$9 (Luo et al., 2016).

This formulation is notable because it avoids the basis-alignment difficulties of fidelity-based diagnostics in DMRG. It is also basis-independent, since the trace distance is unitarily invariant. Its limitation is equally clear: it measures total correlations rather than entanglement alone, so it detects non-analytic restructuring of the many-body state without by itself identifying order parameters or universality classes (Luo et al., 2016).

5. Comparative behavior under tensor products

A distinct but related issue is the non-monotonicity of trace distance under tensor products of its arguments. For a fixed ancilla state $D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$0, the CPTP map $D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$1 preserves the unnormalized trace distance exactly,

$D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$2

because $D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$3 (Maziero, 2015).

The nontrivial phenomenon is different: the relative ordering of distinguishability between two different pairs can flip after tensoring both pairs. There exist quartets $D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$4 such that

$D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$5

but

$D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$6

This is the non-monotonicity under tensor products of the arguments, abbreviated NMuTP (Maziero, 2015).

The phenomenon does not occur for all state classes. For pure states, ordering is preserved under tensor powers because

$D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$7

and both functions are strictly decreasing in the overlap. One-qubit mixed states with collinear Bloch vectors also avoid NMuTP, since the two-copy trace distance is a monotonically increasing function of the one-copy trace distance for such pairs (Maziero, 2015).

Mixed states can nevertheless exhibit NMuTP even in the simplest commuting qubit case. For

$D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$8

$D(p,q)=\frac{1}{2}\sum_i |p_i-q_i|.$9

one has

$\rho$00

but for two copies,

$\rho$01

The ranking therefore reverses (Maziero, 2015).

To quantify the reversal strength, the study introduces

$\rho$02

defined only when an ordering flip occurs. Extensive Monte Carlo sampling over $\rho$03 quartets per experiment for qubits shows that the fraction of quartets exhibiting NMuTP is non-negligible and can be particularly high when one state in a pair is maximally mixed, reaching about $\rho$04. For higher-dimensional qudits, both the fraction of NMuTP quartets and the strength measure $\rho$05 decrease with dimension (Maziero, 2015).

This result places an important restriction on any use of trace distance as a ranking criterion in multi-copy settings. Trace distance remains the correct one-shot operational measure, but its induced ordering across different pairs need not extrapolate faithfully under repeated independent copies (Maziero, 2015).

6. Bounds, algorithms, and extremal regimes

Several developments concern the practical evaluation or extremization of pairwise trace distance, which is often the underlying quantity behind broader “mutual” constructions.

For low-rank states, a strong relation to Hilbert–Schmidt distance is available. Using the convention

$\rho$06

one always has the improved lower bound

$\rho$07

The main upper bound is rank-sensitive:

$\rho$08

A weaker but simpler bound is

$\rho$09

The pure-state identity

$\rho$10

shows the lower bound is tight. Additive linear-entropy bounds are also available,

$\rho$11

and

$\rho$12

with $\rho$13 (Coles et al., 2019).

For direct estimation on quantum hardware, a quantum algorithm based on density matrix exponentiation and improved quantum phase estimation has been proposed for arbitrary pure and mixed states. The construction embeds $\rho$14 into

$\rho$15

so that the positive magnitudes of the eigenvalues of $\rho$16 sum to the target trace distance. The method extracts three scalars, $\rho$17, $\rho$18, and $\rho$19, and reconstructs the estimate through

$\rho$20

The reported overall time complexity is $\rho$21 in the number of qubits $\rho$22. Proof-of-principle simulations and IBM hardware runs of the first-QPE stage were used to validate the approach, while also showing sensitivity to finite-precision effects, especially when the true trace distance is near zero (Ghosh et al., 7 Apr 2026).

An extremal continuous-variable version of the problem has also been solved. For single-mode bosonic Gaussian states with equal mean photon number $\rho$23, the maximally trace-distant pair consists of two pure, isocovariant, equally squeezed states with opposite displacements. Their minimal fidelity is

$\rho$24

so the maximal trace distance is

$\rho$25

The corresponding minimal Helstrom error probability satisfies

$\rho$26

with asymptotic scaling

$\rho$27

For $\rho$28 modes, under the isocovariant constraint, the exact result becomes

$\rho$29

with the optimal pair concentrating energy into a single mode (Volkoff, 2017).

Taken together, these results delimit the modern technical landscape around “mutual trace distance.” The phrase may designate total correlations in a bipartite state, block correlations in a DMRG superblock, or simply the trace distance between two states considered in a composite or comparative setting. What unifies these meanings is not a single universal definition, but a common reliance on the trace norm as a metric of distinguishability whose operational significance is clear, while its geometric and dynamical behavior can be highly context-dependent.