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Toward computability of trace distance discord
F. Ciccarello, T. Tufarelli, V. Giovannetti
TL;DR
Two-qubit trace distance discord was previously analytically computable only for states whose measured marginal is maximally mixed. This paper reduces the general optimization to an explicit two-variable function and derives closed forms for important classes, including arbitrary quantum-classical and X states.
Problem
Analytical trace distance discord was previously available only for Bell-diagonal states and more generally states appearing maximally mixed to the measured party.
Method
The paper transforms the two-qubit optimization into minimization of an explicit function of two variables using Bloch vectors and correlation-matrix singular values.
Results
The minimum is obtained in closed form for relevant classes encompassing arbitrary quantum-classical and X states, extending the prior Bell-diagonal result.
Takeaways & Limitations
The framework combines trace distance discord's mathematical properties with explicit computation for important density-matrix classes and may support further analytic extensions.
Takeaways & Limitations
In the spin-chain application, the comparable analytical quantum-discord expression is possible but yields lengthy and uninformative formulas.
Abstract
from arXiv · showhide
It is known that a reliable geometric quantifier of discord-like correlations can be built by employing the so-called trace distance. This is used to measure how far the state under investigation is from the closest "classical-quantum" one. To date, the explicit calculation of this indicator for two qubits was accomplished only for states such that the reduced density matrix of the measured party is maximally mixed, a class that includes Bell-diagonal states. Here, we first reduce the required optimization for a general two-qubit state to the minimization of an explicit two-variable function. Using this framework, we show next that the minimum can be analytically worked out in a number of relevant cases including quantum-classical and X states. This provides an explicit and compact expression for the trace distance discord of an arbitrary state belonging to either of these important classes of density matrices.
1. Introduction
The paper addresses the difficulty of computing reliable discord-like correlation measures for two qubits. It introduces a framework for making trace distance discord computable beyond the previously solved maximally mixed marginal states.
- Quantum correlations are not fully captured by entanglement, motivating discord-based measures for composite quantum states.
- No general closed formula for quantum discord is known even for two qubits, illustrating the broader computability challenge.
- Trace distance discord had previously been solved analytically only for Bell-diagonal states and states appearing maximally mixed to the measured party.
- The paper reduces general two-qubit trace distance discord to an explicit two-variable minimization and analytically solves relevant cases including quantum-classical and X states.
- X-state quantum-correlation calculations are especially difficult because the class has five independent parameters and existing quantum-discord algorithms can fail for some X states.
2. One-sided TDD for two-qubit states: general case
For two-qubit states, the paper formulates one-sided trace distance discord through trace-norm optimization over projective measurements. By exploiting the correlation matrix structure and local-unitary invariance, it reduces the problem to minimizing an explicit function of two measurement angles.
- One-sided trace distance discord is the minimum trace-norm distance between a bipartite state and classical-quantum states with zero discord relative to measurements on A.
- For a qubit subsystem A, the optimization is over projective measurements associated with completely depolarizing channels on A.
- The two-qubit state is parametrized by the local Bloch vectors and a real 3×3 correlation matrix, while the measurement is represented by a unit Bloch-sphere vector.
- A singular-value decomposition of the correlation matrix and a local unitary rotation on B simplify the trace-norm calculation without changing the discord.
- The arbitrary-state calculation becomes minimization of an explicit function h over the two angles θ and φ defining the measurement direction.
3. Bell diagonal states and states with homogeneous singular values
The paper identifies symmetry classes that simplify the general optimization, including states with a maximally mixed measured subsystem and states with equal singular-value moduli. It recovers known Bell-diagonal results and gives an explicit value for a representative homogeneous-spectrum family.
- The optimization simplifies for states possessing specific symmetries in their correlation matrix and local Bloch vectors.
- Class B includes states with arbitrary local Bloch vector x_A and correlation singular values having equal moduli.
- Class A includes Bell-diagonal states and is characterized by a maximally mixed reduced state for subsystem A.
- For class A, the minimization is attained when the measurement direction points along the principal correlation-frame vector associated with the largest ordered singular value.
- For mixtures pϱ_A ⊗ I_B/2 + (1 − p)|Ψ−⟩⟨Ψ−|, the trace distance discord is (1 − p)/2.
4. Correlation matrix with a single non-zero singular eigenvalue
For states with a correlation matrix having one non-zero singular value, the minimization reduces analytically to a closed formula. This framework yields the trace distance discord for general quantum-classical states and characterizes its maximum.
- The class is defined by γ2 = γ3 = 0, while γ1 = γ and the Bloch vector x_A remains arbitrary subject to physicality.
- The minimization is restricted to the plane spanned by the singular-vector direction w_1 and x_A, because the relevant quantities decrease as the projected component of e grows.
- The resulting trace distance discord is obtained by minimizing an explicit min-max function over the azimuthal angle φ, with the minimum occurring at the crossing points of its two periodic terms.
- Quantum-classical states: Quantum-classical states fit this single-singular-value case, allowing an analytical closed formula for the trace distance discord of the most general state in that class.
- Quantum-classical states: The quantum-classical formula depends on the local Bloch-vector lengths s_0 and s_1, their angle φ, and weights p and 1 − p; its maximum is 1/4 at s_0 = s_1 = 1, p = 1/2, and φ = π/2.
5. X states
For arbitrary two-qubit X states, the optimization is reduced to boundary analysis and yields a compact closed-form TDD expression. The result depends on three state parameters and includes Bell-diagonal states as a special case.
- State structure: An X state has an X-shaped matrix form with four relevant parameters: xA3 and the three correlation coefficients γk.The correlation matrix is diagonal, and |γ1|≥|γ2| always holds.
- Results: For X states, the discord depends only on three parameters, while Bell-diagonal states reduce to half the intermediate value among {|γk|}.The Bell-diagonal result is recovered as a special case of the X-state expression.
- Minimization framework: The angular minimization depends on µ=sin^2θ and ν=sin^2φ, which range over the square S defined by 0≤µ,ν≤1.The objective is h=a+√(a^2−b).
- Minimization framework: The function h has no minimum in the interior of S, so the global minimum is found by analyzing its boundary edges, including singular points.The square-root term makes h non-differentiable where a^2=b, requiring separate treatment.
- Global minimum: The global minimum is achieved on the edge ν=0, leading to the closed-form TDD expression for an arbitrary two-qubit X state.Different parameter regimes determine which boundary expression supplies the minimum.
- Special case: When the relevant correlation matrix has only one nonzero singular value, the TDD is D(→)(ρAB)=|γ1|/2.This equals half the absolute value of the nonzero off-diagonal entry in the corresponding case.
6. Application: propagation of QCs across a spin chain
The paper applies its X-state formula to quantum-correlation propagation through a spin chain. Because the evolving two-qubit state is determined by the transfer amplitude, TDD yields a compact analytic description of its dynamics.
- Physical setting: The application studies bipartite quantum correlations between an external qubit and sites of an XX spin chain as they evolve in time.The end-to-end case corresponds to propagation across the chain.
- Motivation: The quantum-discord formulas are analytically available but lengthy and uninformative, whereas TDD produces simple and informative formulas for the same dynamics.This motivates applying the X-state result to the propagation problem.
- State parameterization: For the relevant two-qubit states, the single-excitation transition amplitude f(t) fully specifies the output state and therefore any corresponding quantum-correlation measure.The dynamics are consequently parameterized by |f(t)|.
- Correlation dynamics: For N=3, quantum discord is non-monotonic in |f|, vanishing at |f|=0 and 1 and reaching one maximum at an intermediate value.The TDD calculation uses γ1=|f(t)|, γ2=γ3=0, and xA3=1−|f(t)|^2.
- Correlation dynamics: The TDD has the same qualitative non-monotonic time behavior as quantum discord, while its dependence on |f| is analytically straightforward.The resulting function is plotted against both time and the transfer amplitude.
- Quantitative result: The TDD reaches a maximum of approximately 0.22 at |f|M≈0.6 and vanishes at |f|=0 and 1.The stationary-point equation has only one root in the interval [0,1].
7. Conclusions
The paper reduces two-qubit TDD computation to an explicit two-variable minimization and obtains closed-form results for important state classes, including arbitrary quantum-classical and X states. It combines TDD's mathematical properties with explicit computation and suggests broader analytical applicability.
- The authors reduce two-qubit TDD calculation to finding the minimum of an explicit two-variable function.
- Closed-form minima are obtained for relevant classes encompassing arbitrary quantum-classical and X states.
- X states include Bell-diagonal states, which were previously the only states with an analytical TDD expression.
- The framework combines TDD's desirable mathematical properties with explicit computation for quantum-classical and X density matrices.
- The developed framework may be further exploited to enlarge the class of quantum states admitting analytical TDD expressions.
Appendix A. Derivation of Eq. (48)
The appendix rewrites f±(φ, α), a linear combination of cosine and sine, as a single cosine and uses the resulting factors to recover Eq. (48).
- f±(φ, α) is rewritten as A± cos(φ−δ±) from its linear combination of cos φ and sin φ.
- The factor A± is obtained using ˜xA1 = xA cos α.
- Substituting the appendix expression into Eq. (47) yields Eq. (48) of the main text.
- Eliminating sin δ± and cos δ± through Eqs. (A.1) and (A.2) completes the derivation of Eq. (48).
Appendix B. Eq. (65) for Bell diagonal states
For Bell-diagonal states, the TDD expression reduces to half the intermediate singular-value magnitude, matching earlier results.
- Bell-diagonal states are mixtures of the four Bell states and have maximally mixed reduced states, with xA = xB = 0.
- Bell-diagonal states are X states because their density matrices are linear combinations of operators with X-form representations.
- For the stated Bell-diagonal case, the square root in Eq. (65) becomes |γ2| and the TDD reduces accordingly.
- TDD equals half the intermediate value among {|γ1|, |γ2|, |γ3|}, agreeing with earlier references and Section 3.1.