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SLOCC classification of n qubits invoking the proportional relationships for spectrums and for standard Jordan normal forms

Dafa Li

arXiv:1703.01598v1quant-ph

TL;DR

SLOCC classification is significant because states in the same class can perform the same quantum-information tasks, while n ≥ 4 qubits have infinitely many classes. The paper studies proportional relationships for spectra and SJNFs of matrices built from coefficient matrices of equivalent states, yielding 12 groups and 34 families.

  • Problem

    For n ≥ 4 qubits, infinitely many SLOCC classes make a finite partition by an invariant criterion desirable.

    Source: §I · p. 1

  • Method

    The paper studies proportional spectra and SJNFs of matrices constructed from coefficient matrices of two SLOCC-equivalent states.

    Source: §I · p. 1

  • Results

    The proportional relationships yield 12 groups and 34 families for pure states of n ≥ 4 qubits, respectively.

    Source: §I · p. 1, §V · p. 4

  • Takeaways & Limitations

    The authors suggest that classifying SJNFs under SLOCC may be useful for classifying linear differential systems.

    Source: §V · p. 4

Abstract

from arXiv · show

We investigate the proportional relationships for spectrums and for SJNFs (Standard Jordan Normal Forms) of the matrices constructed from coefficient matrices of two SLOCC (stochastic local operations and classical communication) equivalent states of $n$ qubits. Invoking the proportional relationships for spectrums and for SJNFs, pure states of $n$ ($\geq 4$) qubits are partitioned into 12 groups and 34 families under SLOCC, respectively. Specially, it is true for four qubits.

I. INTRODUCTION

SLOCC classification matters because states in the same class can perform the same quantum-information tasks, yet n ≥ 4 qubits have infinitely many SLOCC classes. This paper uses proportional spectra and SJNFs to partition those states into 12 groups and 34 families, respectively.

I. INTRODUCTION

  • Because states in the same SLOCC class can perform the same quantum-information tasks, classification distinguishes types of entanglement with shared task capabilities.For n ≥ 4 qubits, there are infinitely many SLOCC classes, motivating a finite family partition by an invariant criterion.Source: §I · p. 1
  • Verstraete et al. used general singular value decomposition to partition pure four-qubit states into nine SLOCC-inequivalent families.Subsequent work extensively studied four-qubit entanglement classification.Source: §I · p. 1
  • Recent work has sought SLOCC-invariant polynomials for classifying and measuring entanglement of n qubits.The passage gives examples of invariant-polynomial degrees for two, three, even n ≥ 4, and odd n ≥ 4 qubits.Source: §I · p. 1
  • SLOCC-invariant ranks of coefficient matrices have also been proposed for classification.Source: §I · p. 1
  • Proportional spectra and SJNFs partition pure states of n ≥ 4 qubits into 12 groups and 34 families, respectively.The authors also report new SLOCC classifications for four qubits.Source: §I · p. 1

A. The proportional relationships for spectrums and for SJNFs

For SLOCC-equivalent n-qubit states, the constructed matrices have spectra and standard Jordan normal forms related by a common nonzero scale. The section illustrates the relationship for four-qubit states and uses it to distinguish an inequivalent pair.

A. The proportional relationships for spectrums and for SJNFs

  • The coefficient matrix is formed from the state’s coefficients after choosing q1,…,qℓ as row bits, and the construction uses the Pauli operator σy and the transpose of C.For the spectrum relationship, q1 and q2 are chosen as the row bits; the resulting matrices are 4 by 4.Source: A. The proportional relationships for… · pp. 1–2
  • SLOCC-equivalent states have constructed matrices whose spectra scale by k, the product of the local operators’ determinants.The same factor multiplies the eigenvalues in the standard Jordan normal form while preserving the Jordan block sizes.Source: A. The proportional relationships for… · p. 2
  • For the four-qubit states ζ4 and ζ5 with a ≠ b, their reported SJNFs differ, so Theorem 1 implies they are SLOCC inequivalent.The forms are reported as J2(b)aa and J2(a)bb, respectively.Source: A. The proportional relationships for… · p. 2

B. Partition pure states of n (≥4) qubits into 12 groups and 34 families

Theorem 1 reduces SLOCC classification for n (≥4) qubits to classifying 4 by 4 complex matrices. The resulting spectrum types define 12 groups, while SJNFs define 34 families, with states in different groups or families SLOCC inequivalent.

B. Partition pure states of n (≥4) qubits into 12 groups and 34 families

A. SLOCC classification of four qubits

For four qubits, the paper applies the proportional relationships to partition states into 12 groups and 34 families. It says both classifications are complete and contrasts them with an earlier nine-family classification reported as incomplete.

A. SLOCC classification of four qubits

  • Four-qubit states are partitioned into 12 groups and 34 families, and both classifications are complete.The paper supplies a corresponding state for each spectrum, characteristic-polynomial, and SJNF type.Source: A. SLOCC classification of four qubits · p. 3
  • Verstraete et al. used a generalization of singular value decomposition with non-standard Jordan blocks to partition four-qubit states into nine families.The paper notes that the nine-family classification was incomplete after Lab3 was identified as SLOCC equivalent to a subfamily of Labc2 [28].Source: A. SLOCC classification of four qubits · p. 3

B. SLOCC classification of three qubits

For three qubits, the paper uses the spectrum and SJNFs of constructed matrices to distinguish the six SLOCC classes. The stated parameter λ2 is the 3-tangle.

B. SLOCC classification of three qubits

C. SLOCC classification of two qubits

For two qubits, the paper distinguishes two cases by whether a0a3 equals a1a2. Their matrix SJNFs support distinguishing two SLOCC classes.

C. SLOCC classification of two qubits

IV. SLOCC CLASSIFICATION OF n QUBITS UNDER Ai ∈SL(2, C)

Under determinant-one SLOCC operations, the matrices associated with equivalent n-qubit states have identical characteristic polynomials, spectra, and standard Jordan normal forms. Differences in any of these properties imply that the states are inequivalent under these operations.

IV. SLOCC CLASSIFICATION OF n QUBITS UNDER Ai ∈SL(2, C)

  • Equivalent states under determinant-one SLOCC operations have the same characteristic polynomial, spectrum, and standard Jordan normal form, up to block order.Source: §IV · p. 4
  • Different characteristic polynomials, spectra, or standard Jordan normal forms imply SLOCC inequivalence under determinant-one operations.The theorem gives this criterion in contrapositive form.Source: §IV · p. 4
  • A SLOCC-equivalence class may contain infinitely many equivalence classes under determinant-one SLOCC operations.Source: §IV · p. 4

V. CONCLUSION

The paper derives proportional spectrum and standard-Jordan-form relationships for SLOCC-equivalent states, then uses them to partition n-qubit states into groups and families. It also identifies determinant-one invariants and notes a possible use of the classification for linear differential systems.

V. CONCLUSION

  • For n ≥ 4 qubits, proportional spectra yield 12 SLOCC groups, while proportional standard Jordan normal forms yield 34 families.Source: §V · p. 4
  • For equivalent states under determinant-one SLOCC operations, the associated matrices have invariant spectra, characteristic polynomials, and standard Jordan normal forms.Source: §V · p. 4
  • The authors suggest that classifying standard Jordan normal forms under SLOCC may be useful for classifying linear differential systems.Source: §V · p. 4

VI. APPENDIX CORRESPONDING STATES OF FOUR QUBITS

The appendix lists representative four-qubit states by parameter conditions across the Gabcd, La2b2, and Labc2 families. These listings distinguish states according to parameter equalities, inequalities, and zero values.

VI. APPENDIX CORRESPONDING STATES OF FOUR QUBITS

  • The Gabcd listing gives eight states, with cases distinguished by which parameters are zero, equal, or distinct.The supplied entries include G1 through G8 and specify conditions for G2 through G8.Source: §VI · pp. 4–5
  • The Labc2 listing gives eleven states, including cases defined by parameter equalities, zero values, and distinctness.The supplied entries specify ζ1–ζ4 and ζ6–ζ12.Source: §VI · p. 5
  • The La2b2 listing gives two states, η1 and η2, under the stated parameter conditions.Source: §VI · p. 5
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