Correlated coherence need not be monogamous for pure three-qubit states

Abstract

A pure state of three qubits has a negative l1 correlated-coherence monogamy difference.

Work in the computational product basis. Fin(n) consists of the indices 0 through n − 1, and a basis index of three qubits is a function from Fin(3) to Fin(2). Qubits 0, 1 and 2 are A, B and C, respectively. Complex absolute values are moduli. The anonymous brackets Fintype n and DecidableEq n supply a finite enumeration and decidable equality for the matrix indices.

Definition 1.1 (Coherence in a fixed basis).

$$\forall (n:\mathrm{Type}),([\mathrm{Fintype}\left(n\right)],([\mathrm{DecidableEq}\left(n\right)],(\forall (\rho :\mathrm{Matrix}\left(n,n,\mathbb{C}\right)),(l1Coherence\left(\rho \right)=\sum _{(i:n)}(\sum _{(j:n),i\ne j}(\left|\rho \left(i,j\right)\right|))))))$$

Formalization. D5/S3/Quantum/Information/CorrelatedCoherenceMonogamyRefutation.l1Coherence (✓ std3).

Citation. Marcos L. W. Basso; Jonas Maziero (2020). Monogamy and trade-off relations for correlated quantum coherence. DOI: 10.1088/1402-4896/abafcb. URL: https://arxiv.org/abs/2006.14037v2.

Commentary.

The l1 coherence is the sum of the moduli of all off-diagonal matrix entries. Each ordered pair of distinct indices is counted once. The expression is defined for every complex square matrix, without positivity or trace assumptions, and is the l1-norm coherence used in Section II.

Definition 1.2 (The seven-term monogamy difference).

$$\forall (\psi :((\mathrm{Fin}\left(3\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),(\mathrm{monogamyGap}\left(\psi \right)=l1Coherence\left(\left({\mathrm{Tr}}_{\mathrm{Fin}\left(3\right)\setminus \left\{0,1,2\right\}}\right)\left((\psi {\psi }^{\ast })\right)\right)+l1Coherence\left(\left({\mathrm{Tr}}_{\mathrm{Fin}\left(3\right)\setminus \left\{0\right\}}\right)\left((\psi {\psi }^{\ast })\right)\right)+l1Coherence\left(\left({\mathrm{Tr}}_{\mathrm{Fin}\left(3\right)\setminus \left\{1\right\}}\right)\left((\psi {\psi }^{\ast })\right)\right)+l1Coherence\left(\left({\mathrm{Tr}}_{\mathrm{Fin}\left(3\right)\setminus \left\{2\right\}}\right)\left((\psi {\psi }^{\ast })\right)\right)-(l1Coherence\left(\left({\mathrm{Tr}}_{\mathrm{Fin}\left(3\right)\setminus \left\{0,1\right\}}\right)\left((\psi {\psi }^{\ast })\right)\right)+l1Coherence\left(\left({\mathrm{Tr}}_{\mathrm{Fin}\left(3\right)\setminus \left\{0,2\right\}}\right)\left((\psi {\psi }^{\ast })\right)\right)+l1Coherence\left(\left({\mathrm{Tr}}_{\mathrm{Fin}\left(3\right)\setminus \left\{1,2\right\}}\right)\left((\psi {\psi }^{\ast })\right)\right)))$$

Formalization. D5/S3/Quantum/Information/CorrelatedCoherenceMonogamyRefutation.monogamyGap (✓ std3).

Citation. Marcos L. W. Basso; Jonas Maziero (2020). Monogamy and trade-off relations for correlated quantum coherence. DOI: 10.1088/1402-4896/abafcb. URL: https://arxiv.org/abs/2006.14037v2.

Commentary.

For a vector ψ and a set S of qubits, the partial trace over Fin(3) outside S of ψ ψ* is its reduced matrix on S. The star on the column vector denotes conjugate transpose; equivalently, the pure matrix has entry ψ(x) times the complex conjugate of ψ(y). The retained basis is the set of functions from S to Fin(2). A trace over the empty set retains the full pure matrix. Write M for monogamyGap. Its four positive terms are the full coherence and the three single-qubit coherences; its three subtracted terms are the two-qubit coherences. This is Eq. (33), the difference between the correlated coherence across A versus BC and the sum of those across AB and AC. The formula itself does not require ψ to have norm one.

Definition 1.3 (The conjecture for three qubits).

$$\mathrm{claim}\iff (\forall (\psi :((\mathrm{Fin}\left(3\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),((\sum _{(x:(\mathrm{Fin}\left(3\right)\to \mathrm{Fin}\left(2\right)))}({\left|\psi \left(x\right)\right|}^2)=1)\Rightarrow (0\le \mathrm{monogamyGap}\left(\psi \right))))$$

Formalization. D5/S3/Quantum/Information/CorrelatedCoherenceMonogamyRefutation.claim (✓ std3).

Citation. Marcos L. W. Basso; Jonas Maziero (2020). Monogamy and trade-off relations for correlated quantum coherence. DOI: 10.1088/1402-4896/abafcb. URL: https://arxiv.org/abs/2006.14037v2.

Commentary.

Section III.A conjectures monogamy for every tripartite pure state. Here it is restricted to three qubits: every complex vector whose squared moduli sum to one has a nonnegative monogamy difference. All reduced matrices use the same computational product basis. The sufficient conditions (22)–(24) are not assumed in this conjecture.

Theorem 1.4 (A pure three-qubit state with a negative difference).

$$\neg (\mathrm{claim})$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/CorrelatedCoherenceMonogamyRefutation.result (✓ std3). ∎

Resolves. Problems/basso-maziero-2020-correlated-coherence-monogamy-refutation (refuted) by D5/S3/Quantum/Information/CorrelatedCoherenceMonogamyRefutation.result.

Source. Repository-derived.

Acknowledgement. Marcos L. W. Basso; Jonas Maziero (2020). Monogamy and trade-off relations for correlated quantum coherence. DOI: 10.1088/1402-4896/abafcb. URL: https://arxiv.org/abs/2006.14037v2.

Commentary.

Take ψ = (−16, 1, 128, −16, −128, −16, 1, −128)/√49922, in the order |000⟩, |001⟩, |010⟩, |011⟩, |100⟩, |101⟩, |110⟩, |111⟩. The first bit is qubit 0. The sum of the squares of the eight integer amplitudes is 49922, so ψ is a unit vector.

For the full state and the marginals A, B, C, AB, AC and BC, respectively, the l1 coherences are 138434/49922, 8416/49922, 288/49922, 288/49922, 48928/49922, 48928/49922 and 50336/49922. Each reduced entry is the sum, over a common configuration of the discarded qubits, of the two corresponding integer amplitudes multiplied together and divided by 49922. Summing the moduli of its off-diagonal entries gives these seven values.

The alternating sum is M(ψ) = −766/49922 = −383/24961, which is negative. Thus a pure three-qubit state contradicts claim and hence the conjecture for all tripartite pure states. This does not contradict Theorem 4: its trade-off hypothesis is not assumed by the conjecture.

References

• Truth anchor: D5/S3/Quantum/Information/CorrelatedCoherenceMonogamyRefutation.claim
• Truth anchor: D5/S3/Quantum/Information/CorrelatedCoherenceMonogamyRefutation.l1Coherence
• Truth anchor: D5/S3/Quantum/Information/CorrelatedCoherenceMonogamyRefutation.monogamyGap
• Truth anchor: D5/S3/Quantum/Information/CorrelatedCoherenceMonogamyRefutation.result
• Dependency: D5/S3/Quantum/Entanglement/PurityTimeReversalOverlapMinimum