Partial transpose criteria for symmetric states

Bohnet-Waldraff, Fabian; Braun, Daniel; Giraud, Olivier

doi:10.1103/physreva.94.042343

Cited by 18 publications

(17 citation statements)

References 37 publications

(69 reference statements)

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“…Recent works like [68] consider the Schmidt decomposition of various bipartitions of Dicke states but do not deal with typicality, randomness and quantum chaos in this context. Also previous works such as [69][70][71][72][73][74] have studied the reduced density matrices of permutationally invariant systems and their entanglement. Our approach here is to focus on generic pure permutation symmetric states with a view of defining an ensemble of them that would be useful in studies of quantum chaos as for example in the kicked top which we subsequently analyze in detail.…”

Section: Permutation Symmetric Statesmentioning

confidence: 99%

Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos

2018

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Many-body states that are invariant under particle relabelling, the permutation symmetric states, occur naturally when the system dynamics is described by symmetric processes or collective spin operators. We derive expressions for the reduced density matrix for arbitrary subsystem decomposition for these states and study properties of permutation symmetric states and their subsystems when the joint system is picked randomly and uniformly. Thus defining a new random matrix ensemble, we find the average linear entropy and von Neumann entropy which implies that random permutation symmetric states are marginally entangled and as a consequence the tripartite mutual information (TMI) is typically positive, preventing information from being shared globally. Applying these results to the quantum kicked top viewed as a multi-qubit system we find that entanglement, mutual information and TMI all increase for large subsystems across the Ehrenfest or log-time and saturate at the random state values if there is global chaos. During this time the out-of-time order correlators (OTOC) evolve exponentially implying scrambling in phase space. We discuss how positive TMI may coexist with such scrambling.

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Section: Permutation Symmetric Statesmentioning

confidence: 99%

Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos

2018

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“…The question arises whether similarly efficient measurements can be found for half-integer spin j. It was recently shown in [41] how the positive-partial-transpose (PPT) separability criterion for symmetric states of multi-qubit systems can be formulated in terms of matrix inequalities based on the tensor representation in Eq. (2).…”

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confidence: 99%

Optimal measurement strategies for fast entanglement detection

Milazzo¹,

Braun²,

Giraud³

2019

Phys. Rev. A

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With the advance of quantum information technology, the question of how to most efficiently test quantum circuits is becoming of increasing relevance. Here we introduce the statistics of lengths of measurement sequences that allows one to certify entanglement across a given bi-partition of a multiqubit system over the possible sequence of measurements of random unknown states, and identify the best measurement strategies in the sense of the (on average) shortest measurement sequence of (multi-qubit) Pauli-measurements. The approach is based on the algorithm of truncated moment sequences that allows one to deal naturally with incomplete information, i.e. information that does not fully specify the quantum state. We find that the set of measurements corresponding to diagonal matrix elements of the moment matrix of the state are particularly efficient. For symmetric states their number grows only like the third power of the number N of qubits. Their efficiency grows rapidly with N , leaving already for N = 4 less than a fraction 10 −6 of randomly chosen entangled states undetected.

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“…It is in fact possible to obtain many more sufficient entanglement criteria from the PPT criteria applied to ρ or to its k-qubit reduced density matrices. As shown in [11], the partial transpose matrices, and their positivity, can be expressed in terms of the x µ1...µN in a simple way. As we saw above, the partial transpose ρ PT 2 can be related with the 4 × 4 matrix (x µ1µ20 ) 0 µ1,µ2 3 and positivity of ρ PT 2 is equivalent to the right-hand side of (26).…”

Section: Other Sufficient Entanglement Criteriamentioning

confidence: 99%

Genuinely entangled symmetric states with no N -partite correlations

2017

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We investigate genuinely entangled N -qubit states with no N -partite correlations in the case of symmetric states. Using a tensor representation for mixed symmetric states, we obtain a simple characterization of the absence of N -partite correlations. We show that symmetric states with no N -partite correlations cannot exist for an even number of qubits. We fully identify the set of genuinely entangled symmetric states with no N -partite correlations in the case of three qubits, and in the case of rank-2 states. We present a general procedure to construct families for an arbitrary odd number of qubits.

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Partial transpose criteria for symmetric states

Cited by 18 publications

References 37 publications

Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos

Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos

Optimal measurement strategies for fast entanglement detection

Genuinely entangled symmetric states with no N -partite correlations

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