Symmetric extendibility for qudits and tolerable error rates in quantum cryptography

We analyze the connection between Bell inequality violations and symmetric extendibility of quantum states. We prove that 2-qubit reduced states of multiqubit symmetric pure states do not violate the Bell Clauser-Horne-Shimony-Holt (CHSH) inequality. We then prove the more general converse that any 2-qubit state that violates the CHSH inequality cannot have a symmetric extension. We extend our analysis to qudits and provide a test for symmetric extendibility of 2-qudit states. We show that if a 2-qudit Bell inequality is monogamous, then any 2-qudit state that violates this inequality does not have a symmetric extension. For the specific case of 2-qutrit states, we use numerical evidence to conjecture that the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality is monogamous. Hence, the violation of the CGLMP inequality by any 2-qutrit state could be a sufficient condition for the non-existence of its symmetric extension.

show abstract

“…in contradiction with Eq. (8). Consequently no twoqubit entangled states satisfy the symmetric extendibility criterion.…”

Section: No Symmetric Extension Of 2-qudit Nonlocal Statesmentioning

confidence: 99%

“…We also see this using the criteria in Eq. (8). tr(ρ 2 AB ) = 1 for pure states and detρ AB = 0 (since pure states have rank 1).…”

Section: No Symmetric Extension Of 2-qudit Nonlocal Statesmentioning

confidence: 99%

Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities

2017

View full text Add to dashboard Cite

show abstract

“…Unfortunately, for any higher dimensions a full characterization involving only spectra is highly unlikely [22]. There have been some efforts made for special cases but no general results found [41][42][43]. Nevertheless, our physical picture based on the convexity of B and the symmetry of the system may shed light on the understanding of symmetric extendibility for higher dimensional systems.…”

Section: Theoremmentioning

confidence: 99%

Symmetric extension of two-qubit states

Chen

Kribs

et al. 2014

Phys. Rev. A

View full text Add to dashboard Cite

A bipartite state ρAB is symmetric extendible if there exists a tripartite state ρ ABB ′ whose AB and AB ′ marginal states are both identical to ρAB. Symmetric extendibility of bipartite states is of vital importance in quantum information because of its central role in separability tests, one-way distillation of EPR pairs, one-way distillation of secure keys, quantum marginal problems, and anti-degradable quantum channels. We establish a simple analytic characterization for symmetric extendibility of any two-qubit quantum state ρAB; specifically, tr(ρ 2 B ) ≥ tr(ρ 2 AB ) − 4 √ det ρAB. Given the intimate relationship between the symmetric extension problem and the quantum marginal problem, our result also provides the first analytic necessary and sufficient condition for the quantum marginal problem with overlapping marginals.PACS numbers: 03.65. Ud, 03.67.Dd, 03.67.Mn The notion of symmetric extendibility for a bipartite quantum state ρ AB was introduced in [1] as a test for entanglement. A bipartite density operator ρ AB is symmetric extendible if there exists a tripartite state ρ ABB ′ such that tr B ′ (ρ ABB ′ ) = tr B (ρ ABB ′ ). A state ρ AB without symmetric extension is evidently entangled, and to decide such an extendibility for ρ AB can be formulated in terms of semi-definite programming (SDP) [2]. This then leads to effective numerical tests and bounds [3][4][5][6] that allow for entanglement detection for some well-known positive-partial-transpose (PPT) states [7][8][9][10][11].States with symmetric extension also have a clear operational meaning for quantum information processing [12]. One simple idea is that if a bipartite state ρ AB is symmetric extendible, then one cannot distill any entanglement from ρ AB by protocols only involving local operations and oneway classical communication (from A to B) [13], because of entanglement monogamy [14]. Furthermore, using the ChoiJamiolkowski isomorphism, symmetric extendibility of bipartite states also provides a test for anti-degradable quantum channels [15], and one-way quantum capacity of quantum channels [13].A similar idea applies to the protocols for quantum key distribution (QKD), which aim to establish a shared secret key between two parties (for a review, see [16]). The corresponding QKD protocols can be viewed as having two phases: in a first phase, the two parties establish joint classical correlations by performing measurements on an untrusted bipartite quantum state, while in a second phase a secret key is being distilled from these correlations by a public discussion protocol (via authenticated classical channels) which typically involves classical error correction and privacy amplification [17][18][19][20]. If the underlying bipartite state ρ AB is symmetric extendible, then no secret key can be distilled by a process involving only one-way communication. Therefore, the foremost task of the public discussion protocol is to break this symmetric extendibility by some bi-directional post-selection process. Failure to find such a protocol me...

show abstract

“…Several quantum cryptographic protocols have been developed in order to capitalize on the usefulness of these states, especially their capability to increase security against eavesdropping attacks and the noise threshold that quantum key distribution protocols can tolerate [1,[14][15][16][17][18][19][20][21][22]. Furthermore, qudits also offer advantages for more efficient quantum gates [23] and for quantum information protocols [24,25].…”

Section: Introductionmentioning

confidence: 99%

Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding

et al. 2010

View full text Add to dashboard Cite

High-dimensional quantum states, or qudits, represent a promising resource in the quantum information field. Here we present the experimental generation of four-dimensional quantum states, or ququarts, encoded in the polarization and orbital angular momentum of a single photon. Our technique, based on the q-plate device, allows the ququart to be prepared and measured in all five mutually unbiased bases. We report the reconstruction of the four-dimensional density matrix through the tomographic procedure for different ququart states

show abstract

Symmetric extendibility for qudits and tolerable error rates in quantum cryptography

Cited by 6 publications

References 18 publications

Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities

Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities

Symmetric extension of two-qubit states

Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding

Contact Info

Product

Resources

About