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arXiv:0904.3551 · Physical Review A

Compact set of invariants characterizing graph states of up to eight qubits

Adán Cabello, Antonio J. López-Tarrida, Pilar Moreno, José R. Portillo

Published Jul 1, 2009• 13 citations
arXivHTMLPDF

Abstract

The set of entanglement measures proposed by Hein, Eisert, and Briegel for $n$-qubit graph states [Phys. Rev. A 69, 062311 (2004)] fails to distinguish between inequivalent classes under local Clifford operations if $n\ensuremath{\ge}7$. On the other hand, the set of invariants proposed by van den Nest, Dehaene, and De Moor (VDD) [Phys. Rev. A 72, 014307 (2005)] distinguishes between inequivalent classes, but contains too many invariants (more than $2\ifmmode\times\else\texttimes\fi{}{10}^{36}$ for $n=7$) to be practical. Here we solve the problem of deciding which entanglement class a graph state of $n\ensuremath{\le}8$ qubits belongs to by calculating some of the state's intrinsic properties. We show that four invariants related to those proposed by VDD are enough for distinguishing between all inequivalent classes with $n\ensuremath{\le}8$ qubits.

DOI
10.1103/physreva.80.012102

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The set of entanglement measures proposed by Hein, Eisert, and Briegel for nnn-qubit graph states [Phys. Rev. A 69, 062311 (2004)] fails to distinguish between inequivalent classes under local Clifford operations if n\ensuremath≥7n\ensuremath{\ge}7n\ensuremath≥7. On the other hand, the set of invariants proposed by van den Nest, Dehaene, and De Moor (VDD) [Phys. Rev. A 72, 014307 (2005)] distinguishes between inequivalent classes, but contains too many invariants (more than 2\ifmmode×\else\texttimes\fi10362\ifmmode\times\else\texttimes\fi{}{10}^{36}2\ifmmode×\else\texttimes\fi1036 for n=7n=7n=7) to be practical. Here we solve the problem of deciding which entanglement class a graph state of n\ensuremath≤8n\ensuremath{\le}8n\ensuremath≤8 qubits belongs to by calculating some of the state's intrinsic properties. We show that four invariants related to those proposed by VDD are enough for distinguishing between all inequivalent classes with n\ensuremath≤8n\ensuremath{\le}8n\ensuremath≤8 qubits.
Graph states are a class of multipartite entangled quantum states that are ubiquitous in quantum information. We study equivalence relations between graph states under local unitaries (LU) to obtain distinguishing methods both in local and in networked settings. Based on the marginal structure of graph states, we introduce a family of easy-to-compute LU invariants. We show that these invariants uniquely identify the entanglement classes of every graph state up to eight qubits and discuss their reliability for larger numbers of qubits. To handle larger graphs, we generalize tools to test for local Clifford (LC) equivalence of graph states that work by condensing large graphs into smaller graphs. In turn, we show that statements on the equivalence of these smaller graphs (which are easier to compute) can be used to infer statements on the equivalence of the original, larger graphs. We analyze LU equivalence in two key settings, with and without allowing for the permutation of qubits. We identify entanglement classes, whose marginal structure does not allow us to distinguish them. As a result, we increase the bound on the number of qubits where the LU-LC conjecture holds from 8 to 10 qubits in the setting where qubit permutations are allowed.
Graph states are a class of multipartite entangled quantum states that are ubiquitous in quantum information. We study equivalence relations between graph states under local unitaries (LU) to obtain distinguishing methods both in local and in networked settings. Based on the marginal structure of graph states, we introduce a family of easy-to-compute LU invariants. We show that these invariants uniquely identify the entanglement classes of every graph state up to eight qubits and discuss their reliability for larger numbers of qubits. To handle larger graphs, we generalize tools to test for local Clifford (LC) equivalence of graph states that work by condensing large graphs into smaller graphs. In turn, we show that statements on the equivalence of these smaller graphs (which are easier to compute) can be used to infer statements on the equivalence of the original, larger graphs. We analyze LU equivalence in two key settings, with and without allowing for the permutation of qubits. We identify entanglement classes, whose marginal structure does not allow us to distinguish them. As a result, we increase the bound on the number of qubits where the LU-LC conjecture holds from 8 to 10 qubits in the setting where qubit permutations are allowed.
The entanglement of graph states up to eight qubits is calculated in the regime of iteration calculation. The entanglement measures could be the relative entropy of entanglement, the logarithmic robustness or the geometric measure. All 146 local inequivalent graphs are classified as two categories: graphs with identical upper LOCC entanglement bound and lower bipartite entanglement bound, graphs with unequal bounds. The late may displays non-integer entanglement. The precision of iteration calculation of the entanglement is less than 10−1410^{-14}10−14.
We show how to prepare any graph state of up to 12 qubits with (a) the minimum number of controlled-ZZZ gates and (b) the minimum preparation depth. We assume only one-qubit and controlled-ZZZ gates. The method exploits the fact that any graph state belongs to an equivalence class under local Clifford operations. We extend up to 12 qubits the classification of graph states according to their entanglement properties, and identify each class using only a reduced set of invariants. For any state, we provide a circuit with both properties (a) and (b), if it does exist, or, if it does not, one circuit with property (a) and one with property (b), including the explicit one-qubit gates needed.
Local-unitary (LU) equivalence asks whether two quantum states differ only by independent changes of basis on their qubits. For graph states, whether this relation can be decided in polynomial time has remained open for over a decade. We give a deterministic algorithm that decides LU equivalence for graphs on nnn labelled vertices in O~(n6.38)\widetilde O(n^{6.38})O(n6.38) bit operations and constructs exact single-qubit unitaries whenever the states are equivalent. Building on Claudet and Perdrix's quasipolynomial algorithm, we replace the enumeration of vertex subsets by a compact system of constraints generated from pairs and triples. The remaining graph transformation is found by solving linear equations over the binary field. These new steps cost O~(n5)\widetilde O(n^5)O(n5) bit operations; the inherited graph preprocessing sets the overall bound. We also count the local-Clifford (LC) classes of graph states within any LU class: their number is a power of two, computable within the same bound. For any given graph state, this decides whether single-qubit Clifford gates reach every graph state in its LU class, and supplies a counterexample when they do not. The method also decides LU equivalence of stabilizer codes encoding one logical qubit.