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arXiv:quant-ph/0505036 · New Journal of Physics

Entanglement on mixed stabilizer states: normal forms and reduction procedures

Koenraad M. R. Audenaert, Martin Bodo Plenio

Published Aug 11, 2005• 89 citations

Abstract

The stabilizer formalism allows the efficient description of a sizeable class of pure as well as mixed quantum states of n -qubit systems. That same formalism has important applications in the field of quantum error correcting codes, where mixed stabilizer states correspond to projectors on subspaces associated with stabilizer codes. In this paper, we derive efficient reduction procedures to obtain various useful normal forms for stabilizer states. We explicitly prove that these procedures will always converge to the correct result and that these procedures are efficient in that they only require a polynomial number of operations on the generators of the stabilizers. On one hand, we obtain two single-party normal forms. The first, the row-reduced echelon form, is obtained using only permutations and multiplications of generators. This form is useful to calculate partial traces of stabilizer states. The second is the fully reduced form, where the reduction procedure invokes single-qubit operations and CNOT operations as well. This normal form allows for the efficient calculation of the overlap between two stabilizer states, as well as of the Uhlmann fidelity between them, and their Bures distance. On the other hand, we also find a reduction procedure of bipartite stabilizer states, where the operations involved are restricted to be local ones. The two-party normal form thus obtained lays bare a very simple bipartite entanglement structure of stabilizer states. To wit, we prove that every bipartite mixed stabilizer state is locally equivalent to a direct product of a number of maximally entangled states and, potentially, a separable state. As a consequence, using this normal form we can efficiently calculate every reasonable bipartite entanglement measure of mixed stabilizer states.

DOI: 10.1088/1367-2630/7/1/170

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The stabilizer formalism allows the efficient description of a sizeable class of pure as well as mixed quantum states of n -qubit systems. That same formalism has important applications in the field of quantum error correcting codes, where mixed stabilizer states correspond to projectors on subspaces associated with stabilizer codes. In this paper, we derive efficient reduction procedures to obtain various useful normal forms for stabilizer states. We explicitly prove that these procedures will always converge to the correct result and that these procedures are efficient in that they only require a polynomial number of operations on the generators of the stabilizers. On one hand, we obtain two single-party normal forms. The first, the row-reduced echelon form, is obtained using only permutations and multiplications of generators. This form is useful to calculate partial traces of stabilizer states. The second is the fully reduced form, where the reduction procedure invokes single-qubit operations and CNOT operations as well. This normal form allows for the efficient calculation of the overlap between two stabilizer states, as well as of the Uhlmann fidelity between them, and their Bures distance. On the other hand, we also find a reduction procedure of bipartite stabilizer states, where the operations involved are restricted to be local ones. The two-party normal form thus obtained lays bare a very simple bipartite entanglement structure of stabilizer states. To wit, we prove that every bipartite mixed stabilizer state is locally equivalent to a direct product of a number of maximally entangled states and, potentially, a separable state. As a consequence, using this normal form we can efficiently calculate every reasonable bipartite entanglement measure of mixed stabilizer states.

The determination of many special types of quantum states has been studied thoroughly, such as the generalized $|\text{GHZ}\ensuremath{\rangle}$ states, stochastic local operations and classical communication equivalent $|W\ensuremath{\rangle}$ states, and Dicke states. In this paper, we study another special entanglement state, that is, stabilizer states, which play an important role in quantum error correcting codes, multipartite purification, and so on. We show that all

n

-qubit stabilizer states are uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices for systems which are the supports of

n

independent generators of the corresponding stabilizer formalisms.

Genuinely entangled subspaces are a class of subspaces in the multipartite Hilbert spaces that are composed of only genuinely entangled states. They are thus an interesting object of study in the context of multipartite entanglement. Here we provide a construction of multipartite subspaces that are not only genuinely entangled but also fully non-positive-partial-transpose (NPT) in the sense that any mixed state supported on them has non-positive partial transpose across any bipartition. Our construction originates from the stabilizer formalism known for its use in quantum error correction. To this end, we first introduce a couple of criteria allowing to assess whether any state from a given non-trivial stabilizer subspace is genuinely multipartite entangled. We then use these criteria to construct genuinely entangled stabilizer subspaces for any number of parties and arbitrary local dimension and conjecture them to be of maximal dimension achievable within the stabilizer formalism. At the same time, we prove that every genuinely entangled subspace is fully NPT in the above sense, which implies a quite surprising fact that no genuinely entangled stabilizer subspace can support PPT entangled states.

Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to 3 parties [Bravyi et al., J. Math. Phys. 47 , 062106 (2006)], we show that for 4 and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for 4 parties up to 10 qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to 2 , which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.

Abstract Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of n parties with local dimension q a power of a prime for all q ⩾ n − 1 . Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every q ⩾ n − 1 prime, we show how to construct stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [ [ n , 1 , n / 2 ] ] q for n even with the highest distance allowed by the quantum Singleton bound.

There is a connection between classical codes, highly entangled pure states (called

k

-uniform or absolutely maximally entangled (AME) states), and quantum error correcting codes (QECCs). This leads to a systematic method to construct stabilizer QECCs by starting from a

k

-uniform state or the corresponding classical code and tracing out one party at each step. We provide explicit constructions for codewords, encoding procedure and stabilizer formalism of the QECCs by describing the changes that partial traces cause on the corresponding generator matrix of the classical codes. We then modify the method to produce another set of stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. This construction produces quantum codes starting from an AME state without tracing out any party. Therefore, quantum stabilizer codes with larger codespace can be constructed.

We present a method for multipartite entanglement purification of any stabilizer state shared by several parties. In our protocol each party measures the stabilizer operators of a quantum error-correcting code on his or her qubits. The parties exchange their measurement results, detect or correct errors, and decode the desired purified state. We give sufficient conditions on the stabilizer codes that may be used in this procedure and find that Steane's seven-qubit code is the smallest error-correcting code sufficient to purify any stabilizer state. An error-detecting code that encodes two qubits in six can also be used to purify any stabilizer state. We further specify which classes of stabilizer codes can purify which classes of stabilizer states.