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arXiv:1712.03277

On Bipartite Operators Defined by Completely Different Permutations

Marek Mozrzymas, Dariusz Chruściński, Gniewomir Sarbicki

Published Dec 8, 2017• 0 citations
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Abstract

We introduce a class of bipartite operators acting on $\mathcal{H} \otimes \mathcal{H}$ ($\mathcal{H}$ being an $n$-dimensional Hilbert space) defined by a set of $n$ Completely Different Permutations CDP. Bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems. It turns out that any set of CDPs gives rise to a certain direct sum decomposition of the total Hilbert space which enables one simple construction of the corresponding bipartite operator. Interestingly, if set of CDPs defines an abelian group then the corresponding bipartite operator displays an additional property -- the partially transposed operator again corresponds to (in general different) set of CDPs. Therefore, our technique may be used to construct new classes of so called PPT states which are of great importance for quantum information. Using well known relation between bipartite operators and linear maps one use also construct linear maps related to CDPs.

DOI
10.48550/arxiv.1712.03277

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We introduce a class of bipartite operators acting on H⊗H\mathcal{H} \otimes \mathcal{H}H⊗H (H\mathcal{H}H being an nnn-dimensional Hilbert space) defined by a set of nnn Completely Different Permutations CDP. Bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems. It turns out that any set of CDPs gives rise to a certain direct sum decomposition of the total Hilbert space which enables one simple construction of the corresponding bipartite operator. Interestingly, if set of CDPs defines an abelian group then the corresponding bipartite operator displays an additional property -- the partially transposed operator again corresponds to (in general different) set of CDPs. Therefore, our technique may be used to construct new classes of so called PPT states which are of great importance for quantum information. Using well known relation between bipartite operators and linear maps one use also construct linear maps related to CDPs.
We introduce a class of bipartite operators acting on H⊗H (H being an n-dimensional Hilbert space) defined by a set of n Completely Different Permutations (CDPs). Bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems. It turns out that any set of CDPs gives rise to a certain direct sum decomposition of the total Hilbert space which enables one simple construction of the corresponding bipartite operator. Interestingly, if set of CDPs defines an abelian group, then the corresponding bipartite operator displays an additional property – the partially transposed operator again corresponds to (in general different) set of CDPs. Therefore, our technique may be used to construct new classes of the so-called PPT (positive under partial transposition) states which are of great importance for quantum information. Using well-known relation between bipartite operators and linear maps, one can use also construct linear maps related to CDPs.
We construct a large class of quantum d ⊗ d states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure — that is way we call them circulant states. It turns out that partial transposition maps any such decomposition into another one and hence both original density matrix and its partially transposed partner share similar cyclic properties. This class contains many well known examples of PPT states from the literature and gives rise to a huge family of completely new states.
Abstract We investigate unitary operators acting on a tensor product space, with the property that the quantum channels they generate, via the Stinespring dilation theorem, are of a particular type, independently of the state of the ancilla system in the Stinespring relation. The types of quantum channels we consider are those of interest in quantum information theory: unitary conjugations, constant channels, unital channels, mixed unitary channels, positive partial transpose channels, and entanglement breaking channels. For some of the classes of bipartite unitary operators corresponding to the above types of channels, we provide explicit characterizations, necessary and/or sufficient conditions for membership, and we compute the dimension of the corresponding algebraic variety. Inclusions between these classes are considered, and we show that for small dimensions, many of these sets are identical.
We first show that partial transposition for pure and mixed two-particle states in a discrete NNN-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter rrr of a Bell state ∣Φ+⟩|Φ^{+} \rangle∣Φ+⟩ and the completely incoherent state, ρ^r=r∣Φ+⟩⟨Φ+∣+(1−r)I^N2\hatρ_r = r |Φ^{+} \rangle \langle Φ^{+}| + (1-r)\frac{\hat{\mathbb{I}}}{N^2}ρ^​r​=r∣Φ+⟩⟨Φ+∣+(1−r)N2I^​: we find that for r0<r<1r_0 < r < 1r0​<r<1, where r0r_0r0​ is obtained as a function of the dimensionality NNN, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.
It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we derive several generally valid purity-type formulas for one-party reductions of rank-1 operators, and we point out relevant relations between the Schmidt decomposition and the Bloch representation of bipartite pure states.