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Bipartite Entanglement, Partial Transposition and the Uncertainty Principle for Finite-Dimensional Hilbert Spaces

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 Added by Yehuda Band
 Publication date 2016
  fields Physics
and research's language is English




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We first show that partial transposition for pure and mixed two-particle states in a discrete $N$-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 $r$ of a Bell state $|Phi^{+} rangle$ and the completely incoherent state, $hat{rho}_r = r |Phi^{+} rangle langle Phi^{+}| + (1-r)frac{hat{mathbb{I}}}{N^2}$: we find that for $r_0 < r < 1$, where $r_0$ is obtained as a function of the dimensionality $N$, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.



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We show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of a momentum-like variable of one of the particles in the Wigner function for the state. This generalizes a result obtained for continuous-variable systems to the discrete-variable system case. We show that, in principle, quantum mechanics allows measuring the expectation value of an observable in a partially transposed state, in spite of the fact that the latter may not be a physical state. We illustrate this result with the example of an isotropic state, which is dependent on a parameter $r$, and an operator whose variance becomes negative for the partially transposed state for certain values of $r$; for such $r$, the original states are entangled.
We present a brief review of discrete structures in a finite Hilbert space, relevant for the theory of quantum information. Unitary operator bases, mutually unbiased bases, Clifford group and stabilizer states, discrete Wigner function, symmetric informationally complete measurements, projective and unitary t--designs are discussed. Some recent results in the field are covered and several important open questions are formulated. We advocate a geometric approach to the subject and emphasize numerous links to various mathematical problems.
The most general quantum object that can be shared between two distant parties is a bipartite channel, as it is the basic element to construct all quantum circuits. In general, bipartite channels can produce entangled states, and can be used to simulate quantum operations that are not local. While much effort over the last two decades has been devoted to the study of entanglement of bipartite states, very little is known about the entanglement of bipartite channels. In this work, we rigorously study the entanglement of bipartite channels as a resource theory of quantum processes. We present an infinite and complete family of measures of dynamical entanglement, which gives necessary and sufficient conditions for convertibility under local operations and classical communication. Then we focus on the dynamical resource theory where free operations are positive partial transpose (PPT) superchannels, but we do not assume that they are realized by PPT pre- and post-processing. This leads to a greater mathematical simplicity that allows us to express all resource protocols and the relevant resource measures in terms of semi-definite programs. Along the way, we generalize the negativity from states to channels, and introduce the max-logarithmic negativity, which has an operational interpretation as the exact asymptotic entanglement cost of a bipartite channel. Finally, we use the non-positive partial transpose (NPT) resource theory to derive a no-go result: it is impossible to distill entanglement out of bipartite PPT channels under any sets of free superchannels that can be used in entanglement theory. This allows us to generalize one of the long-standing open problems in quantum information - the NPT bound entanglement problem - from bipartite states to bipartite channels. It further leads us to the discovery of bound entangled POVMs.
We investigate the features of the entanglement spectrum (distribution of the eigenvalues of the reduced density matrix) of a large quantum system in a pure state. We consider all Renyi entropies and recover purity and von Neumann entropy as particular cases. We construct the phase diagram of the theory and unveil the presence of two critical lines.
A new purification scheme is proposed which applies to arbitrary dimensional bipartite quantum systems. It is based on the repeated application of a special class of nonlinear quantum maps and a single, local unitary operation. This special class of nonlinear quantum maps is generated in a natural way by a hermitian generalized XOR-gate. The proposed purification scheme offers two major advantages, namely it does not require local depolarization operations at each step of the purification procedure and it purifies more efficiently than other know purification schemes.
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