Entangled quantum systems have properties that have fundamentally overthrown the classical worldview. Increasing the complexity of entangled states by expanding their dimensionality allows the ...implementation of novel fundamental tests of nature, and moreover also enables genuinely new protocols for quantum information processing. Here we present the creation of a (100 × 100)-dimensional entangled quantum system, using spatial modes of photons. For its verification we develop a novel nonlinear criterion which infers entanglement dimensionality of a global state by using only information about its subspace correlations. This allows very practical experimental implementation as well as highly efficient extraction of entanglement dimensionality information. Applications in quantum cryptography and other protocols are very promising.
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We determine sufficient conditions for certain classes of (n+k)×n matrices E to have all order-n minors to be nonzero. For a special class of (n+1)×n matrices E, we give the formula for the order-n ...minors. As an application we construct subspaces of Cm⊗Cn of maximal dimension, which do not contain any vector of Schmidt rank less than k and each of which has a basis of Schmidt rank k for k=2,3,4.
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Abstract
We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing ...that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. Our scrutiny reveals that these operators can be used to simulate any bipartite unitary gate via stochastic local operations and classical communication. Furthermore, we establish a one-to-one connection between the set of local diagonal unitary invariant dual unitary operators with maximum entangling power and the set of complex Hadamard matrices. Finally, we discuss distinguishability of unitary operators in the setting of the stated diagonal symmetry.
Abstract The multipartite unitary gates are called genuine if they are not product unitary operators across any bipartition. We mainly investigate the classification of genuine multipartite unitary ...gates of Schmidt rank two (SR-2), by focusing on the multiqubit scenario. For genuine multipartite (excluding bipartite) unitary gates of SR-2, there is an essential fact that their Schmidt decompositions are unique. Based on this fact, we propose a key notion named as singular number (SN) to classify the unitary gates concerned. The SN is defined as the number of local singular operators in the Schmidt decomposition. We then determine the accurate range of SN. For each SN, we formulate the parametric Schmidt decompositions of genuine multiqubit unitary gates under local equivalence. Finally, we extend the study to three-qubit diagonal unitary gates due to the close relation between diagonal unitary gates and SR-2 unitaries. We start with discussing two typical examples of SR-2, one of which is a fundamental three-qubit unitary gate, i.e. the CCZ gate. Then we characterize the diagonal unitary gates of SR greater than two. We show that a three-qubit diagonal unitary gate has SR at most three, and present a necessary and sufficient condition for such a unitary gate of SR-3. This completes the characterization of all genuine three-qubit diagonal unitary gates.
The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen 2011 Phys. Rev. A 84 012325. We ...extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.
It is conjectured that four mutually unbiased bases in dimension 6 do not exist in quantum information. The conjecture is equivalent to the nonexistence of some three
$ 6\times 6 $
6
×
6
complex ...Hadamard matrices (CHMs) with Schmidt rank at least 3. We investigate the
$ 6\times 6 $
6
×
6
CHM U of Schmidt rank 3 containing two nonintersecting identical
$ 3\times 3 $
3
×
3
submatrices V, i.e.
$ U=\frac {1}{\sqrt 2}\left \begin {smallmatrix} W & V \\ V & X\end {smallmatrix}\right $
U
=
1
2
W
V
V
X
. We show that such U exists, V, W, X have rank 2 or 3, and they have rank 2 at the same time. We construct the analytical expressions of U when V is, respectively, of rank 2, unitary and normal. We apply our results to the conjecture by showing that U with some normal V is not one of the three
$ 6\times 6 $
6
×
6
CHMs.
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Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex ...Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation
⊗
and
⊗
. We show that
and
have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log
3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.
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The entangling power of a bipartite unitary operation shows the maximum created entanglement with the product input states. For an arbitrary two-qubit unitary operation, it is sufficient to consider ...its normalized operation U with parameters and c3. We show how to compute the entangling power of U when . In particular we construct the analytical expressions of entangling power of such U for two examples. We also formulate the entangling power of bipartite unitary operations of Schmidt rank two for any dimensions.
Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, it is shown 2(N−1)$2(N-1)$ generalized controlled‐X (GCX) gates, 6 ...single‐qubit rotations about the y‐ and z‐axes, and N+5$N+5$ single‐partite y‐ and z‐rotation‐types which are defined in this paper are sufficient to simulate a controlled‐unitary gate Ucu(2⊗N)$\mathcal {U}_{\text{cu}(2\otimes N)}$ with A$\text{A}$ controlling on C2⊗CN$\mathbb {C}^2\otimes \mathbb {C}^N$. In the scenario of the unitary gate Ucd(M⊗N)$\mathcal {U}_{\text{cd}(M\otimes N)}$ with M≥3$M\ge 3$ that is locally equivalent to a diagonal unitary on CM⊗CN$\mathbb {C}^M\otimes \mathbb {C}^N$, 2M(N−1)$2M(N-1)$ GCX gates and 2M(N−1)+10$2M(N-1)+10$ single‐partite y‐ and z‐rotation‐types are required to simulate it. The quantum circuit for implementing Ucu(2⊗N)$\mathcal {U}_{\text{cu}(2\otimes N)}$ and Ucd(M⊗N)$\mathcal {U}_{\text{cd}(M\otimes N)}$ are presented. Furthermore, it is found that Ucu(2⊗2)$\mathcal {U}_{\text{cu}(2\otimes 2)}$ with A$\text{A}$ controlling has Schmidt rank two, and in other cases the diagonalized form of the target unitaries can be expanded in terms of specific simple types of product unitary operators.
A quantum circuit of the controlled‐unitary operation with side controlling on C2⊗CN$C^{2} \otimes C^{N}$ system is designed by utilizing Cartan decomposition technique. The synthesis is extended to the unitary operations on CM⊗CN$C^{M} \otimes C^{N}$ which are locally equivalent to diagonal unitary operations. Additionally, the possible Schmidt rank of these unitary operations is presented in detail.
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The operator Schmidt rank of an operator acting on the tensor product Cn⊗Cm is the number of terms in a decomposition of the operator as a sum of simple tensors with factors forming orthogonal ...families in their respective matrix algebras. It has been known that for unitary operators acting on two copies of C2, the operator Schmidt rank can only take the values 1, 2, and 4, the value 3 being forbidden. In this paper, we settle an open question, showing that the above obstruction is the only one occurring. We do so by constructing explicit examples of bipartite unitary operators of all possible operator Schmidt ranks, for arbitrary dimensions (n,m)≠(2,2).
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