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.
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.
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.
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.
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.
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).
Can vectors with low Schmidt rank form mutually unbiased bases? Can vectors with high Schmidt rank form positive under partial transpose states? In this work, we address these questions by presenting ...several new results related to Schmidt rank constraints and their compatibility with other properties. We provide an upper bound on the number of mutually unbiased bases of
C
m
⊗
C
n
(
m
≤
n
)
formed by vectors with low Schmidt rank. In particular, the number of mutually unbiased product bases of
C
m
⊗
C
n
cannot exceed
m
+
1
, which solves a conjecture proposed by McNulty et al. Then, we show how to create a positive under partial transpose entangled state from any state supported on the antisymmetric space and how their Schmidt numbers are exactly related. Finally, we show that the Schmidt number of operator Schmidt rank 3 states of
M
m
⊗
M
n
(
m
≤
n
)
that are invariant under left partial transpose cannot exceed
m
-
2
.
We show that any unital qubit channel can be implemented by letting the input system interact unitarily with a four-dimensional environment in the maximally mixed state and then tracing out the ...environment. We also provide an example where the dimension of such an environment has to be at least 3.
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