Maximal coherence in random basis

Given a preferred orthonormal basis $B$ in the Hilbert space of a quantum system, we define a measure of the coherence-generating power of a unitary operation with respect to $B$ . This measure is the average coherence generated by the operation acting on a uniform ensemble of incoherent states. We give its explicit analytical form in any dimension and provide an operational protocol to directly detect it. We characterize the set of unitaries with maximal coherence-generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For a large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence-generating power with respect to any basis. Finally, extensions to general unital quantum operations and the relation to the concept of asymmetry are discussed.Given a preferred orthonormal basis $B$ in the Hilbert space of a quantum system, we define a measure of the coherence-generating power of a unitary operation with respect to $B$. This measure is the average coherence generated by the operation acting on a uniform ensemble of incoherent states. We give its explicit analytical form in any dimension and provide an operational protocol to directly detect it. We characterize the set of unitaries with maximal coherence-generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For a large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence-generating power with respect to any basis. Finally, extensions to general unital quantum operations and the relation to the concept of asymmetry are discussed.

arXiv:1610.00217Physical Review AMay 2, 201765 citations

Coherence generating power of quantum unitary maps

Paolo Zanardi A, Georgios Styliaris, Lorenzo Campos Venuti

Given a preferred orthonormal basis $B$ in the Hilbert space of a quantum system we define a measure of the coherence generating power of a unitary map with respect to $B$ . We give its explicit analytical form and provide an operational protocol to directly detect it. This measure vanishes if and only if the unitary map transforms all the incoherent states into themselves and is non-increasing under pre- or post-processing by incoherent operations. We characterize the set of unitaries with maximal coherence generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence generating power with respect to any basis. Finally, extensions to general unital completely positive maps and the relation to the concept of asymmetry are discussed.Given a preferred orthonormal basis $B$ in the Hilbert space of a quantum system we define a measure of the coherence generating power of a unitary map with respect to $B$. We give its explicit analytical form and provide an operational protocol to directly detect it. This measure vanishes if and only if the unitary map transforms all the incoherent states into themselves and is non-increasing under pre- or post-processing by incoherent operations. We characterize the set of unitaries with maximal coherence generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence generating power with respect to any basis. Finally, extensions to general unital completely positive maps and the relation to the concept of asymmetry are discussed.

Oct 2, 20160 citations

Average versus maximal coherence

Shunlong Luo, Yuan Sun

Physics Letters AJun 21, 201933 citations

Maximal coherence in a generic basis

Yao Yao, Guohui Dong, Li Ge +2 more

Since quantum coherence is an undoubted characteristic trait of quantum physics, the quantification and application of quantum coherence has been one of the long-standing central topics in quantum information science. Within the framework of a resource theory of quantum coherence proposed recently, a fiducial basis should be preselected for characterizing the quantum coherence in specific circumstances, namely, the quantum coherence is a basis-dependent quantity. Therefore, a natural question is raised: what are the maximum and minimum coherences contained in a certain quantum state with respect to a generic basis? While the minimum case is trivial, it is not so intuitive to verify in which basis the quantum coherence is maximal. Based on the coherence measure of relative entropy, we indicate the particular basis in which the quantum coherence is maximal for a given state, where the Fourier matrix (or more generally, complex Hadamard matrices) plays a critical role in determining the basis. Intriguingly, though we can prove that the basis associated with the Fourier matrix is a stationary point for optimizing the ${l}_{1}$ norm of coherence, numerical simulation shows that it is not a global optimal choice.Since quantum coherence is an undoubted characteristic trait of quantum physics, the quantification and application of quantum coherence has been one of the long-standing central topics in quantum information science. Within the framework of a resource theory of quantum coherence proposed recently, a fiducial basis should be preselected for characterizing the quantum coherence in specific circumstances, namely, the quantum coherence is a basis-dependent quantity. Therefore, a natural question is raised: what are the maximum and minimum coherences contained in a certain quantum state with respect to a generic basis? While the minimum case is trivial, it is not so intuitive to verify in which basis the quantum coherence is maximal. Based on the coherence measure of relative entropy, we indicate the particular basis in which the quantum coherence is maximal for a given state, where the Fourier matrix (or more generally, complex Hadamard matrices) plays a critical role in determining the basis. Intriguingly, though we can prove that the basis associated with the Fourier matrix is a stationary point for optimizing the ${l}_{1}$ norm of coherence, numerical simulation shows that it is not a global optimal choice.

arXiv:1611.01740Physical Review ADec 29, 201651 citations

More About Robustness of Coherence

Piyu Li, Feng Liu, Yan-Qin Xu +1 more

International Journal of Theoretical PhysicsMar 10, 20180 citations

Given a preferred orthonormal basis

B

in the Hilbert space of a quantum system, we define a measure of the coherence-generating power of a unitary operation with respect to

B

. This measure is the average coherence generated by the operation acting on a uniform ensemble of incoherent states. We give its explicit analytical form in any dimension and provide an operational protocol to directly detect it. We characterize the set of unitaries with maximal coherence-generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For a large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence-generating power with respect to any basis. Finally, extensions to general unital quantum operations and the relation to the concept of asymmetry are discussed.

Given a preferred orthonormal basis

B

in the Hilbert space of a quantum system we define a measure of the coherence generating power of a unitary map with respect to

B

. We give its explicit analytical form and provide an operational protocol to directly detect it. This measure vanishes if and only if the unitary map transforms all the incoherent states into themselves and is non-increasing under pre- or post-processing by incoherent operations. We characterize the set of unitaries with maximal coherence generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence generating power with respect to any basis. Finally, extensions to general unital completely positive maps and the relation to the concept of asymmetry are discussed.

{l}_{1}

norm of coherence, numerical simulation shows that it is not a global optimal choice.