Diamond standard

With the help of the diamond standard , distances between two quantum channels are often specified in quantum mechanics . With the Jamiolkowski isomorphism , this distance can also be extended to states . With quantum mechanical measurements the system is disturbed and the state of the quantum changes. In theoretical considerations, a quantum mechanical measurement is carried out by applying a linear operator to the previous state. The eigenvalues , which the product of the operator and eigenvectors has, are all possible measured values that can occur during a measurement. After the measurement, the question arises as to how far apart the states in which the quanta are now. One way of determining this distance is given by the diamond standard.

Let a quantum channel and its state be: ${\ displaystyle \ Phi}$ ${\ displaystyle J (\ Phi)}$

{\ displaystyle J (\ Phi) = {\ frac {1} {n}} \ sum _ {1 \ leq i, j \ leq n} \ Phi \ left (\ left | i \ right \ rangle \ left \ langle j \ right | \ right) \ otimes \ left | i \ right \ rangle \ left \ langle j \ right |}

If the maps of the canal correspond to any two and with , the matrix is also referred to as the Choi-Jamiolkowski representation . Now let two quantum channels be, then the distance in the diamond norm corresponds to: ${\ displaystyle M_ {m} (\ mathbb {C})}$ ${\ displaystyle M_ {n} (\ mathbb {C})}$ ${\ displaystyle m, n \ in \ mathbb {R} ^ {+}}$ ${\ displaystyle J (\ Phi)}$ ${\ displaystyle \ Phi _ {1}, \ Phi _ {2}}$

{\ displaystyle \ left \ | \ Phi _ {1} - \ Phi _ {2} \ right \ | _ {\ diamond} = \ sup _ {\ rho} \ left \ | (\ Phi _ {1} \ otimes \ operatorname {Id} _ {k}) (\ rho) - (\ Phi _ {2} \ otimes \ operatorname {Id} _ {k}) (\ rho) \ right \ | _ {1}}

with identity channel mapped by at yourself, track standard , the upper bound of any and all density operators from . This supremum is always calculable for a firmly chosen one . The correspondingly calculated distance also expresses the probability of error for a misinterpretation of the result of two quantum channels and is often used for corresponding comparative calculations: In many cases, the pure use of the trace standard helps, but sometimes it does not provide satisfactory results, especially when viewed of quantum channels. ${\ displaystyle \ operatorname {Id} _ {k}}$ ${\ displaystyle M_ {k} (\ mathbb {C})}$ ${\ displaystyle \ | \ cdot \ | _ {1}}$ ${\ displaystyle k \ geq 1}$ ${\ displaystyle \ rho}$ ${\ displaystyle M_ {nk} = M_ {n} \ otimes M_ {k}}$ ${\ displaystyle k}$

literature

Avraham Ben-Aroya, Amnon Ta-Shma: On the complexity of approximating the diamond norm . In: Quantum Physics . 2009, arxiv : 0902.3397 (English).
Walter Noll: Finite-Dimensional Spaces: Algebra, Geometry and Analysis . Springer Science & Business Media, 2012, ISBN 978-94-010-9335-4 , pp. 168 ff . (English, limited preview in Google Book Search).

Individual evidence

↑ Andrew Childs, Michele Mosca: Theory of Quantum Computation, Communication and Cryptography: 4th Workshop, TQC 2009, Waterloo, Canada, May 11-13. Revised Selected Papers Springer, March 10, 2010, p. 68
↑ James M. McCracken: Negative Quantum Channels: An Introduction to Quantum Maps that are Not Completely Positive Morgan & Claypool Publishers, 2014, p. 121

[1] Andrew Childs, Michele Mosca: Theory of Quantum Computation, Communication and Cryptography: 4th Workshop, TQC 2009, Waterloo, Canada, May 11-13. Revised Selected Papers Springer, March 10, 2010, p. 68

[2] James M. McCracken: Negative Quantum Channels: An Introduction to Quantum Maps that are Not Completely Positive Morgan & Claypool Publishers, 2014, p. 121