Fidelity of quantum states

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In quantum mechanics, notably in quantum information theory, fidelity quantifies the "closeness" between two density matrices. It expresses the probability that one state will pass a test to identify as the other. It is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.

Definition

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The fidelity between two quantum states   and  , expressed as density matrices, is commonly defined as:[1][2]

 

The square roots in this expression are well-defined because both   and   are positive semidefinite matrices, and the square root of a positive semidefinite matrix is defined via the spectral theorem. The Euclidean inner product from the classical definition is replaced by the Hilbert–Schmidt inner product.

As will be discussed in the following sections, this expression can be simplified in various cases of interest. In particular, for pure states,   and  , it equals: This tells us that the fidelity between pure states has a straightforward interpretation in terms of probability of finding the state   when measuring   in a basis containing  .

Some authors use an alternative definition   and call this quantity fidelity.[2] The definition of   however is more common.[3][4][5] To avoid confusion,   could be called "square root fidelity". In any case it is advisable to clarify the adopted definition whenever the fidelity is employed.

Motivation from classical counterpart

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Given two random variables   with values   (categorical random variables) and probabilities   and  , the fidelity of   and   is defined to be the quantity

 .

The fidelity deals with the marginal distribution of the random variables. It says nothing about the joint distribution of those variables. In other words, the fidelity   is the square of the inner product of   and   viewed as vectors in Euclidean space. Notice that   if and only if  . In general,  . The measure   is known as the Bhattacharyya coefficient.

Given a classical measure of the distinguishability of two probability distributions, one can motivate a measure of distinguishability of two quantum states as follows: if an experimenter is attempting to determine whether a quantum state is either of two possibilities   or  , the most general possible measurement they can make on the state is a POVM, which is described by a set of Hermitian positive semidefinite operators  . When measuring a state   with this POVM,  -th outcome is found with probability  , and likewise with probability   for  . The ability to distinguish between   and   is then equivalent to their ability to distinguish between the classical probability distributions   and  . A natural question is then to ask what is the POVM the makes the two distributions as distinguishable as possible, which in this context means to minimize the Bhattacharyya coefficient over the possible choices of POVM. Formally, we are thus led to define the fidelity between quantum states as:

 

It was shown by Fuchs and Caves[6] that the minimization in this expression can be computed explicitly, with solution the projective POVM corresponding to measuring in the eigenbasis of  , and results in the common explicit expression for the fidelity as 

Equivalent expressions

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Equivalent expression via trace norm

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An equivalent expression for the fidelity between arbitrary states via the trace norm is:

 

where the absolute value of an operator is here defined as  .

Equivalent expression via characteristic polynomials

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Since the trace of a matrix is equal to the sum of its eigenvalues

 

where the   are the eigenvalues of  , which is positive semidefinite by construction and so the square roots of the eigenvalues are well defined. Because the characteristic polynomial of a product of two matrices is independent of the order, the spectrum of a matrix product is invariant under cyclic permutation, and so these eigenvalues can instead be calculated from  .[7] Reversing the trace property leads to

 .

Expressions for pure states

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If (at least) one of the two states is pure, for example  , the fidelity simplifies to This follows observing that if   is pure then  , and thus 

If both states are pure,   and  , then we get the even simpler expression: 

Properties

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Some of the important properties of the quantum state fidelity are:

  • Symmetry.  .
  • Bounded values. For any   and  ,  , and  .
  • Consistency with fidelity between probability distributions. If   and   commute, the definition simplifies to  where   are the eigenvalues of  , respectively. To see this, remember that if   then they can be diagonalized in the same basis:  so that  
  • Explicit expression for qubits.

If   and   are both qubit states, the fidelity can be computed as [1] [8]

 

Qubit state means that   and   are represented by two-dimensional matrices. This result follows noticing that   is a positive semidefinite operator, hence  , where   and   are the (nonnegative) eigenvalues of  . If   (or  ) is pure, this result is simplified further to   since   for pure states.


Unitary invariance

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Direct calculation shows that the fidelity is preserved by unitary evolution, i.e.

 

for any unitary operator  .

Relationship with the fidelity between the corresponding probability distributions

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Let   be an arbitrary positive operator-valued measure (POVM); that is, a set of positive semidefinite operators   satisfying  . Then, for any pair of states   and  , we have   where in the last step we denoted with   and   the probability distributions obtained by measuring   with the POVM  .


This shows that the square root of the fidelity between two quantum states is upper bounded by the Bhattacharyya coefficient between the corresponding probability distributions in any possible POVM. Indeed, it is more generally true that   where  , and the minimum is taken over all possible POVMs. More specifically, one can prove that the minimum is achieved by the projective POVM corresponding to measuring in the eigenbasis of the operator  .[9]

Proof of inequality

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As was previously shown, the square root of the fidelity can be written as  which is equivalent to the existence of a unitary operator   such that

 Remembering that   holds true for any POVM, we can then write where in the last step we used Cauchy-Schwarz inequality as in  .

Behavior under quantum operations

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The fidelity between two states can be shown to never decrease when a non-selective quantum operation   is applied to the states:[10]  for any trace-preserving completely positive map  .

Relationship to trace distance

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We can define the trace distance between two matrices A and B in terms of the trace norm by

 

When A and B are both density operators, this is a quantum generalization of the statistical distance. This is relevant because the trace distance provides upper and lower bounds on the fidelity as quantified by the Fuchs–van de Graaf inequalities,[11]

 

Often the trace distance is easier to calculate or bound than the fidelity, so these relationships are quite useful. In the case that at least one of the states is a pure state Ψ, the lower bound can be tightened.

 

Uhlmann's theorem

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We saw that for two pure states, their fidelity coincides with the overlap. Uhlmann's theorem[12] generalizes this statement to mixed states, in terms of their purifications:

Theorem Let ρ and σ be density matrices acting on Cn. Let ρ12 be the unique positive square root of ρ and

 

be a purification of ρ (therefore   is an orthonormal basis), then the following equality holds:

 

where   is a purification of σ. Therefore, in general, the fidelity is the maximum overlap between purifications.

Sketch of proof

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A simple proof can be sketched as follows. Let   denote the vector

 

and σ12 be the unique positive square root of σ. We see that, due to the unitary freedom in square root factorizations and choosing orthonormal bases, an arbitrary purification of σ is of the form

 

where Vi's are unitary operators. Now we directly calculate

 

But in general, for any square matrix A and unitary U, it is true that |tr(AU)| ≤ tr((A*A)12). Furthermore, equality is achieved if U* is the unitary operator in the polar decomposition of A. From this follows directly Uhlmann's theorem.

Proof with explicit decompositions

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We will here provide an alternative, explicit way to prove Uhlmann's theorem.

Let   and   be purifications of   and  , respectively. To start, let us show that  .

The general form of the purifications of the states is: were   are the eigenvectors of  , and   are arbitrary orthonormal bases. The overlap between the purifications is where the unitary matrix   is defined as The conclusion is now reached via using the inequality  :  Note that this inequality is the triangle inequality applied to the singular values of the matrix. Indeed, for a generic matrix  and unitary  , we have where   are the (always real and non-negative) singular values of  , as in the singular value decomposition. The inequality is saturated and becomes an equality when  , that is, when   and thus  . The above shows that   when the purifications   and   are such that  . Because this choice is possible regardless of the states, we can finally conclude that 

Consequences

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Some immediate consequences of Uhlmann's theorem are

  • Fidelity is symmetric in its arguments, i.e. F (ρ,σ) = F (σ,ρ). Note that this is not obvious from the original definition.
  • F (ρ,σ) lies in [0,1], by the Cauchy–Schwarz inequality.
  • F (ρ,σ) = 1 if and only if ρ = σ, since Ψρ = Ψσ implies ρ = σ.

So we can see that fidelity behaves almost like a metric. This can be formalized and made useful by defining

 

As the angle between the states   and  . It follows from the above properties that   is non-negative, symmetric in its inputs, and is equal to zero if and only if  . Furthermore, it can be proved that it obeys the triangle inequality,[2] so this angle is a metric on the state space: the Fubini–Study metric.[13]

References

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  1. ^ a b R. Jozsa, Fidelity for Mixed Quantum States, J. Mod. Opt. 41, 2315--2323 (1994). DOI: http://doi.org/10.1080/09500349414552171
  2. ^ a b c Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. doi:10.1017/CBO9780511976667. ISBN 978-0521635035.
  3. ^ Bengtsson, Ingemar (2017). Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge, United Kingdom New York, NY: Cambridge University Press. ISBN 978-1-107-02625-4.
  4. ^ Walls, D. F.; Milburn, G. J. (2008). Quantum Optics. Berlin: Springer. ISBN 978-3-540-28573-1.
  5. ^ Jaeger, Gregg (2007). Quantum Information: An Overview. New York London: Springer. ISBN 978-0-387-35725-6.
  6. ^ C. A. Fuchs, C. M. Caves: "Ensemble-Dependent Bounds for Accessible Information in Quantum Mechanics", Physical Review Letters 73, 3047(1994)
  7. ^ Baldwin, Andrew J.; Jones, Jonathan A. (2023). "Efficiently computing the Uhlmann fidelity for density matrices". Physical Review A. 107: 012427. arXiv:2211.02623. doi:10.1103/PhysRevA.107.012427.
  8. ^ M. Hübner, Explicit Computation of the Bures Distance for Density Matrices, Phys. Lett. A 163, 239--242 (1992). DOI: https://doi.org/10.1016/0375-9601%2892%2991004-B
  9. ^ Watrous, John (2018-04-26). The Theory of Quantum Information. Cambridge University Press. doi:10.1017/9781316848142. ISBN 978-1-316-84814-2.
  10. ^ Nielsen, M. A. (1996-06-13). "The entanglement fidelity and quantum error correction". arXiv:quant-ph/9606012. Bibcode:1996quant.ph..6012N. {{cite journal}}: Cite journal requires |journal= (help)
  11. ^ C. A. Fuchs and J. van de Graaf, "Cryptographic Distinguishability Measures for Quantum Mechanical States", IEEE Trans. Inf. Theory 45, 1216 (1999). arXiv:quant-ph/9712042
  12. ^ Uhlmann, A. (1976). "The "transition probability" in the state space of a ∗-algebra" (PDF). Reports on Mathematical Physics. 9 (2): 273–279. Bibcode:1976RpMP....9..273U. doi:10.1016/0034-4877(76)90060-4. ISSN 0034-4877.
  13. ^ K. Życzkowski, I. Bengtsson, Geometry of Quantum States, Cambridge University Press, 2008, 131