Basis Entropy

Projective measurement can increase the entropy of a state $\rho$, the increased entropy is not only up to the basis of projective measurement, but also has something to do with the properties of the state itself. In this paper we define this increased entropy as basis entropy. And then we discuss the usefulness of this new concept by showing its application in explaining the success probability of Grover's algorithm and the existence of quantum discord. And as shown in the paper, this new concept can also be used to describe decoherence.

Introduction-Projective measurement can increase the entropy of a state ρ [1] [3].And for different states, the increased entropy is different, and it is up to two factors 1.The orthogonal projectors of projective measurement 2. The state itself.
Every state has its increased entropy after a projective measurement.This increased entropy is actually a quite useful physical quantity.The aim of this paper is to show the usefulness of this increased entropy by providing some theorems about this increased entropy.
A mentionable merit of this increased entropy is that it is highly related to quantum discord, and it can be used to explain the existence of quantum discord.
Basis entropy-First we should give this increased entropy a proper name.Since projective measurement is dependent on its basis, we suggest using 'basis entropy' to describe this increased entropy of the state.The meaning of which is, with the knowledge of the state(the knowledge here means the information which equals to the von Neumann entropy of this state), the ignorance of a state's projective measurement result.And according to the definition of basis entropy, we can calculate basis entropy by using the following formula where P i is a complete set of orthogonal projectors.
A good example which can illustrate the physical meaning of basis entropy is Grove's algorithm [4].First let us review the procedure of Grover's algorithm [1].
The first state is the initial state, where n represents the qubit number, by applying H ⊗n to initial n qubits we get the state in step two, and then we apply the Grover iteration (2|ψ ψ| − I)O[6], we get the state in step three.After applying the Grover iteration for about R ≈ π √ 2 n /4 times, we get the wanted state |x 0 , the detail of Grover's algorithm refer [1].
For Grover's algorithm, we actually search in the state (∑ , and measure it with projectors {|0 0|, |1 1|}, before apply Grover iteration, the basis entropy of the database state is n, this is our ignorance of the measurement result, which means without applying Grover iteration, we need n bits to describe the measurement result.After apply one time of Grover iteration, the basis entropy decreased, which means our ignorance of the measurement result decreased, so the probability of finding the target state |x 0 increased.As shown in Figure 1, with the decreasing of the basis entropy, the successful probability is increasing. Figure 1 is the basis entropy change of Grover's algorithm, here we set n = 20, clearly we can see from this figure that, the more time we apply Grover iteration(under the desired time R ≈ π √ 2 n /4 , see appendix A), the basis entropy will be smaller and the success probability will be higher.So basis entropy is the ignorance of the measurement result with the knowledge of the state.A state's basis entropy is dependent on different measurement projectors, so the number of its basis entropy is actually infinite, but we are only interested in its maximum and minimum basis entropy.For a state's maximum basis entropy, we have the following theorem.
Theorem 1. Iff a state's maximal basis entropy is log 2 D, the state is a pure state.
where D is the dimension of Hilbert space.If a state's maximal basis entropy is log 2 D, which means Since the entropy of a state is non-negative and in a D-dimensional Hilbert space, the state's von Neumann entropy is at most log 2 D [1], so if the basis entropy of a state is log 2 D, then S(ρ) must be zero, which means the state is a pure state.Let's prove the other direction of the theorem.It seems only for pure state like has log 2 D basis entropy, while for pure state like ( √ 3|0 + |1 )/2, its basis entropy will be smaller than log 2 D. We will prove that for any pure state, its maximum basis entropy will be log 2 D, as long as we choose right projectors to measure them.For simplicity, we just prove the case D = 2.For any pure state, we can write its density matrix as where a, b and c are coefficients and σ 1 , σ 2 , σ 3 are Pauli matrices.We only need to prove that where, is the complete set of orthogonal projectors.And V is a 2-dimensional unitary transformation.It has be proven that for any 2-dimensional pure state, its maximum basis entropy is log 2 2 = 1 (for those who are interested in the details ,see appendix B).And now we complete the proof of theorem 1.The proof can be easily extended to higher Hilbert space.
From theorem 1, it is natural to get the following corollary: Corollary 1.Only for states like ρ = ∑ D i |i i|/D, they have no basis entropy, it means no projective measurement can increase their entropy.This corollary is easy to prove, since for a state like ρ = ∑ D i |i i|/D, its von Neumann has reached the maximal in its Hilbert space, so no projective measurement can increase its entropy.
We can use maximum basis entropy to judge whether a state is mixed or not.According to theorem 1, if a state's maximum basis entropy is log 2 D, then the state is a pure state.If its maximum basis entropy is smaller than log 2 D, then it's a mixed state.If its maximum basis entropy is zero, it has no basis entropy, then the state is ρ = ∑ D i |i i|/D.
Basis entropy and quantum discord-For most states, regardless of pure states or mixed states, their minimum basis entropy is zero, which means there exists a complete set of orthogonal projectors, under these projectors, we can get the full knowledge of this state.But there are some states, their minimum basis entropy are not zero, which means no matter what projectors we use to measure them, there is always some information we can not get.This inaccessible information, we will show next, is quantum discord.
Quantum discord comes from the projective measurement [7], just as the basis entropy.For a bell state, its basis entropy is a constant, so it's easy for us to calculate it minimum basis entropy.So let's take a bell state as an example to explain why the quantum discord is related to minimum basis entropy.For bell state like (|00 + |11 )/ √ 2, the subsystem is A and B, the discord of this state is where I(A : B) is the quantum mutual information between A and B, and J(A : B) {Π B i } is the measurement mutual information between A and B. From this equation we can see that the quantum discord of this state comes from S(B).Then let's calculate the basis entropy of the joint system AB, measure AB by projectors {|00 00|, |01 01|, |10 10|, |11 11|}, and after this projective measurement the whole system AB will become and Therefore the basis entropy of the joint system AB is equal to the quantum discord between its subsystems.The equivalence of these two concept is not a coincidence, because S(B) = tr A (ρ AB ), while doing this partial trace to A, we are measuring A with projectors {|0 0|, |1 1|}, since A and B are fully entangled, it is equivalent to measuring A and B at the same time, which means doing a partial trace to this joint system AB is equal to measure it with measurement projectors {|00 00|, |01 01|, |10 10|, |11 11|}, so its basis entropy equals the quantum discord between its subsystems.From the above analysis, we can see that in bell state β 00 , the quantum discord between two subsystems is a manifestation of basis entropy of the joint system.Bell state β 00 is a special state, it's easy for us to detail it in the above way.For general joint state, it's not easy for us to do the same thing, but the quantum discord between its subsystems still equals to its minimum basis entropy due to the same reason.For example, let's consider the discord of Werner state [8] ρ = (1 − z)I/4 + z|ψ ψ| with |ψ = (|00 + |11 )/ √ 2. One can calculate that the quantum discord of this Werner state is equal to its basis entropy.For example as shown in Figure 2, when z = 1/3, the value of quantum discord is 0.1258, it's basis entropy is also 0.1258.

Quantum discord Basis entropy
The quantum discord of Bell states or Werner state are exactly equal to their basis entropy, no matter which orthogonal projectors we used to measure them, like {|0 0|, |1 1|} or {|+ +|, |− −|}.This is because Bell states or Werner state are invariant under local rotations.Their basis entropy is invariant under different orthogonal projectors, thus their quantum discord is equal to their minimum basis entropy no matter what projectors we use.For ordinary two-qubit system ρ = (I + ∑ 3 i=1 c i σ i ⊗ σ i )/4 [9], its basis entropy is not a constant.We have shown in appendix C that, its quantum discord is still equal to its minimum basis entropy.
Then for the minimum basis entropy of a state, we have the following theorem.Theorem 3. If a state's minimum basis entropy is nonzero, there must be quantum discord between its subsystems.
From the above example we can see that, quantum discord it's a symmetric physical quantity for states like ρ = (I + ∑ 3 i=1 c i σ i ⊗ σ i )/4.For asymmetric quantum discord, the theorem still holds, see discussion in appendix C Basis entropy and coherence-One can see that basis entropy is also highly related to quantum coherence [10].Except the formula we use to measure basis entropy showed up naturally.One can find that the formula use to quantify coherence is actually a special case of basis entropy, it's a basis settled basis entropy.If a state has no coherence, we can't guarantee its maximum basis entropy is zero, but if a state's maximum basis entropy is zero, then it must have no coherence.This difference between these two concepts has physical meaning.For example for state |0 , it has no coherence, we can't decohere it, since its density matrix has no off-diagonal elements.But this state's maximum basis entropy is 1, from this perspective, we can still further decohere it.The decoherence about coherence is basis-dependent [14], means the state can only be seen as decohered under specific basis.The decoherence about maximum basis entropy is not basisdependent, means if we decohere a state's maximum basis entropy, the state will be fully decohered and it can't be further decohered.Take the following state as an example.
We decohere this state under basis {|0 , |1 }, it will become According to corollary 1, this state has basis entropy, so it can be further decohered.The corresponding projectors are After this , the state becomes According to corollary 1, this state has no basis entropy, so it can't be further decohered.
Therefore the maximum basis entropy can be used as a criterion to measure how much the environment can maximum decohere a quantum system, for pure states, they are highly isolated states and they have the maximum basis entropy, so they are easy to be decohered.And for state like ρ = ∑ D i |i i|/D, is actually a classical state, its maximum basis entropy is zero, so it has been fully decohered, the environment can't further decohere it.
Conclusions-In this paper we have shown that, the increased entropy of a state after a projective measurement is a useful physical quantity, we named this increased entropy as basis entropy and then we showed some usefulness of this new concept.For example it can be used to explain the increase of success probability in Grover's algorithm and it can be used to give a reasonable interpretation of the existence of quantum discord.At the end of this paper, we also discussed the difference between basis entropy and coherence, and showed that basis entropy can be used to describe decoherence.we begin with this state.It's not difficult to check that under measurement projectors {|0 0|, |1 1|}, its basis entropy is n.No matter how many times we apply the Grover iteration, the state G k |ψ is a pure state, its von Neumann entropy is zero, so we just need to calculate the entropy of the state after projective measurement, which is which can be write as With the increase of k, the basis entropy of the state G k |ψ is decreasing, while the success probability is increasing, as shown in Figure 1. ;

Figure 1 :
Figure 1: Basis entropy and success probability change of Grover's algorithm curve of quantum discord and basis entropyBasis entropy