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Lecture Notes Lecture Notes edX Quantum Cryptography: Week 3 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Licence. Contents 3.1 When are two quantum states almost the same?3 3.1.1 Trace distance...................................................3 3.1.2 Fidelity.........................................................5 3.2 Measuring uncertainty: the min-entropy6 3.2.1 The min-entropy..................................................6 3.2.2 The conditional min-entropy........................................7 3.3 What it means to be ignorant9 3.4 Uncertainty principles: a bipartite guessing game 11 3.4.1 Analysis: winning probability of the guessing game..................... 13 3.5 Extended uncertainty relation principles: A tripartite guessing game 14 3.5.1 Analysis: winning probability of the tripartite guessing game.............. 16 3.1 When are two quantum states almost the same? 3 We have seen in Week 1 an example of communication between Alice and Bob, where the transmitted message is hidden from any eavesdropper Eve. There, we have seen the importance of using a large key K shared between Alice and Bob, but looks completely random from Eve’s perspective. In the next few lectures, we will concern ourselves with how to establish such a key. In this week, we will first learn about ways to quantify quantum information, which will be crucial in formulating what does it mean to be secure in cryptographic protocols. 3.1 When are two quantum states almost the same? It will be important for us to have some notion of what it means to approximately produce a particular quantum state. 3.1.1 Trace distance One measure of closeness that is of extreme importance in quantum cryptography, and also in the design of quantum circuits is the trace distance. Let us suppose, we would like to implement a protocol or algorithm that produces state rideal. Unfortunately, due to imperfections, our protocol produces the state rreal. If we now use this protocol or algorithm as a subroutine in a much larger protocol or computation, how is this larger protocol affected if we can only make rreal instead of rideal? Intuitively, it is clear that if rreal and rideal are nearly impossible to distinguish, then it should not matter much in the large protocol which one we use. We would thus like a distance measure that is directly related to how well we can distinguish the two states. To this end, let us suppose that we really don’t know whether we have the real or ideal state. Imagine that we are given rreal and rideal each with probability 1=2, and we are challenged to distinguish them. To this end, we can perform a measurement using some operators Mreal and Mideal = I − Mreal. The probability of distinguishing the two states is then 1 1 1 1 p = tr[M r ] + tr[M r ] = + tr[M (r − r )] : (3.1) succ 2 real real 2 ideal ideal 2 2 real real ideal To find the best measurement, we can optimize the term Mreal above over all measurement operators. We know (see Week 1 lecture notes, section on POVMs) that 0 ≤ Mreal ≤ I, i.e. Mreal’s eigenvalues all lie between 0 and 1. Thus the maximum success probability is given by max 1 1 psucc = + max tr[M (rreal − rideal)] : (3.2) 2 2 0≤M≤I What is, then, the operator M that would maximize the trace quantity tr[M (rreal − rideal)]? This question has been analyzed in [Hel76], and the optimal M is the projector onto the positive eigenspace of rreal − rideal. More concretely, consider the diagonalized form of the linear operator rreal − rideal, and denote this diagonal matrix as D = ∑i dijdiihdij. Furthermore, denote the set S+ = f jjd j > 0g. The optimal M is then given by Mopt = ∑ jd jihd jj: (3.3) j2S+ It turns out the the trace distance precisely captures this idea of distinguishing states. Definition 3.1.1 — Trace distance. The trace distance between two quantum states rreal and rideal is given by D(rreal;rideal) = max tr[M (rreal − rideal)] : (3.4) 0≤M≤I 4 The trace distance can also be written as 1 hp i D(r ;r ) = tr A†A ; (3.5) real ideal 2 where A = rreal − rideal. In the literature, you will also see the trace distance written using the following notation 1 1 D(r ;r ) = kr − r k = kr − r k : (3.6) real ideal 2 real ideal tr 2 real ideal 1 If two states are close in trace distance, then there exists no measurement - no process in the universe - that can tell them apart very well. It also means that if we use a subroutine that makes rreal instead of rideal and the two are close in trace distance, then we can safely conclude that also the surrounding larger protocol cannot see much difference. Otherwise, we could use the large protocol to tell the two states apart, but we know this cannot be. Definition 3.1.2 — Closeness in terms of trace distance. Two quantum states r and s are e-close, if D(r;s) ≤ e. We also write this as r ≈e s. Proposition 3.1.1 The trace distance is a metric, that is, a proper distance measure that corresponds to our intuitive notions of distance. We have the following properties for all states r;s;t: 1. Non-negative: D(r;s) ≥ 0, where equality is achieved if and only if r = s. 2. Symmetric: D(r;s) = D(s;r). 3. Triangle inequality: D(r;s) ≤ D(r;t) + D(t;s). 4. Convexity: D(∑i piri;s) ≤ ∑i piD(ri;s). Example 3.1.1 Consider r1 = j0ih0j and r2 = j+ih+j. Firstly, calculate 1 0 1 1 1 1 1 −1 r − r = − = : (3.7) 1 2 0 0 2 1 1 2 −1 −1 Therefore, the trace distance is equal to s 2 s 1 1 1 −1 1 1 2 0 1 D(r1;r2) = · tr = · tr = p : (3.8) 2 2 −1 −1 2 2 0 2 2 Another way to do so is to first consider the diagonalization of r1 − r2, which can be done by first calculating its eigenvalues, solving the following equation: 1 1 2 − l − 2 det 1 1 = 0: (3.9) − 2 − 2 − l The solutions are given by l = ± p1 . One can also find the eigenvector je i = (x y)T correspond- 2 + ing to l = p1 , 2 1 1 −1x 1 x x −1 = p =) = p : (3.10) 2 −1 −1 y 2 y y 2 − 1 On the other hand, normalization condition gives x2 + y2 = 1, and the solution is found to be p p x = cos ; y = sin : (3.11) 8 8 The optimal measurement operator that distinguishes r1;r2 is then given by Mopt = je+ihe+j, while 1 tr Mopt (r1 − r2) = p : (3.12) 2 3.1 When are two quantum states almost the same? 5 Since states which are e−close to each other cannot be distinguished well, it will later be helpful to have the notion of a set of states which are all e−close to a particular state r. This is often called the e−ball of r. Definition 3.1.3 — e−ball of r. Given any density matrix r, the e−ball of r is defined as the set of all states r0 which are e−close to r in terms of trace distance, i.e. Be (r) := fr0 j r0 ≥ 0;tr(r0) = 1;D(r;r0) ≤ eg: (3.13) 3.1.2 Fidelity Although we have not seen this in the lectures, there is another common measure for closeness of states is known as the fidelity, which for pure states is directly related to their inner prod- uct. Definition 3.1.4 — Fidelity. Given density matrices r1 and r2, the fidelity between r1 and r2 is qp p F(r1;r2) = tr r1r2 r1 : (3.14) For pure states r1 = jY1ihY1j and r2 = jY2ihY2j the fidelity takes on a simplified form: F(r1;r2) = jhY1jY2ij : (3.15) If only one of the states r1 = jY1ihY1j is pure, we have p F(r1;r2) = hY1jr2jY1i : (3.16) Although the fidelity is not a metric (since F(r1;r2) = 0 does not imply that r1 = r2), it does have an intuitive interpretation, if we were to verify whether we managed to produce a desired target state jYi. Suppose that we want to build a machine that produces jYihYj, yet we are only able to produce some state r. Let us suppose we now measure r to check for success. We can do this (theoretically) by measuring Msucc = jYihYj ; (3.17) Mfail = I − jYihYj : (3.18) The success probability is directly related to the fidelity between the true output r and the target state jYi as 2 tr[Msuccr] = hYjrjYi = F(jYi;r) : (3.19) It is interesting to note that another way to write the fidelity is as max jhrAPjsAPij ; (3.20) jrAPi;jsAPi where jrAPi and jsAPi are purifications of the states rA and sA using a purifying system P. Proposition 3.1.2 For any two quantum states r;s, the fidelity satisfies the following properties 1. Between 0 and 1: 0 ≤ F(r;s) ≤ 1. 2. Symmetric: F(r;s) = F(s;r). 3. Multiplicative under tensor product: F(r1 ⊗ r2;s1 ⊗ s2) = F(r1;s1) + F(r2;s2). 4. Invariant under unitary operations: F(r;s) = F(UrU†;UsU†). p 5. Relation to trace distance: 1 − F(r;s) ≤ D(r;s) ≤ 1 − F2(r;s). Conversely, we also p have that 1−D(r;s) ≤ F(r;s) ≤ 1 − D2(r;s).
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