Kapitza Fall 2017 NOISY QUANTUM MEASUREMENT

Transcription

1 Kapitza Fall 2017 NOIY QUANTUM MEAUREMENT Ben altzman Q.I.T. TERM PAPER Professor arada Raeev December 10 th 2017

2 1 Introduction Not only does God play dice but... where they can t be seen. he sometimes throws them tephen Hawking Despite Albert Einstein s insistence that the world is deterministic, quantum mechanics introduces a probabilistic world view. Measuring a quantum becomes a delicate task: measurements inherently affect the system in question. In this paper, based on a lecture given to fellow Kapitza members on December 3 rd, 2017, I discuss the formulation of measurement in a noisy quantum system. 2 A New Description of Measurement Measurement is commonly described by defining a set of proection operators that are complete, i.e. Π = I Introducing a probe system, we can get information about the original system from the probe. Measurement can be described by the unitary interaction of the system with the probe, with operator U P. uppose we have a system in state Ψ and a probe P with dimension d and orthonormal basis { 0 P, 1 P,..., d 1 P }. Defining the probe s initial state as 0 P, we can write the initial state of the whole system as and measurement operators Ψ { P } {0,1,...,d l} The operator can be written in the probe s basis as U P =,k M,k k P (1 for operators {M,k }. The probability of outcome from the probe is p J ( = Ψ 0 P U P (I P U P Ψ (2 1

3 and the state of the system after measurement is (I P (U P Ψ pj ( (3 Note that I P is the measurement operator on the full system, since only the probe is measured. ince U P is unitary, we have U P U P = I P = I I P. Consider the case k = 0. For clarity, define M M,0. ( U P U k=0 ( P M 0 P M 0 P M M 0 0 P = = M M 0 0 P (4 But U P U P k=0 I 0 0 P. Using (4, we conclude that I = M M (5 Using the definition of U P, we can simplify (2 and (3. ubstituting (1 into (2: ( p J ( = Ψ 0 P M,k k P (I P ( M,k k P Ψ = Ψ M,k 0 k P (I P M,k Ψ k 0 P = Ψ M I P M Ψ P = Ψ M M Ψ 2

4 and therefore p J ( = Ψ M M Ψ (6 imilarly, we can simplify the post-measurement state to M Ψ P pj ( (7 The state of can be read off from (7 easily, since and P are in a pure product state. Measurement can therefore be described as a set of measurement operators {M }, instead of {Π }, that satisfy (5. ee Appendix A for discussion on the application of this process to ensembles. When transferring classical data over a quantum channel, the receiver doesn t need the post-measurement state to process the information in a quantum fashion. The relevant probability is the probability of error. For any such situation where the probability of an outcome matters and the postmeasurement state does not, we can describe a positive operator-valued measure (POVM with a set of operators {Λ } = {M M } that are non-negative and complete. Clearly, proection is a type of POVM. The probability of success of the POVM is p X (x Tr Λ x ρ x x X where ρ x is the density matrix for state ψ x. 3 Composite ystems uppose we have two indepedent ensembles, ε A = {p X (x, ψ x } and ε B = {p Y (Y, φ y }. The density matrix for the oint state ψ x φ y is E X,Y ( ψ X φ Y ( ψ X φ Y = E X,Y ψ X ψ X φ Y φ Y = x,y p X (xp Y (y ψ x ψ x φ y φ y = x p X (x ψ x ψ x y φ y φ y = ρ σ (8 where ρ and σ are the density matrices for ε A and ε B respectively. Now suppose we have a oint ensemble in which systems A and B are correlated classically. We d like a formulation to express this ensemble similarly to the independent situation above. To do this, we introduce a new random variable Z that X and Y are conditioned on. The two ensembles 3

5 are ε A = {p X Z (x z, ψ x,z } (density matrix ρ z and ε B = {p Y Z (y z, φ y,z } (density matrix σ z, and X Z and Y Z are independent. Using the same procedure as we did with (8, we obtain the density matrix of the total state: E X,Y,Z ( ψ X,Z φ Y,Z ( ψ X,Z φ Y,Z = x,y,z p Z (zp X Z (x zp Y Z (y z ψ x,z ψ x,z φ y,z φ y,z (9 Define a new random variable W = X Y Z. We can write the density matrix in (9 as p W (w φ w φ w φ w φ w (10 w o, we can write any state with the properties discussed in this paragraph as a product of pure states. This type of state is termed separable, and contains no entanglement. In other words, a separable state can always be prepared classically. ee Appendix B for an application involving separable states. 4 Local Density Operators uppose systems A and B are in an entangled Bell state Φ + AB. Take a POVM Λ on A. The measurement operators for the system are Λ A I B. The probability of outcome is p J ( = Φ + Λ A I B Φ + AB = 1 1 kk Λ A 2 I B ll AB k,l=0 1 = 1 k Λ A 2 l A k I B l B k,l=0 = Tr Λ 1 1 A ( k k 2 A k=0 = Tr Λ A π A (11 where the local density operator for A is the maximally mixed state π A = k=0 k k A. This process goes similarly for B. Thus, the following global state gives the same predictions in local measurements as Φ + AB : π A π B 4

6 We d like to define what we mean by a local density operator in order to describe the results of local measurements. To do this, we need to define the partial trace operation. uppose { k A } and { l B } are orthonormal bases for the Hilbert spaces of A and B. Then { k A l B } is an orthonormal basis for the product of the Hilbert spaces. For density operator ρ AB, the probability of outcome is p J ( = Tr (Λ A I Bρ AB = k,l k A l B (Λ A I Bρ AB k A l B = k A I A l B (Λ A I Bρ AB I A l B k A k,l = k A Λ A (I A l B ρ AB (I A l B k A k l = Tr (I A l B ρ AB (I A l B (12 Λ A l o, we define the partial trace for B as Tr B X AB = (I A l B X AB (I A l B l (13 and the local density operator for A as ρ A = Tr B ρ AB (14 Therefore, (12 becomes p J ( = Tr Λ A ρ A (15 Alice can predict the outcome of local measurements with (15. 5 Classical-Quantum Ensembles uppose Alice prepares a quantum system with density matrix ρ x A and probability distribution p X (x. he passes this ensemble to Bob, who must learn about it. There is a loss of information in X after preparation which is minimized if the state is pure. ρ A = x p x(x x x A for an orthonormal basis { x } x X. For a mixed state, ρ A = x p X(xρ x A is more difficult to extract information from. 5

7 One solution is for Alice is to prepare a classical-quantum ensemble: {p X (x, x x X ρ x A} x X This ensemble is so-called because system X is classical, while system A is quantum. The density operator for the entire system is ρ XA = x p X (x x x X ρ x A (16 uppose Bob makes a measurement of the system with {I X Λ A }. This is akin to Bob measuring an isolated system A with {Λ A }. Why? Tr ρ XA (I X Λ A = Tr ( p X (x x x X ρ x A(I X Λ A x = Tr p X (x( x x X I X ρ x AΛ A = x = x x = Tr ρ A Λ A Tr x x X I X Tr p X (xρ x AΛ A Tr p X (xρ x AΛ A o Bob can extract information about A from the whole system with a local measurement on A. 6 Conclusion At the heart of quantum mechanics is a rule that sometimes governs politicians or CEOs - as long as no one is watching, anything goes. Lawrence M. Krauss Measurement of quantum systems is tricky, and matters only get more complicated when you consider composite quantum systems. With udicious choices of measurement operators and careful preparation of a system or composite system, one can make quantum measurement seem more akin to the classical case. 6

8 Appendix A: Redefining Measurement for Ensembles uppose we have an ensemble {p X (x, ψ x } x X with density operator ρ = x X p X (x ψ x ψ x Using the same procedure as outlined in section 2: p J ( = ( p X (x ψ x 0 P M,k k P (I P x X ( M,k k P ψ x = p X (x ψ x M,k 0 k P (I P x X M,k ψ x k 0 P = p X (x ψ x M I P M ψ x P x X = x X p X (x ψ x M M ψ x p J ( = x X p X (x ψ x M M ψ x = M M = Tr M M ρ This is a reformulation of the Born Rule. The post-measurement state is M ρm p J ( 7

9 Appendix B: The CHH Game Consider a game with two players, Alice (A and Bob (B. Alice and Bob cannot communicate with each other, but they can communicate with a referee. The referee sends out two bits: x to Alice and y to Bob. Each player sends a bit back to the referee (a and b respectively. The win condition is x y = a b It is known that the maximum probability of success for any strategy that Alice and Bob agree to use beforehand is 0.75 for classical strategies and cos 2 (π/ for quantum strategies. The form of a classical strategy is p AB XY (a, b x, y = dλp Λ (λp A ΛX (a λ, xp B ΛY (b λ, y where Λ is a continuous index for their strategies. The density operator for the state of the system is ρ AB = dλp Λ (λ ψ λ ψ λ φ λ φ λ In this case, the classical strategy is p AB XY (a, b x, y = Tr (Π (x a Π (y b ρ AB = Tr dλp Λ (λ Π (x a ψ λ ψ λ A Π (y b φ λ φ λ B = dλp Λ (λ φ λ Π (x a φ λ φ λ Π (y b φ λ Therefore, we have p A ΛX (a λ, x = φ λ Π (x a φ λ and p B ΛY (b λ, y = φ λ Π (y b φ λ The above are classical strategies that simulate quantum strategies, provided that the initial system is in a separable state. This implies that the upper limit on the probability of winning for a separable states is 0.75, not cos 2 (π/8. References 1 Wilde, M. M. (2013. Quantum information theory. Cambridge University Press. 2 ándor, I. and Ferenc, B. Quantum Computing and Communications - An Engineering Approach, Chapter 3: Measurements. Accessed December