Abstract
In this chapter, we introduce the general theory of QM with two formulations. The first one is based on the Postulates (Axioms) of QM, which rather follows a traditional view, while the second one is devised for an application of quantum information science in a way to clarify the most general classes of quantum states, measurements, and time evolutions of a given quantum system under the possible presence of environment. By specifying the underling preconditions, we carefully explain their logical and physical connections between superficially different formulations.
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- 1.
For the mathematical simplicity, we restrict ourselves to the cases where the associated Hilbert spaces are finite-dimensional. However, a formal generalization to the infinite-dimensional cases follows in the parallel manner [1] subject to the careful treatments of topological issues and domains of operators [2].
- 2.
The symbol “\( |\)” with “\(s\)” implies that the probability is considered as the conditional probability conditioned that the state is \(s\). (see the footnote 18 for the conditional probability). Precisely speaking, we are identifying the random variable and the measurement and use the same symbol \(M\).
- 3.
Using a measure theoretical languages, one can formulate the general measurement theory in a parallel manner including continuous outcomes (see e.g. [8]).
- 4.
With an ONB \(\{|\psi _i\rangle \}_{i=1}^d\) of \(\mathop {\mathcal {H}}\nolimits \), let \((a_1,\,\ldots ,\,a_d)^T:= (\langle \psi _1 | \psi \rangle ,\,\ldots ,\,\langle \psi _d | \psi \rangle ), (b_1,\,\ldots ,\,b_d)^T := (\langle \psi _1 | \phi \rangle ,\,\ldots ,\,\langle \psi _d | \phi \rangle ) \in \mathop {\mathbb {C}}\nolimits ^d\) be representations (A.4) of vectors \(|\psi \rangle \) and \(|\phi \rangle \). Then, the matrix representation of \(A:=|\phi \rangle \langle \psi | \) with the same ONB reads \(a_{ij}:= \langle \psi _i | A \psi _j \rangle = \langle \psi _i | \phi \rangle \langle \psi | \psi _j \rangle = b_i \overline{a_j}\) (see (5.6) in Exercise 5.1).
- 5.
A linear map from \(\mathop {\mathcal {H}}\nolimits \) to \(\mathop {\mathbb {C}}\nolimits \) (or \(\mathbb {R}\)) is called a linear functional on \(\mathop {\mathcal {H}}\nolimits \).
- 6.
- 7.
By the Born rule (5.9), we have \(\mathrm {Pr}( A = a \ | \ |\psi \rangle ) = \langle e^{i \theta }\phi | P_a e^{i\theta }\phi \rangle = e^{-i\theta } e^{i\theta } \langle \phi | P_a \phi \rangle = \mathrm {Pr}(A = a \ | \ |\phi \rangle )\) for any physical quantity \(A\).
- 8.
For those who are interested in this problem, we recommend to read [13].
- 9.
Using (2.7), we have \(|\phi ^x_+ \rangle \langle \phi ^x_+| = \frac{1}{2} \left( \begin{array}{cc}1 &{} 1 \\ 1 &{} 1 \end{array}\right) \), \(|\phi ^x_- \rangle \langle \phi ^x_-| = \frac{1}{2} \left( \begin{array}{cc}1 &{} -1 \\ -1 &{} 1 \end{array}\right) \), from which we obtain \(|\phi ^x_+ \rangle \langle \phi ^x_+| - |\phi ^x_- \rangle \langle \phi ^x_-| = \left( \begin{array}{cc}0 &{} 1 \\ 1 &{} 0 \end{array}\right) = \sigma _x\).
- 10.
Notice that the multiplicities of eigenvalues \(0\) and \(6\) are \(1\) and \(2\), respectively. A unit eigenvector belonging to the eigenvalue \(0\) is \(|\xi _0\rangle = \frac{1}{\sqrt{6}}(1,-1,-2)^T\), and thus one can compute \(P_0 = |\xi _0 \rangle \langle \xi _0|\) by using (2.7). On the other hand, the eigenvalue \(6\) has a multiplicity \(2\), where one can find two orthogonal unit eigenvectors, e.g., \(|\xi _6\rangle := \frac{1}{\sqrt{2}}(1,1,0)^T\) and \(|\xi ^\prime _6\rangle := \frac{1}{\sqrt{3}}(1,-1,1)\). From these eigenvectors, we obtain \(P_6= |\xi _6 \rangle \langle \xi _6| + |\xi ^\prime _6 \rangle \langle \xi ^\prime _6| = \frac{1}{6}A\). Observe here the difference between an eigenvalue decomposition \(A = 0 |\xi _0 \rangle \langle \xi _0| + 6 |\xi _6 \rangle \langle \xi _6| + 6 |\xi '_6 \rangle \langle \xi '_6| = \sum _{i=1}^3 a_i |\phi _i \rangle \langle \phi _i|\) where \(a_1= 0,a_2=a_3 = 6\), \(|\phi _1\rangle := |\xi _0\rangle ,|\phi _2\rangle := |\xi _6\rangle ,|\phi _3\rangle := |\xi '_6\rangle \) and the spectral decomposition \(A = 0 P_0 + 6 P_6 = \sum _{a=0,6} a P_a\).
- 11.
Note that \(P_a Q_a = P_a (1-P_a) = P_a - P_a^2 = 0\). Check also that \(Q_a = Q_a^2 = Q_a^\dagger \).
- 12.
A pair of a position and a momentum is another typical example of the complementary pair of physical quantities, although we need an infinite-dimensional Hilbert space to describe them.
- 13.
The second equality follows since \(\langle \psi | C \psi \rangle \) is real for any Hermitian operator \(C\) (see Proposition A.5).
- 14.
The marginal (probability) distribution gives the local probability distribution of a subset of random variables (physical quantities) without reference to other random variables. For instance, with the joint probability distribution \(\mathrm {Pr}(A=a,B=b)\) of \(A\) and \(B\), the marginal distribution of \(A\) is calculated as \(\sum _{b} \mathrm {Pr}(A=a,B=b) \) by the sum rule of the probability for mutually exclusive events: If events \(E_1\) and \(E_2\) are mutually exclusive, i.e., if one of the occurrence implies non-occurrence of other events, then the probability \(\mathrm {Pr}(E_1 \cup E_2)\) for the sum event \(E_1 \cup E_2\) is the sum of the probabilities \(\mathrm {Pr}(E_1)\) and \(\mathrm {Pr}(E_2)\).
- 15.
Note that we have to collect up all the eigenvalues \(a\) satisfying \(b=f(a)\) for the case where \(f\) is not injective.
- 16.
By Proposition A.8, there exists an ONB \(\{|\phi _i\rangle \}\) which simultaneously diagonalizes \(A\) and \(B\): \(A = \sum _{i=1}^d a_i |\phi _i \rangle \langle \phi _i|,B = \sum _{j=1}^d b_j |\phi _j \rangle \langle \phi _j|\). Letting \(c_i \ (i=1,\ldots ,d)\) be all distinct real numbers, define a Hermitian operator \(C\) by \(C:= \sum _i c_i |\phi _i \rangle \langle \phi _i|\). By choosing real functions \(f\) and \(g\) such that \(a_i = f(c_i),b_i = f(c_i) \ (i=1,\ldots ,d)\), we have \(A=f(C)\) and \(B= g(C)\).
- 17.
For those who are not satisfied with using classical probabilistic events in the theory of QM, just replace a coin to some qubit system in a state, say \(|+\rangle = \frac{1}{\sqrt{2}}(1,1)^T\), and measure a computational basis \(\{|0\rangle ,|1\rangle \}\) to get probabilities \(1/2\).
- 18.
Remind that a probability under a certain conditioning of an event is called a conditional probability . Under a conditioning event \(B\) with non-zero probability \(\mathrm {Pr}(B)\), the conditional probability of \(A\) given \(B\), denoted by \(\mathrm {Pr}(A|B)\), is defined by \(\mathrm {Pr}(A|B) := \frac{\mathrm {Pr}(A \cap B)}{\mathrm {Pr}(B)}\), where \(\mathrm {Pr}(A \cap B)\) is the joint probability of \(A\) and \(B\).
- 19.
One can show this by contradiction. Suppose that \(s\) is described by some unit vector \(|\xi \rangle \in \mathop {\mathbb {C}}\nolimits ^2\). Let \(|\xi \rangle ^\perp \) be an orthogonal vector to \(|\xi \rangle \) so that \(\{|\xi \rangle ,|\xi \rangle ^\perp \}\) forms an ONB of \(\mathop {\mathbb {C}}\nolimits ^2\). Then, the basis measurement \(\{|\xi \rangle ,|\xi \rangle ^\perp \}\) under state \(|\xi \rangle \) gives a contradiction: The probability to get the outcome corresponding to \(|\xi \rangle \) is \( |\langle \xi | \xi \rangle |^2= 1\), which is inconsistent with (5.31).
- 20.
In classical mechanics, a pure state is described by positions and momentums of the particles, namely a point in the phase space. A mixed state is described by a probability distribution on the phase space.
- 21.
To obtain the second equality, one can substitute \(|\pm \rangle = \frac{1}{\sqrt{2}}(|0\rangle \pm |1\rangle )\). Alternatively, just use the arbitrariness of ONBs in the definition of the trace operation: \(\mathop {\mathrm{Tr}}\nolimits P_a = \langle 0|P_a|0\rangle + \langle 1|P_a|1\rangle = \langle +|P_a|+\rangle + \langle -|P_a|-\rangle \) (see Appendix A.3.8).
- 22.
By putting \(A = |\psi \rangle \langle \psi |\) with an arbitrary \(|\psi \rangle \in \mathop {\mathcal {H}}\nolimits \), we have \(\langle \psi | \rho \psi \rangle = \mathrm {Pr}(A = 1 | \rho ) = \mathrm {Pr}(A = 1 | \rho ') = \langle \psi | \rho ' \psi \rangle \). From Proposition A.3-(ii), we have \(\rho = \rho '\).
- 23.
For unit vectors \(|\psi \rangle \) and \(|\phi \rangle \) such that \(|\phi \rangle = e^{i\theta } |\psi \rangle \), the corresponding density operators are the same: \(|\phi \rangle \langle \phi | = e^{i \theta } |\psi \rangle \langle \psi | e^{-i \theta } = |\psi \rangle \langle \psi |\).
- 24.
Notice that this method may not be the only preparation of \(\rho \). As is shown later, a mixed state has always non-unique (and indeed infinitely many) state-preparations, while a pure state has the unique state-preparation (see Proposition 5.7).
- 25.
- 26.
Take the inner product between \(\sigma _i \ (i=0,1,2,3)\) and \(A = \sum _{j=0}^3 x_j \sigma _j\) (the expansion of \(A\) with respect to the basis) and use condition (5.46) to get \(x_i = \frac{1}{2} \mathop {\mathrm{Tr}}\nolimits (A \sigma _i)\).
- 27.
By putting \(\rho = |\psi \rangle \langle \psi |\) with the parametrization (2.15), one can get \(b_1 = \sin \theta \cos \phi ,b_2 = \sin \theta \sin \phi , b_3 = \cos \theta \), which give the polar coordinates of \({\mathbf b}\).
- 28.
For instance, for \(X \in \mathop {\mathcal {L}}\nolimits (\mathop {\mathcal {H}}\nolimits _\mathrm{E}), Y \in \mathop {\mathcal {L}}\nolimits (\mathop {\mathcal {H}}\nolimits _\mathrm{SE})\), \(\mathop {\mathrm{Tr}}\nolimits _\mathrm{E} X \in \mathop {\mathbb {C}}\nolimits \) is a usual trace of \(X\) but \(\mathop {\mathrm{Tr}}\nolimits _\mathrm{E} Y \in \mathop {\mathcal {L}}\nolimits (\mathop {\mathcal {H}}\nolimits _\mathrm{S})\) is a partial trace of \(Y\).
- 29.
Moreover, it has recently been shown that the purification is one of the essential properties to single out QM among all operationally valid probabilistic theories [17].
- 30.
Note, however, that there also exist classically correlated states (separable states) which are not entangled states but have non-zero correlations (see Chap. 7).
- 31.
By the same argument of the derivation of (5.24) under a density operator, we get \(\mathrm {Pr}(A = a, B = b | |\psi \rangle ) = \mathop {\mathrm{Tr}}\nolimits _\mathrm{SE}(P_a \otimes Q_b \rho )\), where \(A = \sum _a a P_a\) and \(B= \sum _b b Q_b\) are spectral decomposition of \(A\) and \(B\). Substitute this into \(\mathrm {Pr}(A = a, B = b | \rho ) = \mathrm {Pr}(A = a | \rho _\mathrm{S}) \mathrm {Pr}(B = b | \rho _\mathrm{E})\), we have \(\mathop {\mathrm{Tr}}\nolimits _\mathrm{SE}(P_a \otimes Q_b \rho ) = (\mathop {\mathrm{Tr}}\nolimits _\mathrm{S} P_a \rho _\mathrm{S})(\mathop {\mathrm{Tr}}\nolimits _\mathrm{E} Q_b \rho _\mathrm{E}) = \mathop {\mathrm{Tr}}\nolimits _\mathrm{SE}(P_a \otimes Q_b \rho _\mathrm{S} \otimes \rho _\mathrm{E})\). As \(A\) and \(B\) are arbitrary, we obtain \(\rho = \rho _\mathrm{S}\otimes \rho _\mathrm{E}\).
- 32.
- 33.
Alternatively, if \(\rho \) is prepared as a reduced density operator (even with probabilistic mixtures), one can realize the map (5.51) by means of a local unitary evolution. Note that based on Postulate \(2\), the total unitary evolution is given by a unitary operator \(U \otimes \mathop {I}\nolimits \) on a total system.
- 34.
Preparing the state by a probabilistic mixture \(\{p,1-p;s_1,s_2\}\), we have \(\mathrm {Pr}(M = m \ | \ \{p,1-p;s_1,s_2\}) = \mathrm {Pr}(``M = m\text {''} \cap s_1) + \mathrm {Pr}(``M = m\text {''} \cap s_2) = p \mathrm {Pr}(M = m |s_1) + (1-p) \mathrm {Pr}(M = m | s_2)\). (see also the derivation of (5.31)).
- 35.
Let \(f\) be an affine function from a convex subset \(W\) of a vector space \(V\) to a vector space \(V'\). A linear extension \(\tilde{f}\) of \(f\) is a linear map from \(V\) to \(V'\) satisfying \(f(w) = \tilde{f}(w)\) for all \(w \in W\).
- 36.
- 37.
By the positivity of \(E_m\), we have \(\mathop {\mathrm{Tr}}\nolimits (E_m \rho ) = \sum _i q_i \langle \phi _i | E_m \phi _i \rangle \ge 0\), where \(\rho = \sum _i q_i |\phi _i \rangle \langle \phi _i|\) is an eigenvalue decomposition of \(\rho \). Also, by the condition \(\sum _m E_m = \mathop {I}\nolimits \), we have \(\sum _m \mathop {\mathrm{Tr}}\nolimits (E_m \rho ) = \mathop {\mathrm{Tr}}\nolimits ((\sum _m E_m) \rho ) = \mathop {\mathrm{Tr}}\nolimits \rho = 1\).
- 38.
By \(P_m=P_m^2\) and \(\sum _n P_n = \mathop {I}\nolimits \), we have \(P_m = P_m \mathop {I}\nolimits P_m = P_m (\sum _n P_n) P_m = P_m + \sum _{n \ne m} P_m P_n P_m\) for any \(m\), thus \( \sum _{n \ne m} P_m P_n P_m = 0\). As the sum of positive operators \(P_m P_n P_m\) is zero, we have \(P_m P_n P_m = 0 \ (n \ne m)\). Moreover, as \(P_mP_nP_m = P_m P_n P_n P_m = (P_n P_m)^\dagger (P_n P_m)\), we obtain \(P_n P_m = 0\) for any \(n \ne m\). (Show that \(A^\dagger A = 0\) implies \(A = 0\).)
- 39.
For any \(A \ge 0\) and any \(|\psi \rangle \), we have \(\langle \psi | (U^\dagger A U)\psi \rangle = \langle (U \psi ) | A(U \psi ) \rangle \ge 0\).
- 40.
For \(E \ge 0\) we can define \(\sqrt{E}:= \sum _e \sqrt{e} P_e\) where \(E = \sum _e e P_e \ (e \ge 0)\) is the spectral decomposition of \(E\). Note that \((\sqrt{E})^\dagger = \sqrt{E} \ge 0, (\sqrt{E})^2 = E\).
- 41.
For any \(|\psi \rangle \otimes |\phi _1\rangle ,|\psi '\rangle \otimes |\phi _1\rangle \in W\), we have \(\langle U (\psi \otimes \phi _1) | U (\psi ' \otimes \phi _1) \rangle = \sum _{j,k=1}^n \langle \sqrt{E_j} \psi \otimes \phi _j | \sqrt{E_k} \psi ' \otimes \phi _k \rangle = \sum _{j,k=1}^n \langle \psi | \sqrt{E_j}^\dagger \sqrt{E_k}\psi ' \rangle \langle \phi _j | \phi _k \rangle = \sum _{j} \langle \psi | E_j \psi ' \rangle = \langle \psi | ( \sum _{j} E_j) \psi ' \rangle = \langle \psi | \psi ' \rangle \).
- 42.
In general, \(\rho \) and \(\sigma \) can be distinguished by the following procedure: First, we perform a POVM measurement \(\{F_k\}_{k \in \mathcal K}\). Then, we decide whether the true state is \(\rho \) or \(\sigma \) from the obtained outcome \(k \in \mathcal{K}\) by using a two-valued decision function \(f\!:\!\mathcal{K} \rightarrow \{1,2\}\). (One can use a mixed strategy as well, but in this case, it is enough to use a two-valued decision function.) Under a state \(\xi =\rho \) or \(\sigma \), the probability to obtain the final outcome \(i=1,2\) is given by \(\sum _{k; f(k)=i}\mathop {\mathrm{Tr}}\nolimits (\xi F_k) = \mathop {\mathrm{Tr}}\nolimits (\xi (\sum _{k; f(k)=i}F_k))\). Therefore, by putting \(E_i\!:=\! \sum _{k; f(k)=1}F_k \ (i=1,2)\), the above procedure can be reduced to the application of the two-valued POVM \(\{E_1,E_2\}\). That is, the condition to distinguish the two states \(\rho \) and \(\sigma \) is the existence of a POVM \(\{E_1,E_2\}\) satisfying that \(\mathop {\mathrm{Tr}}\nolimits \rho E_1 = 1, \mathop {\mathrm{Tr}}\nolimits \sigma E_2 = 1 \ (\Leftrightarrow \mathop {\mathrm{Tr}}\nolimits \rho E_2 = 0, \mathop {\mathrm{Tr}}\nolimits \sigma E_1 = 0)\).
- 43.
One can consider the affine property for time-evolution map as one of the preconditions.
- 44.
Any operator \(A \in \mathop {\mathcal {L}}\nolimits (\mathop {\mathcal {H}}\nolimits _\mathrm{A})\) can be written as \(A = \sum _{k=0}^3 i^k p_k \rho _k\) with non-negative real numbers \(p_k \ge 0\) and density operators \(\rho _k \in \mathop {\mathcal {S}}\nolimits (\mathop {\mathcal {H}}\nolimits _\mathrm{A})\) (\(k=0,1,2,3\)). (see the solution of Exercise 5.14.) By the linearity of the trace, we have \(\mathop {\mathrm{Tr}}\nolimits _\mathrm{B} (\Lambda '(A)) = \sum _k i^k p_k \mathop {\mathrm{Tr}}\nolimits _\mathrm{B}(\Lambda (\rho _k)) = \sum _k i^k p_k \mathop {\mathrm{Tr}}\nolimits _\mathrm{A}(\rho _k) = \mathop {\mathrm{Tr}}\nolimits _\mathrm{A} (A)\). (Note that \(\mathop {\mathrm{Tr}}\nolimits _\mathrm{B}(\Lambda '(\rho _k)) = 1 = \mathop {\mathrm{Tr}}\nolimits _\mathrm{A}(\rho _k) \) since \(\rho _k\) and \(\Lambda (\rho _k)\) are density operators.) Since any positive operator \(A \in \mathop {\mathcal {L}}\nolimits (\mathop {\mathcal {H}}\nolimits _\mathrm{A})\) can be written as \(A = a \rho \) with a non-negative real number \(a := \mathop {\mathrm{Tr}}\nolimits A\) and a density operator \(\rho :=A /a\), we have \(\Lambda '(A) = a \Lambda (\rho )\). As \(a \ge 0\) and \( \mathop {\mathcal {S}}\nolimits (\mathop {\mathcal {H}}\nolimits _\mathrm{B}) \ni \Lambda (\rho ) \ge 0\), \(\Lambda '(A)\) is a positive operator.
- 45.
Note, however, that there is a longstanding argument for the validity of this matter. It is usually discussed with reference to the presence of initial correlations with an environment. However, the root problem includes the validity to deal with a time evolution by means of a map (see [21] and references therein).
- 46.
The theory of measurement processes with operational point of view was initiated by Davies and Lewis [22], and completed by Kraus [6] and Ozawa [23] by adding the notion of complete positivity. In particular, a careful consideration of the necessary and sufficient conditions for the description of measurement processes was thoroughly given by Ozawa.
- 47.
Any discrete CP instrument \(\{\Lambda _m\}_m\) can be described by measurement operators as follows: Let \(\Lambda _m (A) = \sum _{k=1}^{l_m} V^{(m)}_k A {V^{(m)}_k}^\dagger \) be the (Kraus) representation. By \(\sum _{m,k} {V^{(m)}_k}^\dagger V^{(m)}_k = \mathop {I}\nolimits \), we have a tuple \(\{V^{(m)}_k\}_{k,m}\) of measurement operators. By performing a non-selective measurement \(M'\) of \(\{V^{(m)}_k\}_{k,m}\) and outputting only \(m\) with discarding \(k\), we obtain (5.80).
- 48.
- 49.
One can use the same proof for Theorem 5.6 by letting a TPCP map represented by \(\Lambda (X) = \sum _m V_m X V_m^\dagger \) (Kraus representation). Only the difference is that we don’t perform the final measurement \(|m \rangle \langle m|\).
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M. Hayashi, Quantum Information: An Introduction (Springer, Berlin, 2006) Originally published in Japanese in 2004
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Hayashi, M., Ishizaka, S., Kawachi, A., Kimura, G., Ogawa, T. (2015). Foundations of Quantum Mechanics and Quantum Information Theory. In: Introduction to Quantum Information Science. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43502-1_5
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